Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
We introduce some multiple integrals that are expected to have the same singularities as the singularities of the $ n$-particle contributions $\chi^{(n)}$ to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differenti…
Authors: ** *원문에 저자 정보가 명시되어 있지 않음.* (일반적으로 J‑M Maillard, B. Nickel, N. Zenine
Singularities of n -fold in tegrals of the Ising lass and the theory of ellipti urv es S. Boukraa † , S. Hassani § , J.-M. Maillard ‡ and N. Zenine § † LPTHIRM and Départemen t d'Aéronautique, Univ ersité de Blida, Algeria Cen tre de Re her he Nuléaire d'Alger, 2 Bd. F ran tz F anon, BP 399, 16000 Alger, Algeria ‡ LPTMC, Univ ersité de P aris 6, T our 24, 4ème étage, ase 121, 4 Plae Jussieu, 75252 P aris Cedex 05, F rane E-mail: maillardlptm.jussieu.fr , maillardlptl.jussieu.f r, boukraamail.univ-blida. dz, njzenineyahoo.om Abstrat. W e in tro due some m ultiple in tegrals that are exp eted to ha v e the same singularities as the singularities of the n -partile on tributions χ ( n ) to the suseptibilit y of the square lattie Ising mo del. W e nd the F u hsian linear dieren tial equation satised b y these m ultiple in tegrals for n = 1 , 2 , 3 , 4 and only mo dulo some primes for n = 5 and 6 , th us pro viding a large set of (p ossible) new singularities of the χ ( n ) . W e disuss the singularit y struture for these m ultiple in tegrals b y solving the Landau onditions. W e nd that the singularities of the asso iated ODEs iden tify (up to n = 6 ) with the leading pin h Landau singularities. The seond remark able obtained feature is that the singularities of the ODEs asso iated with the m ultiple in tegrals redue to the singularities of the ODEs asso iated with a nite numb er of one dimensional inte gr als . Among the singularities found, w e underline the fat that the quadrati p olynomial ondition 1 + 3 w + 4 w 2 = 0 , that o urs in the linear dieren tial equation of χ (3) , atually orresp onds to a remark able prop ert y of seleted ellipti urv es, namely the o urrene of omplex m ultipliation. The in terpretation of omplex m ultipliation for ellipti urv es as omplex xed p oin ts of the seleted generators of the renormalization group, namely isogenies of ellipti urv es, is sk et hed. Most of the other singularities o urring in our m ultiple in tegrals are not related to omplex m ultipliation situations, suggesting an in terpretation in terms of (motivi) mathematial strutures b ey ond the theory of ellipti urv es. P A CS : 05.50.+q, 05.10.-a, 02.30.Hq, 02.30.Gp, 02.40.Xx AMS Classiation s heme n um b ers : 34M55, 47E05, 81Qxx, 32G34, 34Lxx, 34Mxx, 14Kxx Key-w ords : Suseptibilit y of the Ising mo del, singular b eha viour, F u hsian linear dieren tial equations, apparen t singularities, Landau singularities, pin h singularities, mo dular forms, Landen transformation, isogenies of ellipti urv es, omplex m ultipliation, Heegner n um b ers, mo duli spae of urv es, p oin ted urv es. 1. In tro dution The suseptibilit y χ of the square lattie Ising mo del has b een sho wn b y W u, MCo y , T ray and Barou h [ 1 ℄ to b e expressible as an innite sum of holomorphi funtions, Singularities of n -fold inte gr als 2 giv en as m ultiple in tegrals, denoted χ ( n ) , that is k T · χ = P χ ( n ) . B. Ni k el found [2, 3℄ that ea h of these χ ( n ) 's is atually singular on a set of p oin ts lo ated on the unit irle | s | = | sinh(2 K ) | = 1 , where K = J/ k T is the usual Ising mo del temp erature v ariable. These singularities are lo ated at solution p oin ts of the follo wing equations: 2 · s + 1 s = u k + 1 u k + u m + 1 u m u 2 n +1 = 1 , − n ≤ m, k ≤ n (1) F rom no w on, w e will all these singularities of the Ni k elian t yp e, or simply Ni k elian singularities. The aum ulation of this innite set of singularities of the higher-partile omp onen ts of χ ( s ) on the unit irle | s | = 1 , leads, in the absene of m utual anellation, to some onsequenes regarding the non holonomi (non D- nite) harater of the suseptibilit y , p ossibly building a natural b oundary for the total χ ( s ) . Ho w ev er, it should b e noted that new singularities that are not of the Ni k elian t yp e w ere diso v ered as singularities of the F u hsian linear dieren tial equation asso iated [4 , 5, 6℄ with χ (3) and as singularities of χ (3) itself [7℄ but seen as a funtion of s . They orresp ond to the quadrati p olynomial 1 + 3 w + 4 w 2 where 2 w = s/ (1 + s 2 ) . In on trast with this situation, the F u hsian linear dieren tial equation, asso iated [8℄ with χ (4) , do es not pro vide an y new singularities. Some remark able Russian-doll struture as w ell as diret sum deomp ositions w ere found for the orresp onding linear dieren tial op erators for χ (3) and χ (4) . In order to understand the true nature of the suseptibilit y of the square lattie Ising mo del, it is of fundamen tal imp ortane to ha v e a b etter understanding of the singularit y struture of the n -partile on tributions χ ( n ) , and also of the mathematial strutures asso iated with these χ ( n ) , namely the innite set of (probably F u hsian) linear dieren tial equations asso iated with this innite set of holonomi funtions. Finding more F u hsian linear dieren tial equations ha ving the χ ( n ) 's as solutions, b ey ond those already found [4 , 8℄ for χ (3) and χ (4) , probably requires the p erformane of a large set of analytial, mathematial and omputer programming tours-de-fore. As an alternativ e, and in order to b ypass this temp orary obstrution, w e ha v e dev elop ed, in parallel, a new strategy . W e ha v e in tro dued [7 ℄ some single (or m ultiple) mo del in tegrals as an ersatz for the χ ( n ) 's as far as the lo us of the singularities is onerned. The χ ( n ) 's are dened b y ( n − 1) -dimensional in tegrals [3, 9 , 10 ℄ (omitting the prefator ‡ ) ˜ χ ( n ) = (2 w ) n n ! n − 1 Y j =1 Z 2 π 0 dφ j 2 π n Y j =1 y j · R ( n ) · G ( n ) 2 (2) where G ( n ) = n Y j =1 x j ( n − 1) / 2 Y 1 ≤ i T c and (1 − s − 4 ) 1 / 4 for T < T c . Singularities of n -fold inte gr als 3 with x i = 2 w 1 − 2 w cos( φ i ) + q (1 − 2 w cos( φ i )) 2 − 4 w 2 , (5) y i = 1 q (1 − 2 w cos( φ i )) 2 − 4 w 2 , n X j =1 φ j = 0 . (6) The t w o families of in tegrals w e ha v e onsidered in [7℄ are v ery rough appro ximations of the in tegrals (2 ). F or the rst family † , w e onsidered the n -fold in tegrals orresp onding to the pro dut of (the square ‡ of the) y i 's, in tegrated o v er the whole domain of in tegration of the φ i (th us getting rid of the fators G ( n ) and R ( n ) ). Here, w e found a subset of singularities o urring in the χ ( n ) as w ell as the quadrati p olynomial ondition 1 + 3 w + 4 w 2 = 0 . F or the seond family , w e disarded the fator G ( n ) and the pro dut of y i 's, and w e restrited the domain of in tegration to the prinipal diagonal of the angles φ i ( φ 1 = φ 2 = · · · = φ n − 1 ). These simple in tegrals (o v er a single v ariable), w ere denoted [7 ℄ Φ ( n ) D : Φ ( n ) D = − 1 n ! + 2 n ! Z 2 π 0 dφ 2 π 1 1 − x n − 1 ( φ ) · x (( n − 1) φ ) (7) where x ( φ ) is giv en b y (5). Remark ably these v ery simple in tegrals b oth r epr o du e al l the singularities , disussed b y Ni k el [2 , 3 ℄, as w ell as the quadrati ro ots of 1 + 3 w + 4 w 2 = 0 found [4, 5 ℄ for the linear ODE of χ (3) . One should ho w ev er note that, in on trast with the χ ( n ) , no Russian-doll or diret sum deomp osition struture is found for the linear dieren tial op erators orresp onding to these Φ ( n ) D . Another approa h has b een in tro dued as a simpliation of the suseptibilit y of the Ising mo del b y onsidering a magneti eld restrited to one diagonal of the square lattie [11 ℄. F or this diagonal suseptibilit y mo del [11 ℄, w e b eneted from the form fator de omp osition of the diagonal t w o-p oin t orrelations C ( N , N ) , that has b een reen tly presen ted [12 ℄, and subsequen tly pro v ed b y Lyb erg and MCo y [13 ℄. The orresp onding n -fold in tegrals χ ( n ) d w ere found to exhibit remark able diret sum strutures inherited from the diret sum strutures of the form fator [11, 12 ℄. The linear dieren tial op erators of the form fator [ 12℄ b eing losely link ed to the seond order dieren tial op erator L E (resp. L K ) of the omplete ellipti in tegrals E (resp. K ), this diagonal suseptibilit y mo del [11 ℄ is losely link ed to the ellipti urv es of the t w o-dimensional Ising mo del. By w a y of on trast, w e note that the singularities of the linear ODE's for these n -fold in tegrals [11 ℄ χ ( n ) d are quite elemen tary (onsisting of only n -th ro ots of unit y) in omparison with the singularities w e enoun ter for the in tegrals on a single v ariable (7). These t w o approa hes orresp onding to t w o dieren t sets of n -fold in tegrals of the Ising lass [14 ℄ are omplemen tary: (7) is more dediated to repro due the non- trivial head p olynomials eno ding the lo ation of the singularities of the χ ( n ) , but fails † Denoted Y ( n ) ( w ) in [7℄. ‡ Surprisingly the in tegrand with ( Q n j =1 y j ) 2 yields seond order linear dieren tial equations [7 ℄, and onsequen tly , w e ha v e b een able to totally deipher the orresp onding singularit y struture. By w a y of on trast the in tegrand with the simple pro dut ( Q n j =1 y j ) yields linear dieren tial equations of higher order, but with iden tial singularities [7℄. Singularities of n -fold inte gr als 4 to repro due some remark able (Russian-doll, diret sum deomp osition) algebraio- dieren tial strutures of the orresp onding linear dieren tial op erators, while the other one [11 ℄ preserv es these non-trivial strutures of the orresp onding linear dieren tial op erators but pro vides a p o orer represen tation of the lo ation of the singularities ( n -th ro ots of unit y). In this pap er, w e return to the in tegrals (2) where, this time, the natural next step is to onsider the follo wing family of n -fold in tegrals Φ ( n ) H = 1 n ! · n − 1 Y j =1 Z 2 π 0 dφ j 2 π · n Y j =1 y j · 1 + Q n i =1 x i 1 − Q n i =1 x i (8) whi h amoun ts to getting rid of the (fermioni) fator ( G ( n ) ) 2 in the n -fold in tegral (2). This family is as lose as p ossible to ( 2), for whi h w e kno w that nding the orresp onding linear dieren tial ODE's is a h uge task. The idea here is that the metho ds and te hniques w e ha v e dev elop ed [4, 5℄ for series expansions alulations of χ (3) and χ (4) , seem to indiate that the quite in v olv ed fermioni term ( G ( n ) ) 2 in the in tegrand of ( 2) should not impat greatly on the lo ation of singularities of these n -fold in tegrals (2 ). This is the b est simpliation of the in tegrand of (2 ) for whi h w e an exp et to retain m u h exat information ab out the lo ation of the singularities of the original Ising problem. Ho w ev er, w e ertainly do not exp et to reo v er from the n -fold in tegrals (8) the lo al singular b eha vior (exp onen ts, amplitudes of singularities, et ...). Getting rid of the (fermioni) fator ( G ( n ) ) 2 are w e mo ving a w a y from the ellipti urv es of the t w o-dimensional Ising mo del ? Could it b e p ossible that w e lose the strong (Russian-doll, diret sum deomp osition) algebraio-dieren tial strutures of the orresp onding linear dieren tial op erators inherited from the seond order dieren tial op erator L E (resp. L K ) of the omplete ellipti in tegrals E (resp. K ) but k eep some haraterization of ellipti urv es through more primitiv e (univ ersal) features of these n -fold in tegral lik e the lo ation of their singularities ? In the sequel, w e giv e the expressions of Φ (1) H , Φ (2) H and the F u hsian linear dieren tial equations for Φ ( n ) H for n = 3 and n = 4 . F or n = 5 , 6 , the omputation (linear ODE sear h of a series) b eomes m u h harder. Consequen tly w e use a mo dulo prime metho d to obtain the form of the orresp onding linear ODE with totally expliit singularit y struture. These results pro vide a large set of andidate singularities for the χ ( n ) . F rom the resolution of the Landau onditions [7℄ for (8), w e sho w that the singularities of (the linear ODEs of ) these m ultiple in tegrals atually redue to the onatanation of the singularities of (the linear ODEs of ) a set of one-dimensional in tegrals. W e disuss the mathemati al, as wel l as physi al, interpr etation of these new singularities. In partiular w e will see that they orresp ond to pinhe d L andau- like singularities as previously notied b y Ni k el [15 ℄. Among all these p olynomial singularities, the quadrati n um b ers 1 + 3 w + 4 w 2 = 0 are highly seleted. W e will sho w that these seleted quadrati n um b ers are related to omplex multipli ation for the el lipti urves parameterizing the square Ising mo del. The pap er is organized as follo ws. Setion (2) presen ts the m ultidimensional in tegrals Φ ( n ) H and the singularities of the orresp onding linear ODE for n = 3 , · · · , 6 , that w e ompare with the singularities obtained from the Landau onditions. W e sho w that the set of singularities asso iated with the ODEs of the m ultiple in tegrals Φ ( n ) H redue to the singularities of the ODEs asso iated with a nite numb er of one- dimensional inte gr als . Setion (3) deals with the omplex multipli ation for the el lipti Singularities of n -fold inte gr als 5 urves related to the singularities giv en b y the zeros of the quadrati p olynomial 1 + 3 w + 4 w 2 = 0 . Our onlusions are giv en in setion ( 4). 2. The singularities of the linear ODE for Φ ( n ) H F or the rst t w o v alues of n , one obtains Φ (1) H = 1 1 − 4 w (9) and Φ (2) H = 1 2 · 1 1 − 16 w 2 · 2 F 1 (1 / 2 , − 1 / 2 ; 1; 16 w 2 ) . (10) F or n ≥ 3 , the series o eien ts of the m ultiple in tegrals Φ ( n ) H are obtained b y expanding in the v ariables x i and p erforming the in tegration (see App endix A). One obtains Φ ( n ) H = 1 n ! · ∞ X k =0 ∞ X p =0 (2 − δ k, 0 ) · (2 − δ p, 0 ) · w n ( k + p ) · a n ( k , p ) (11) where a ( k , p ) is a 4 F 3 h yp ergeometri series dep enden t on w . The adv an tage of using these simplied in tegrals (8) instead of the original ones (2) is t w ofold. Using (11) the series generation is straigh tforw ard ompared to the omplexit y related to the χ ( n ) . As an illustration note that on a desk omputer, Φ ( n ) H are generated up to w 200 in less than 10 seonds CPU time for all v alues of n , while the simplest ase of the χ ( n ) , namely χ (3) , to oks three min utes to generate the series up to w 200 . This dierene b et w een the Φ ( n ) H and χ ( n ) inreases rapidly with inreasing n and inreasing n um b er of generated terms. W e note that for the Φ ( n ) H quan tities and for a xed order, the CPU time is dereasing ♯ with inreasing n . F or χ ( n ) the opp osite is the ase. The seond p oin t is that, for a giv en n , the linear ODE an b e found with less terms in the series ompared to the linear ODE for the χ ( n ) . Indeed for χ (3) , 360 terms w ere needed while 150 terms w ere enough for Φ (3) H . The same feature holds for χ (4) and Φ (4) H (185 terms for χ (4) and 56 terms ¶ for Φ (4) H ). With the fully in tegrated sum (11 ), a suien t n um b er of terms is generated to obtain the linear dieren tial equations. W e sueeded in obtaining the linear dieren tial equations, resp etiv ely of minimal order v e and six, orresp onding to Φ (3) H and Φ (4) H . These linear ODE's are giv en in App endix B. F or Φ ( n ) H ( n ≥ 5 ), the alulations, in order to get the linear ODEs b eome really h uge ‡ . F or this reason, w e in tro due a mo dular strategy whi h amoun ts to generating large series mo dulo a prime and then deduing the ODE mo dulo that prime. Note that the ODE of minimal order is not ne essarily the simplest one as far as the required n um b er of terms in the series expansion to nd the linear ODE is onerned. W e ha v e already enoun tered su h a situation [8, 11 ℄. F or Φ (5) H (resp. Φ (6) H ), the linear ODE ♯ This an b e seen from the series expansion (11 ). Denoting R 0 the xed order, one has n · ( p + k ) ≤ R 0 , while the CPU time for the series generation of a n ( k, p ) is not strongly dep enden t on n . ¶ F rom no w on, for ev en n , the n um b er of terms stands for the n um b er of terms in the v ariable x = w 2 . ‡ Exept the generation of large series whi h remains reasonable. Singularities of n -fold inte gr als 6 of minimal order is of order 17 (resp. 27) and needs 8471 (resp. 9272) terms in the series expansion to b e found. A tually , for Φ (5) H (resp. Φ (6) H ), w e ha v e found the orresp onding linear ODEs of order 28 (resp. 42) with only 2208 (resp. 1838) terms from whi h w e ha v e dedued the minimal ones. The form of these t w o minimal order linear ODEs obtained mo dulo primes is sk et hed in App endix B . In partiular, the singularities (giv en b y the ro ots of the head p olynomial in fron t of the highest order deriv ativ e), are giv en with the orresp onding m ultipliit y in App endix B . Some details ab out the ODE sear h are also giv en in App endix B . W e ha v e also obtained v ery long series (20000 o eien ts) mo dulo primes for Φ (7) H , but, unfortunately , this has not b een suien t to iden tify the linear ODE (mo d. prime) up to order 100. The singularities of the linear ODE for the rst Φ ( n ) H are resp etiv ely zeros of the follo wing p olynomials (b esides w = ∞ ): n = 3 , w · 1 − 16 w 2 (1 − w ) ( 1 + 2 w ) 1 + 3 w + 4 w 2 , n = 4 , w · 1 − 16 w 2 1 − 4 w 2 , n = 5 , w · 1 − 16 w 2 1 − w 2 (1 + 2 w ) 1 + 3 w + 4 w 2 1 − 3 w + w 2 1 + 2 w − 4 w 2 1 + 4 w + 8 w 2 1 − 7 w + 5 w 2 − 4 w 3 1 − w − 3 w 2 + 4 w 3 1 + 8 w + 20 w 2 + 15 w 3 + 4 w 4 , (12) n = 6 , w · 1 − 16 w 2 1 − 4 w 2 1 − w 2 1 − 25 w 2 1 − 9 w 2 1 + 3 w + 4 w 2 1 − 3 w + 4 w 2 1 − 10 w 2 + 29 w 4 . (13) F or n = 7 and n = 8 , b esides mo dulo primes series alulations desrib ed ab o v e, w e also generated v ery large series from whi h w e obtained in oating p oin t form, the p olynomials giv en in App endix C (using generalised dieren tial P adé metho ds). If w e ompare the singularities for Φ ( n ) H to those obtained with the Diagonal mo del ♯ presen ted in [7℄, i.e. Φ ( n ) D , one sees that the singularities of the linear ODE for the Diagonal mo del are iden tial to those of the linear ODE of the Φ ( n ) H for n = 3 , 4 (and are a prop er subset to those of Φ ( n ) H for n = 5 , 6 ). The additional singularities for n = 5 , 6 are zeros of the p olynomials: n = 5 , 1 + 3 w + 4 w 2 1 + 4 w + 8 w 2 × 1 − 7 w + 5 w 2 − 4 w 3 , n = 6 , 1 + 3 w + 4 w 2 1 − 3 w + 4 w 2 1 − 25 w 2 . F or n = 7 , the zeros of the follo wing p olynomials (among others) are singularities whi h are not of Ni k el's t yp e ( 1) and do not o ur for Φ ( n ) D : 1 + 8 w + 1 5 w 2 − 21 w 3 − 60 w 4 + 16 w 5 + 96 w 6 + 64 w 7 , 1 − 4 w − 1 6 w 2 − 48 w 3 + 32 w 4 − 128 w 5 . ♯ Not to b e onfused with the diagonal suseptibilit y and the orresp onding [11℄ n -fold in tegrals χ ( n ) d . Singularities of n -fold inte gr als 7 The linear ODEs of the m ultiple in tegrals Φ ( n ) H th us displa y additional singularities for n = 5 , 6 and n = 7 ( n = 8 see b elo w) ompared to the linear ODE of the single in tegrals Φ ( n ) D . W e found it remark able that the linear ODEs for the in tegrals Φ ( n ) D displa y all the Ni k elian singularities, as w ell as the new quadrati n um b ers 1 + 3 w + 4 w 2 = 0 found for χ (3) . It is th us in teresting to see ho w the singularities for Φ ( n ) D are inluded in the singularities for Φ ( n ) H and whether the new (with resp et to Φ ( n ) D ) singularities an b e giv en b y one-dimensional in tegrals similar to Φ ( n ) D . Let us men tion that the singularities of the linear ODE for Φ (3) H (resp etiv ely Φ (4) H ) ar e r emarkably also singularities of the linear ODE for Φ (5) H (resp etiv ely Φ (6) H ). In the follo wing, w e will sho w ho w this omes ab out and ho w it generalizes. F or this, w e solv e in the sequel the Landau onditions for the n -fold in tegrals (8). 2.1. L andau onditions for the Φ ( n ) H W e remind the reader that the Landau onditions [7℄ are ne essary onditions for singularities to b e the singularities of the inte gr al r epr esentation itself . In a previous pap er [7 ℄, w e ha v e sho wn for partiular in tegral represen tations b elonging to the Ising lass in tegrals [14 ℄, that in fat the solutions of Landau onditions iden tify for sp ei ¶ ongurations (see b elo w) with the singularities of the ODE asso iated with the quan tit y under onsideration. The Landau onditions [7℄ amoun t to arrying out algebrai alulations [7 ℄ on the in tegrand (8) to get singularities of these n -fold in tegrals or ev en, as w e will see in the sequel, singularities of the orresp onding linear ODE [7℄. In the sequel w e use the follo wing in tegral represen tation [ 1, 2℄: y j x n j = Z 2 π 0 dψ j 2 π · exp( i n ψ j ) 1 − 2 w · (cos( φ j ) + cos( ψ j )) . (14) Dening D ( φ j , ψ j ) = 1 − 2 w · (cos( φ j ) + cos( ψ j )) , (15) the in tegral Φ ( n ) H (see its expansion (A.1) in App endix A), b eomes Φ ( n ) H = 1 n ! · Z 2 π 0 n Y j =1 dφ j 2 π dψ j 2 π (16) × D − 1 ( φ j , ψ j ) · δ n X j =1 φ j δ n X j =1 ψ j , where the Dira delta's are in tro dued to tak e are of the onditions n X j =1 φ j = 0 , n X j =1 ψ j = 0 mo d . 2 π (17) on b oth the angles φ j and the auxillary angles ψ j . ¶ In that resp et one m ust reall the notion of leading singularities in on trast with the subleading singularities (see page 54 in [16℄.) Singularities of n -fold inte gr als 8 The Landau onditions [16 , 17 ℄ an easily b e written [7 ℄: α j · D ( φ j , ψ j ) = 0 , j = 1 , · · · , n , (18) β j · φ j = 0 , γ j · ψ j = 0 , j = 1 , · · · , n − 1 , (19) α j · sin( φ j ) − α n · sin( φ n ) + β j = 0 , j = 1 , · · · n − 1 , (20) α j · sin( ψ j ) − α n · sin( ψ n ) + γ j = 0 , j = 1 , · · · , n − 1 (21) together with (17 ). The Landau singularities are obtained b y solving these equations ♯ in all the unkno wns, where the parameters α j , β j , γ j should not b e all equal to zero. In this pap er, our aim is not to nd all the solutions of the ab o v e equations but to sho w that the singularities of the linear ODE for the Φ ( n ) H are solutions of the Landau onditions. F urthermore, in w orking out v arious Ising lass in tegrals [ 14 ℄ and the t w o mo dels of [7℄ (see App endix D), w e remark ed that the singularities of the linear ODE are, in fat, inluded in a partiular onguration. What w e mean b y onguration is the set of v alues (equal to zero or not) of the parameters α j , β j , γ j . The onguration w e onsider α j 6 = 0 , β j = γ j = 0 , (22) orresp onds to pinh singularities on the manifolds D ( φ j , ψ j ) = 0 . One ma y also b e on vined to tak e β j = γ j = 0 , sine the in tegrand is p erio di †† in φ j and ψ j . Let us stress that the the onguration onsidered where all the Lagrange m ultipliers of the singularit y manifolds D ( φ, ψ ) are dieren t from zero ( α j 6 = 0 , for an y j ) leads to the so-alled le ading L andau singularities follo wing the terminology of page 54 of [16 ℄. The Landau onditions b eome: 1 − 2 w · (cos( φ j ) + cos ( ψ j )) = 0 , j = 1 , · · · , n, (23) α j sin( φ j ) − α n sin( φ n ) = 0 , j = 1 , · · · , n − 1 , (24) α j sin( ψ j ) − α n sin( ψ n ) = 0 , j = 1 , · · · , n − 1 . (25) and: n X j =1 φ j = 0 , n X j =1 ψ j = 0 mo d . 2 π (26) The Landau singularities are solutions of these onditions (see App endix E for details). Note that the rst three onditions (23), (24 ), (25 ) are in v arian t b y the transformation: w − → − w, φ j − → φ j + π , ψ j − → ψ j + π . (27) but the Landau onditions ( 23 ), (24 ), (25 ) together with (26 ) are in v arian t b y transformation (27 ) if and only if n is even . This distintion b et w een ev en and o dd in teger n (orresp onding to the symmetry breaking of w ↔ − w ) is reminisen t of the distintion b et w een ev en and o dd in teger n for the χ ( n ) asso iated with the distintion b et w een lo w and high temp erature regimes. The Landau onditions yield t w o families of singularities expressed in terms of Cheb yshev p olynomials of the rst and seond kind. The rst family reads: T 2 p 1 (1 / 2 w + 1) = T n − 2 p 1 − 2 p 2 (1 / 2 w − 1) , (28) 0 ≤ p 1 ≤ [ n/ 2] , 0 ≤ p 2 ≤ [ n/ 2] − p 1 ♯ Note that onditions (19 ), β j · φ j = 0 , γ j · ψ j = 0 , j = 1 , · · · , n − 1 ha v e to b e onsidered in the general Landau onditions. They do not o ur if one restrits oneself to pin h singularities. †† B. Ni k el, priv ate omm uniation. Singularities of n -fold inte gr als 9 The seond family is giv en b y the elimination of z from: T n 1 ( z ) − T n 2 4 w − z 1 − 4 w z = 0 , (29) T n 1 1 2 w − z − T n 2 1 2 w − 4 w − z 1 − 4 w z = 0 , U n 2 − 1 ( z ) · U n 1 − 1 1 2 w − 4 w − z 1 − 4 w z − U n 2 − 1 1 2 w − z · U n 1 − 1 4 w − z 1 − 4 w z = 0 with n 1 = p 1 , n 2 = n − p 1 − 2 p 2 , (30) 0 ≤ p 1 ≤ n, 0 ≤ p 2 ≤ [( n − p 1 ) / 2] . (31) One reognizes in the rst set of equations (28 ), a generalization of the singularities giv en b y Ni k el [15 ℄ for the pin h singularities oming from the pro dut of the y j 's, and also deriv ed for our m ultiple in tegral denoted Y ( n ) in [7℄. These ha v e b een written as [7 , 15 ℄: T k (1 / 2 w + 1) = T n − k (1 / 2 w − 1) (32) Note that, omparativ ely to (28 ), the in teger k should b e ev en † . The seond set of equations (29) is a generalization of the singularities w e deriv ed for Φ ( n ) D in [7℄. In b oth form ulae, one notes the o urrene of a seond v arying in teger p 2 , leading to a b etter understanding of the singularities of these in tegrals. Indeed with p 2 running, the linear ODE for Φ ( n ) H will automatially on tain all the singularities of the linear ODEs for Φ ( n − 2) H , Φ ( n − 4) H , · · · , Φ ( n − 2 q ) H . F or n = 7 , w e ha v e he k ed that the singularities sp ei to n = 7 ( p 2 = 0 in (28), (29 )) also app ear as singularities of the linear ODE in oating p oin t form (see App endix D for details). F or p 2 = 1 , part of the singularities app ear in oating p oin t form, while for p 2 = 2 (i.e. singularities of Φ (3) H ), no singularities app ear in oating p oin t form. Similarly , for n = 8 , w e ha v e he k ed that the singularities sp ei to n = 8 ( p 2 = 0 in (28 ), (29 )) also app ear as singularities of the linear ODE in oating p oin t form (see App endix D for details). F or p 2 ≥ 1 , no singularities app ear in oating p oin t form. Let us remark that the non observ ation of some singularities in oating p oin t form is not really signian t. Indeed, w e ha v e used 1250 (resp. 1200 terms) for Φ (7) H (resp. Φ (8) H ) while the Φ (7) H and Φ (8) H linear ODEs need more than 20000 terms. Figure 1 sho ws the rst family of singularities ( 28) displa y ed in the omplex s plane lose enough to the unit s -irle. This gure learly sho ws a quite ri h struture for these set of p oin ts. This gure lo oks lik e a net w ork of no dal p oin ts link ed together b y (ardioid-lik e) urv es that an, at rst sigh t, hardly b e distinguished from ars of irles. In partiular the seleted p oin ts 1 + 3 w + 4 w 2 = 0 as w ell as the singularities for Φ (5) , lik e 1 + 8 w + 20 w 2 + 15 w 3 + 4 w 4 = 0 an b e seen to o ur quite learly as some of these no dal p oin ts. † This is a onsequene of (23 ), (24 ), (25 ), (26 ) yielding k · π = 0 mo d. 2 π (see App endix E.1). Singularities of n -fold inte gr als 10 –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1 Figure 1. First family of singularities (28 ) in the omplex s plane ( n ≤ 51 ). –3 –2 –1 0 1 2 3 –2 –1 0 1 2 3 Figure 2. First family of singularities (28 ) in the omplex s plane far from the unit irle ( n ≤ 51 ). Singularities of n -fold inte gr als 11 –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1 Figure 3. First and seond family of singularities ( 28 ), (29 ) in the omplex s plane ( n ≤ 16 ). –3 –2 –1 0 1 2 3 –2 –1 0 1 2 3 Figure 4. First and seond family of singularities ( 28 ), (29 ) in the omplex s plane far from the unit irle ( n ≤ 16 ). Singularities of n -fold inte gr als 12 Figure 2 sho ws the rst family of singularities (28) far from the unit irle. Figure 3 sho ws all the singularities altogether (rst and seond family) lose to the unit s - irle. Finally gure 4 sho ws all the singularities together ((28), (29)) that are not so lose to the unit s -irle. The aum ulation of singularities one an see on gure 1 near s = i and s = − i seem to onrm the statemen t made in Orri k et al [18 ℄ that these t w o p oin ts are t w o quite unpleasan t p oin ts for the suseptibilit y of the Ising mo del for whi h the series expansions are not ev en asymptotially on v ergen t. Besides repro duing exatly the singularities of the linear ODE for Φ ( n ) H , it is remark able to see from the form ula (28 ), (29 ), ho w to tra k where ea h singularit y- p olynomial omes from. This allo ws one to understand ho w the singularities of the Ising lik e in tegrals Y ( n ) and Φ ( n ) D (see [7 ℄) and ev en the Ni k elian singularities (1 ) emerge in these m ultiple in tegrals ( 8). This omes simply from the partition (30 ) and the equiv alen t one in ( 28). 2.2. Singularities: fr om n -fold inte gr als to one dimensional inte gr als Consider for instane the singularities 1 − 7 w + 5 w 2 − 4 w 3 = 0 o urring in Φ (5) H , whi h are giv en b y (28 ) for n = 5 , p 1 = 1 and p 2 = 0 . As far as onditions on the in tegration angles (see (33) b elo w), this arises from a situation where t w o angles are equal and the three others are equal. Reall that the Φ ( n ) D in tegrals are onstruted with the follo wing restritions on the angles: φ 1 = φ 2 = · · · = φ, φ n = − ( n − 1) φ. (33) One sees that a generalization of this mo del (33 ) is simply: φ 1 = φ 2 = · · · = φ k , φ k +1 = φ k +2 = · · · = φ n , k = 0 , 1 , · · · , [ n / 2] . (34) By the ondition on the angles, this ase is inde e d one dimensional , with: φ n = − ( n − k ) k · φ + 2 j π k , j integer . (35) The mo del (33 ) is ob viously giv en b y (34) for k = 1 . The Ni k elian singularities are also giv en b y (34) for k = 0 , but this time, the underlying mo del is zero- dimensional. The mo del onstruted along the same lines as in [ 7 ℄ orresp onds to an in tegrand: n − 1 X j =0 1 1 − x n 2 π j n . (36) The Ni k elian singularities arise as p oles. F or k ≥ 2 , the singularities giv en b y the mo del (34 ), whi h app ear in (8), are th us giv en neither b y ( 1) nor b y Φ ( n ) D . Consider one v ariable of in tegration su h as (7), where the in tegrand is: 1 1 − x n − 1 ( φ ) · x (( n − 1) φ ) − → 1 1 − x n − k ( φ ) · x k ( φ n ) . (37) and denote b y Φ ( n ) k su h in tegrals (one then has Φ ( n ) 1 = Φ ( n ) D ). Singularities of n -fold inte gr als 13 Fix n = 5 and k = 2 . The onstrain t (35 ) on the angles reads: φ 5 = − 3 2 φ 1 + j π , j integer (38) with one in tegration v ariable. The series of o eien ts of Φ (5) 2 is generated along the same lines as for Φ ( n ) D (see App endix A). The F u hsian linear dieren tial equation is of order six and this order is indep enden t of the v alue of j in (38 ). The singularities of the linear ODE are zeros of the follo wing p olynomials: w · 1 − 16 w 2 (1 + w ) 1 − 3 w + w 2 1 + 2 w − 4 w 2 × 1 + 4 w + 8 w 2 1 − 7 w + 5 w 2 − 4 w 3 . (39) W e obtain singularities (from the last t w o p olynomials) app earing for Φ (5) H and not o urring for Φ (5) D . The o urrene of the singularities 1 + 3 w + 4 w 2 = 0 for (the linear ODE of ) Φ (5) H but not for (the linear ODE of ) Φ (5) D is explained along similar lines. Note that these singularities are ommon to (the linear ODE of ) Φ (3) H , Φ (5) H and Φ (6) H . The p olynomial 1 + 3 w + 4 w 2 app ears for (the linear ODE of ) Φ (5) H from (28), namely: T 2 p 1 (1 / 2 w + 1) = T n − 2 p 2 − 2 p 1 (1 / 2 w − 1) . (40) One sees that the p olynomial 1 + 3 w + 4 w 2 will app ear for all om binations of n , p 1 and p 2 su h that: 2 p 1 = 2 , n − 2 p 2 − 2 p 1 = 1 . (41) In other w ords, the p olynomial that arises for giv en n and p 1 , wil l also app e ar for the same v alue of p 1 and for n − 2 p 2 . The singularities orresp onding to 1 + 3 w + 4 w 2 = 0 o ur for Φ (5) H with n = 5 , p 1 = 1 and p 2 = 1 , but (41 ) is also satised for n = 3 , p 1 = 1 and p 2 = 0 whi h sho ws a situation with three angles,with t w o of them equal. This is preisely the in tegrand in (7), i.e. in Φ (3) D . Consider no w the ase n = 6 and k = 2 . This amoun ts to onsidering the n -fold in tegral Φ (6) 2 with: φ 6 = − 2 φ 1 + j π , j integer . (42) The results are dep enden t on the in teger j . F or instane, the series around w = 0 reads: Φ (6) 2 = 1 + w 6 + 32 w 8 ± w 9 + 659 w 10 (43) ± 1296 w 11 + 1169 1 w 12 + · · · With the + sign in the series (43), the linear dieren tial equation is of order v e and the singularities are giv en b y the zeros of the p olynomials: w · 1 − 16 w 2 (1 − w ) ( 1 + 2 w ) 1 − 9 w 2 × 1 − 25 w 2 1 + 3 w + 4 w 2 . (44) The results orresp onding to the hoie of a min us sign in the series (43 ) are ob viously obtained b y § w → − w . W e obtain the singularities 1 − 25 w 2 and 1 ± 3 w + 4 w 2 = 0 o urring for (the linear ODE of ) Φ (6) H but not for (the linear ODE of ) Φ (6) D . Similarly , for n = 7 , ( k go es to 3), one obtains for k = 2 , the singularities as zeros of the follo wing p olynomial 1 + 8 w + 15 w 2 − 21 w 3 − 60 w 4 + 16 w 5 + 96 w 6 + 64 w 7 , § The last ase for n = 6 , i.e. k = 3 do es not pro vide singularities other than Ni k el's. Singularities of n -fold inte gr als 14 whi h has indeed b een found n umerially in the linear ODE sear h on a large series orresp onding to Φ (7) H (see App endix C). W e ha v e the remark able fat that the singularities of the linear ODE for the m ultiple in tegral Φ ( n ) H are giv en b y a nite set of singularities of linear ODEs of a set of one-dimensional in tegrals, namely , N ( N + 1) / 2 one-dimensional in tegrals, with N = [ n/ 2] . F or instane, the singularities of the four-dimensional in tegral Φ (5) H iden tify with those of, at most, three one-dimensional in tegrals. This app ears, simply , from the ouple of in tegers in (28 ) whi h read (2 p 1 , n − 2 p 2 − 2 p 1 ) . F or xed n , when p 2 v aries, one sees that w e are in fat onsidering all the lo w er in teger v alues n − 2 p 2 . The same situation holds for (29 ). This iden tiation leads, ob viously , to partiular strutures in the singularities for dieren t n . This is what w e sho w in the sequel. 2.3. Singularity strutur es of n -fold inte gr als and p artiular sets of one-dimensional inte gr als The Landau singularities giv en in App endix E are he k ed against the singularities of the linear ODE for Φ ( n ) H ( n = 3 , · · · , 6 ), and ar e found to b e identi al . Assume that these form ulae do indeed repro due all the singularities of the linear ODE for Φ ( n ) H , for an y n . In this ase, w e an he k whether the singularities app earing at n = m also o ur for n = m + 1 , n = m + 2 , · · · W e ha v e found that the singularities at order 2 n will also b e singularities at order 2 n + 2 p , where p is a p ositiv e in teger. Similarly , the singularities at order 2 n + 1 will also b e presen t at the follo wing o dd orders. What is remark able is the fat that the singularities at o dd order also app ear at ev en orders. The rule is: al l the singularities at o dd or der n also app e ar in the higher or ders (o dd and even) ex ept for the rst ( n − 1) / 2 even or ders . F or instane, the singularities app earing at n = 3 will o ur for all n , exept the rst ev en order, i.e. 4 . The singularities app earing at n = 5 will o ur for all n , exept the rst t w o ev en orders, i.e. 6 and 8 . The onsequene of this emb e dding of the singularities is the o urrene of some singularities at pr e dene d or ders . The singularit y 1 + 2 w = 0 is presen t at any or der n . The singularit y 1 − 2 w = 0 is presen t for an y ev en order 2 n . The singularit y 1 + w = 0 o urs at an y order n ≥ 5 . The singularit y 1 − w = 0 o urs at an y order n , exept for n = 4 . All these singularities are Ni k elian. The rst non Ni k elian singularit y 1 + 3 w + 4 w 2 = 0 app e ars at al l or ders n , exept for n = 4 . Moreo v er, w e ha v e giv en in [ 7℄ the Landau singularities for the (linear ODEs of the) in tegrals Φ ( n ) D . These singularities ha v e b een found to b e iden tial with the singularities of the linear ODE for Φ ( n ) D obtained exatly up to n = 8 and mo dulo a prime up to n = 1 4 . W e ha v e seen that all the singularities of the linear ODE of Φ ( n ) D in the v ariable s lie in the ann ulus dened b y t w o onen tri irles of radius √ 2 and 1 / √ 2 . The radii of the t w o onen tri irles are the ro ots, in the v ariable s , of the p olynomial 1 + 3 w + 4 w 2 = 0 , that is s 2 + s + 2 = 0 and 1 + s + 2 s 2 = 0 . With the m ultiple in tegrals Φ ( n ) H , one sees that some of the singularities ar e not onne d to this ann ulus an ymore. Thanks to the Landau onditions, one an no w understand this struture from the redution of the m ultiple in tegrals Φ ( n ) H to a set of one-dimensional in tegrals Φ ( n ) k as far as the lo ation of singularities is onerned. F or k = 0 , whi h orresp onds to the Singularities of n -fold inte gr als 15 Ni k elian singularities, the ann ulus is the unit irle. F or k = 1 orresp onding to the in tegrals Φ ( n ) D , one has the ann ulus of radii √ 2 and its in v erse. F or ea h k , one exp ets the singularities to lie in an ann ulus with a onen tri struture. F or these ann uli the larger radius inreases (smaller radius dereases) as k inreases. F rom the redution of the singularities of Φ ( n ) H to these Φ ( n ) k , all the singularities for xed p 1 = k in (28 ) and for xed p 1 = k in (29 ) will b e onned to one ann ulus. F or instane for k = 2 , all the singularities o urring in the linear ODE for Φ ( n ) k , (i.e. for all n ), or, equiv alen tly , all the singularities obtained b y ( 28 ) for p 1 = 1 and b y (29 ) for p 1 = 2 will b e onned to the ann ulus of radii 2 . 79 · · · and its in v erse. This v alue is the ro ot, in the v ariable s , of 1 − 7 w + 5 w 2 − 4 w 3 = 0 o urring for Φ (5) H . F or k = 3 , one remarks that the ann ulus will not b e obtained from (28) whi h is restrited b y 2 p 1 , an ev en in teger. In fat this is general. The radii of the ann uli are giv en b y (28 ) for k ev en and b y ( 29 ) for k o dd. The ro ot in the v ariable s that will dene the ann ulus o urs at o dd order n giv en b y 2 k + 1 . The piture no w, is as follo ws. The singularities of the linear ODE for the in tegrals Φ ( n ) H are partionned in to families indexed b y the in teger k . The singularities for k = 0 are Ni k elian and lie on the unit irle, sa y , r 0 = 1 . The singularities for k = 1 lie in the ann ulus r 1 = √ 2 , 1 / √ 2 (w e disard from no w on, the smaller radius). The singularities for k = 2 will b e onned in the ann ulus r 2 . The singularities for k = N will b e in the ann ulus r N . These onen tri ann uli are su h that r 0 < r 2 < · · · < r 2 N and r 1 < r 3 < · · · < r 2 N +1 , (with r 2 k < r 2 k +1 ). As k gro ws, the radii of t w o neigh b oring irles b eha v e as r 2 k +2 − r 2 k → 0 and r 2 k +3 − r 2 k +1 → 0 . This derease is not enough to reate an aum ulation of irles. W e he k ed with k = 300 irles that the derease go es as k − α with α < 1 prev en ting an y on v ergene. F or n large these radii div erge: r N → ∞ when N → ∞ . Note that these families, (i.e. the index k ) ome from the resolution of the Landau onditions and from the redution of the singularities for Φ ( n ) H to the ones of Φ ( n ) k , ( k = 0 , 1 , · · · [ n/ 2] ). W e ha v e no idea as to ho w these families an b e seen diretly from the m utiple in tegrals Φ ( n ) H . If the singularities for Φ ( n ) H happ en to b e iden tial with those o urring in the linear ODE for χ ( n ) , it ma y b eome imp ortan t to see whether this piture p ersists and whether this piture is sho wing another partition of the suseptibilit y χ instead of the kno wn sum on χ ( n ) . Figure 5, 6 and 7 sho w ho w the rst family of singularities (28 ) in the s omplex plane is deomp osed aording to the in teger k in (34 ). Figure 5 sho ws singularities (28) for a given o dd value of k , namely k = 5 for any o dd values of n up to 91 . Figure 6 sho ws singularities (28) for a giv en ev en v alue of k , namely k = 2 for an y o dd v alues of n up to 71 . Figure 7 sho ws singularities (28 ) for a giv en ev en v alue of k , namely k = 6 for an y ev en v alues of n up to 80 . The gures orresp onding to the ltration of the singularities of the rst family (28 ) in terms of the in teger k (previously displa y ed altogether with gures 1 and 2 ) deserv e some ommen ts. First, one sees that the v arious resen t orresp onding to dieren t v alues of k are v ery similar. Seondly one sees from gure 5 that the o dd n , o dd k resen t break the s ↔ − s symmetry (for ev en n , ev en k , the equations for the set of singularities are funtions of s 2 , see gure 7 ) in a quite dramati w a y: the singularities in the resen t of gure 5 all lie only in the left half s -omplex plane. Similarly the singularities in the resen t of gure 6 all lie in the righ t half s -omplex plane. Along this s ↔ − s symmetry line it is w orth realling that the lo w-temp erature Singularities of n -fold inte gr als 16 –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1 Figure 5. Cresen t in the omplex s plane giv en b y ( 28 ): k = 5 , n ≤ 91 , n o dd. –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1 Figure 6. Cresen t in the omplex s plane giv en b y ( 28 ): k = 2 , n ≤ 71 , n o dd. Singularities of n -fold inte gr als 17 –1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1 Figure 7. Cresen t in the omplex s plane giv en b y ( 28 ): k = 6 , n ≤ 80 , n ev en. suseptibilit y of the Ising mo del has this s ↔ − s symmetry (the lo w-temp erature suseptibilit y is a funtion of s 2 or w 2 ) but the high-temp erature suseptibilit y breaks that s ↔ − s symmetry , and this is also the ase for the n -fold in tegral χ ( n ) with n o dd. Our n -fold in tegrals (8) are in tro dued to pro vide an eduated guess as to the lo ation of the singularities of the χ ( n ) . As far as lo ation of singularities of the χ ( n ) are onerned, it is not totally lear for n o dd if the s ↔ − s (resp. w ↔ − w ) symmetry will not b e partially restored on the global set of singularities with the o urene for a singularit y P n ( w ) = 0 for a giv en v alue of n , of the opp osite v alue for, p erhaps, a dieren t v alue of n : P m ( − w ) = 0 . Remark: Quite often, in this pap er, w e use (b y abuse of language) the w ords singularities of an n -fold in tegral to desrib e a larger set of singularities, namely the singularities of the linear ODEs that the n -fold in tegral satises. A rigorous study w ould require, for an y singularit y, to p erform the (dieren tial Galois group and onnetion matrix) analysis w e ha v e p erformed in [6 ℄. It amoun ts to getting extremely large series, dedued from the obtained linear ODE, that oinide with the series expansion of the n -fold in tegral w e are in terested in, and nd out if these series atually exhibit these singularities. With this tedious, but straigh tforw ard, pro edure w e an extrat the singularities of a sp ei n -fold in tegral among the larger set of singularities of the orresp onding linear dieren tial equation. In view of the large n um b er of singularities w e displa y in this pap er, w e ha v e not p erformed su h a systemati analysis, that w ould ha v e b een quite h uge. F urthermore it is imp ortan t to note that this onnetion matrix approa h [6℄ requires to ha v e the linear ODE of the n -fold in tegral. A kno wledge of the linear ODE mo dulo a prime is not suient . W e ould ha v e p erformed this analysis for Φ (3) H and Φ (4) H , but, in that ase, w e already Singularities of n -fold inte gr als 18 ha v e a deep er result [6℄ namely the onnetion matrix analysis for χ (3) and χ (4) , pro viding an understanding of the singularities of these n -fold in tegrals themselv es (in w and also in s ). Righ t no w, the only singularities found for the χ ( n ) , other than Ni k elian, are the quadrati ro ots of 1 + 3 w + 4 w 2 = 0 , (i.e. the rst ann ulus) whi h app ear at all orders (exept n = 4 for Φ ( n ) H ). Let us sho w, in the sequel, ho w this p olynomial an b e sp eial. 3. T o w ards a mathematial in terpretation of the singularities In a set of pap ers [ 19 , 20 ℄, w e ha v e underlined the en tral role pla y ed b y the el lipti p ar ametrization of the Ising mo del, in partiular the role pla y ed b y the seond order linear dieren tial op erators orresp onding to the omplete ellipti in tegral E (or K ), and the o urrene of an innite n um b er of mo dular urves [ 12 ℄, anonially asso iated with el lipti urves . The deep link b et w een the theory of ellipti urv es and the theory of mo dular forms is no w w ell established [21 ℄. Consequen tly , it ma y b e in teresting to seek sp eial v alues of the mo dulus k , (singularities of the χ ( n ) ) that migh t ha v e a ph ysial meaning, as w ell as a mathematial in terpretation. F or that purp ose, reall that the mo dular group requires one to in tro due the ellipti nome, dened in terms of the p erio ds of the ellipti funtions, q = exp − π K (1 − k 2 ) K ( k 2 ) = exp( i π τ ) (45) and the half p erio d ratio ¶ τ . W e write the omplete ellipti in tegral K as K ( k ) = 2 F 1 1 / 2 , 1 / 2 ; 1; k . (46) Relations b et w een K ( k ) ev aluated at t w o dieren t mo duli an b e found in, e.g. [ 22 ℄. 3.1. Some iso genies of el lipti urves se en as gener ators of the r enormalization gr oup The argumen ts in K in these iden tities are related b y the so-alled, resp etiv ely , desending Landen and asending Landen (or Gauss) transformations: k − → k − 1 = 1 − √ 1 − k 2 1 + √ 1 − k 2 (47) k − → k 1 = 2 √ k 1 + k (48) These transformations (or orresp ondenes [23 , 24 ℄), derease or inrease the mo dulus resp etiv ely . Iterating (47 ) or (48 ), one on v erges to k = 0 or k = 1 resp etiv ely . The half p erio d ratio transforms through ( 47), (48 ), as τ → 2 τ , τ → 1 2 τ (49) resp etiv ely . The r e al xed p oin ts of the transformations (47 ) and ( 48 ) are k = 0 (the trivial innite or zero temp erature p oin ts) and k = 1 (the ferromagneti and ¶ In the theory of mo dular forms q 2 is also sometimes used instead of q . In n um b er theory literature the half-p erio d ratio is tak en as − i τ . Singularities of n -fold inte gr als 19 an tiferromagneti ritial p oin t of the square Ising mo del). In terms of the half p erio d ratio, this reads τ = ∞ and τ = 0 resp etiv ely , whi h orr esp ond to a de gener ation of the el lipti p ar ametrization into a r ational p ar ametrization . In view of these xed p oin ts, it is natural to iden tify the transformations (47 ) or (48), and more generally an y transformation § τ → n · τ or τ → τ / n ( n in teger), as exat gener ators of the r enormalization gr oup of the t w o-dimensional Ising mo del ‡ . One do es not need to restrit the analysis to the real xed p oin ts of the transformations. If one onsiders the Landen transformation ( 48 ) as an algebrai transformation of the omplex variable k and if one solv es k 2 1 − k 2 = 0 , one obtains: k · (1 − k ) · ( k 2 + 3 k + 4) = 0 . (50) The quadrati ro ots k 2 + 3 k + 4 = 0 , (51) are (up to a sign) xe d p oints of (48). W e th us see the o urrene of additional non-trivial omplex sele te d values of the mo dulus k , b ey ond the w ell-kno wn v alues k = 1 , 0 , ∞ (orresp onding to degeneration of the ellipti urv e in to a r ational urve ). Ph ysially , these w ell-kno wn v alues k = 1 , 0 , ∞ orresp ond to the riti al Ising mo del ( k = 1 ) and to (high-lo w temp erature) trivializations of the mo del ( k = 0 , ∞ ). 3.2. Complex multipli ation for el lipti urves as ( omplex) xe d p oints of the r enormalization gr oup W e ome no w to our p oin t. The rst unexp eted singularities 1 + 3 w + 4 w 2 = 0 found [4 , 5℄ for the F u hsian linear dieren tial equation of χ (3) , and also in other n -fold in tegrals of the Ising lass [7 ℄, reads in the v ariable k = s 2 as ( k 2 + 3 k + 4) (4 k 2 + 3 k + 1) = 0 . (52) The rst p olynomial ‡ orresp onds to xe d p oints of the Landen transformation (see (50)). In other w ords w e see that the seleted quadrati v alues 1 + 3 w + 4 w 2 = 0 , o urring in the (high-temp erature) suseptibilit y of the Ising mo del as singularities of the three-partile term χ (3) , an b e seen as xe d p oints of the r enormalization gr oup when extende d to omplex values of the mo dulus k . F or ellipti urv es in elds of harateristi zero, the only w ell-kno wn seleted set of v alues for k orresp onds to the v alues for whi h the ellipti urv e has omplex multipli ation [26 ℄. Complex m ultipliation for ellipti urv es orresp onds to algebrai in teger v alues (in tegers in the ase of the Heegner n um b ers, see App endix F) of the mo dular j -funtion, whi h orresp onds to Klein's absolute in v arian t m ultiplied b y (12) 3 = 1728 : j ( k ) = 256 · 1 − k 2 + k 4 3 k 4 · (1 − k 2 ) 2 . (53) A straigh tforw ard alulation of the ellipti nome (45 ) giv es, for the p olynomials (52), resp etiv ely , an exat v alue for τ , the half p erio d ratio, as v ery simple quadr ati numb ers : τ 1 = ± 3 + i √ 7 4 , τ 2 = ± 1 + i √ 7 2 (54) § See relation (1.3) in [25℄. ‡ A similar iden tiation of these isogenies τ → n · τ with exat generators of the renormalization group an b e in tro dued for an y lattie mo del with an ellipti parametrization (Baxter mo del, ...). ‡ Note that the t w o p olynomials in (52 ) are related b y the Kramers-W annier dualit y k → 1 /k . Singularities of n -fold inte gr als 20 These quadrati n um b ers atually orresp ond to omplex multipli ation of the ellipti urv e and for b oth one has j = ( − 1 5 ) 3 . These t w o quadrati n um b ers are su h that 2 τ 1 ∓ 1 = τ 2 . Let us fo us on τ 2 for whi h w e an write: τ = 1 − 2 τ . (55) T aking in to aoun t the t w o mo dular group in v olutions τ → 1 − τ and τ → 1 /τ , w e nd that 1 − 2 /τ is, up to the mo dular gr oup , equiv alen t to τ / 2 . The quadrati relation τ 2 − τ + 2 = 0 th us amoun ts to lo oking at the xed p oin ts of the Landen transformation τ → 2 τ up to the mo dular gr oup . This is, in fat a quite general statemen t. The omplex multipli ation v alues an all b e seen as xed p oin ts, up to the mo dular gr oup , of the generalizations of Landen transformation, namely τ → n τ for n in teger, τ 2 − τ + n = 0 or τ = 1 − n τ ≃ n · τ , where ≃ denotes the equiv alene up to the mo dular gr oup . App endix G presen ts an alternativ e view b y onsidering the solutions as xed p oin ts under Landen transformations of the mo dular j − funtion. In view of the remark able mathematial (and ph ysial) in terpretation of the quadrati v alues 1 + 3 w + 4 w 2 = 0 in terms of omplex multipli ation for el lipti urves, or xe d p oints of the r enormalization gr oup , it is natural to see if su h a omplex m ultipliation of ellipti urv es in terpretation also exists for other singularities of χ ( n ) , and as a rst step, for the singularities of the linear dieren tial equations of our n -fold in tegrals (8), that w e exp et to b e iden tial, or at least ha v e some o v erlap, with the singularities of the χ ( n ) . Noting that the mo dular j − funtion is a funtion of s 2 or w 2 (see (F.2) in App endix F) the o urene of 1 + 3 w + 4 w 2 = 0 as a seleted quadrati p olynomial ondition means, at the same time, the o urene of the other quadrati p olynomial ondition 1 − 3 w + 4 w 2 = 0 (see App endix F and App endix G.2). Besides 1 − 3 w + 4 w 2 = 0 , w e ha v e found t w o other p olynomial onditions whi h orresp ond to remark able in teger v alues of the mo dular j -funtion. The singularities 1 − 8 w 2 = 0 orresp ond to j = (12) 3 and τ = ± 1 + i (see App endix F). They orresp ond to Ni k elian singularities for χ (8) (and th us Φ (8) H ) and to non-Ni k elian singularities for Φ (10) H . Another p olynomial ondition is 1 − 32 w 2 = 0 , whi h giv es non-Ni k elian singularities that b egin to app ear at n = 10 for Φ (10) H . These singularities orresp ond to the in teger v alue of the mo dular j -funtion, j = (66) 3 and to τ = 2 i or τ = − 4 / 5 + 2 i/ 5 . 3.3. Beyond el lipti urves Among the singularities of the linear ODE for Φ ( n ) H giv en in ( 12 ), (13 ) or obtained from the form ula giv en in App endix E up to n = 1 5 , w e ha v e found no other singularit y iden tied with seleted algebrai v alues of the mo dular j -funtion orresp onding to omplex m ultipliation for ellipti urv es. Could it b e that the (non-Ni k elian) singularities (12 ), (13 ), whi h do not mat h with omplex m ultipliation for ellipti urv es, are atually remark able seleted situations for mathematial strutures more omplex than ellipti urv es ? With these new singularities, are w e p ossibly exploring some remark able seleted situations of some mo duli sp a e of urves orr esp onding to p ointe d (marke d) urves [27 ℄, instead of simple ellipti urv es [ 28 ℄? In pratie this just orresp onds to onsidering a pro dut of n times a rational, or ellipti, urv e Singularities of n -fold inte gr als 21 min us some sets of remark able o dimension-one algebrai v arieties [11 ℄, x i x j = 1 , x i x j x k = 1 , h yp erplanes x i = x j , · · · W e try to fully understand the singularities of the n -fold in tegrals orresp onding to the χ ( n ) , that is to sa y partiular n -fold in tegrals link ed to the theory of ellipti urv es. These n -fold in tegrals are more in v olv ed than the (simpler) n -fold in tegrals in tro dued b y Beuk ers, V asily ev [ 29, 30 ℄ and Sorokin [31, 32 ℄, or the Gon haro v-Manin in tegrals [33 ℄ whi h o ur in some mo duli sp a e of urves [34 , 35 ℄ simply orresp onding to a pr o dut of r ational urves ( C P 1 × C P 1 · · · × C P 1 ). An example of su h in tegrals, link ed ¶ to ζ (3) , is displa y ed ♯ in App endix H . Let us lose this setion b y noting that Heegner n um b ers and, more generally , omplex multipli ation ha v e already o urred in other on texts, ev en if the statemen t w as not expliit. In the framew ork of the onstrution of Liouville eld theory , Gerv ais and Nev eu ha v e suggested [41 ℄ new lasses of ritial statistial mo dels, where, b esides the w ell-kno wn N -th ro ot of unit y situation, they found the follo wing seleted v alues of the multipli ative r ossing t [42℄: t = e i π (1+ i √ 3) / 2 = i · e − π √ 3 / 2 , (56) t = e i π (1+ i ) = − e − π . (57) If one w an ts to see this m ultipliativ e rossing as a mo dular nome, the t w o previous situations atually orresp ond to seleted v alues of the mo dular j -funtion namely j ((1 + i √ 3) / 2) = (0) 3 for (56 ), and j (1 + i ) = (12) 3 for (57), whi h atually orresp ond to He e gner numb ers and, mor e gener al ly, omplex multipli ation [26 ℄. It is ho w ev er imp ortan t not to feed the onfusion already to o prev alan t in the literature, b et w een a temp er atur e-like nome su h as (45 ) and a multipli ative r ossing mo dular nome . In the Baxter mo del [43 , 44 ℄, the rst is denoted b y q and the seond one b y x . In fat one probably has, not one, but two mo dular gr oups taking plae, one ating on the temp erature-lik e nome q and the other ating on the m ultipliativ e rossing x . W e will not go further along this quite sp eulativ e line whi h amoun ts to in tro duing el lipti quantum gr oups [45 ℄ and el lipti gamma funtions † (generalization of theta funtions †† ). 4. Conlusion The ultimate goal of our Ising lass in tegrals is to get some insigh t in to the χ ( n ) and, hop efully , in to the suseptibilit y of the Ising mo del. F or that purp ose w e ha v e in tro dued n -fold in tegrals (8 ) su h that w e exp et the singularities of the orresp onding linear ODE to o v erlap, as m u h as p ossible, with the singularities of the linear ODE for the χ ( n ) . W e ha v e obtained the linear dieren tial equations for ¶ Note that ζ (or the p olyzeta) funtion ev aluated at in teger v alues ( ζ (3) , ζ (5) , ...) do o ur in our more in v olv ed n -fold in tegrals, in partiular in the represen tation of the onnetion matries [6℄ of the dieren tial Galois group of the F u hsian linear ODEs of χ ( n ) . ♯ These n -fold in tegrals [36, 37 , 38 , 39 , 40 ℄ lo ok almost the same as the ones w e ha v e in tro dued and analyzed in the study of the diagonal suseptibilit y of the Ising mo del [11 ℄ for whi h n -th ro ot of unit y singularities o ur. † Whi h an b e seen [46℄ as automorphi forms of degree 1 when the Jaobi mo dular forms are automorphi forms of degree 0 and are asso iated (up to simple semi-diret pro duts) with S L (3 , Z ) instead of S L (2 , Z ) †† The partition funtion of the Baxter mo del an b e seen as a ratio and pro dut of ellipti gamma funtions and theta funtions. It is th us naturally expressed as a double innite pro dut. Similar double, and ev en triple, pro duts app ear in orrelation funtions of the eigh t v ertex mo del [47, 48℄. Singularities of n -fold inte gr als 22 these n -fold in tegrals Φ ( n ) H , up to n = 4 and up to n = 6 mo dulo a prime. F rom these exat results together with an exhaustiv e Landau singularit y analysis, w e pro vided a quite omplete desription of the singularities of these linear ODEs. F rom the Landau onditions, the singularit y strutures are explained. The singularities orresp onding to Φ ( n ) H are found to also o ur at a higher predened order p > n . With these m ultiple in tegrals and the asso iated Landau onditions, w e ha v e b een able to understand wh y the simple in tegrals Φ ( n ) D ha v e sueeded repro duing the Ni k elian singularities and the new quadrati 1 + 3 w + 4 w 2 = 0 . These simple in tegrals app ear to b e "a rst appro ximation" to Φ ( n ) H . Other one-dimensional in tegrals p op up to aoun t for the additional singularities not o urring for Φ ( n ) D . W e ha v e then a remark able nding that, the singularities for the m ultiple in tegrals an b e asso iated with the singularities for a nite n um b er of one dimensional in tegrals. If the singularities, asso iated with these n -fold in tegrals ( 8), happ en to b e iden tial with (or to o v erlap) the singularities asso iated with the χ ( n ) , it b eomes imp ortan t to understand this me hanism for the χ ( n ) themselv es. If this me hanism of singularit y em b edding o urs for χ ( n ) , it migh t b e explained b y a Russian doll struture for the same linear dieren tial op erators. W e kno w that the linear dieren tial op erator for χ (1) (resp etiv ely χ (2) ) is on tained in (righ tdivides) the linear dieren tial op erator for χ (3) (resp etiv ely χ (4) ), and furthermore w e ev en ha v e diret sum deomp osition prop erties. F or the Φ ( n ) H , it is not these me hanisms whi h are at w ork. Our primary goal in this study is to iden tify as man y singularities as p ossible for the χ ( n ) . The singularities of the ODEs asso iated with the Φ ( n ) H quan tities orresp ond, in the Landau equations framew ork, to le ading pinh singularities (relativ ely to the singularities manifolds D ( φ, ψ ) = 0 ). F or the other quan tities previously studied [7℄ whi h b elong to the Ising lass in tegrals, the same feature holds. A t this step, the natural questions arising are: whether the s heme, from the Landau singularities p oin t of view, whi h holds for Φ ( n ) H , still holds for χ ( n ) and whether the singularities of Φ ( n ) H an b e onsidered as singularities of the χ ( n ) ? F rom the Landau singularities viewp oin t, the F ermioni determinan t G ( n ) 2 is going to in tro due new manifolds of singularities. When the Lagrange m ultipliers relativ e to the singularities manifolds in tro dued b y the F ermioni determinan t are all set equal to zero, one deals with the Landau equations of the Φ ( n ) H quan tities. Th us the singularities obtained for the Φ ( n ) H quan tities are also solutions of the Landau equations of the χ ( n ) . Ho w ev er this feature do es not mean that the singularities of the Φ ( n ) H quan tities will neessarly app ear as singularities of the χ ( n ) ODEs. Indeed some seletion rules ma y tak e plae and ma y rejet some of them. F or instane, one exp ets singularities link ed to the Q y i to o ur for the Landau singularities of the Φ ( n ) H . One nds that some seletion rules exlude them. Our eduated guess is that all the Landau singularities of the Φ ( n ) H will b e in the Landau singularities of the χ ( n ) , ho w ev er w e do not exlude the p ossibilit y that the χ ( n ) will ha v e more Landau singularities than the Φ ( n ) H . Another eduated guess is that the Landau singularities of the χ ( n ) will exhibit a similar em b edding that the one w e found for the Φ ( n ) H . This naturally raises the question already onsidered in [ 8 ℄, of a strong Russian doll struture for the linear dieren tial op erators of the χ ( n ) , namely that the linear dieren tial op erator of χ (3) (resp. χ (4) ) ould righ t-divide the linear dieren tial op erator of χ (5) (resp. χ (6) ), and so on. Singularities of n -fold inte gr als 23 This kno wledge of the singularities will help in the sear h for the orresp onding linear ODE. F or instane, w e ha v e 24 head p olynomial andidates for χ (5) and 19 andidates for χ (6) that an, from the outset, b e put in fron t of the higher order deriv ativ e of the unkno wn linear ODE. F rom the kno wledge w e ha v e gained from all these n -fold in tegrals of the Ising lass, one an guess the order of magnitude of the m ultipliit y of some singularities. F urthermore, as sho wn for the linear ODE for Φ (5) H and Φ (6) H (and also from previous ODEs), w e kno w that the ost (in terms of the n um b er of series o eien ts) will b e m u h less for a non minimal order linear ODE than for the minimal order one. Conerning the non Ni k elian singularities that the m ultiple in tegrals Φ ( n ) H ha v e giv en, w e fo ussed on 1 + 3 w + 4 w 2 = 0 whi h atually o urs for the linear ODE of χ (3) , or for χ (3) seen as a funtion of s . As far as a mathemati al interpr etation is onerned, w e ha v e sho wn that this quadrati p olynomial ondition orresp onds to a seleted situation for ellipti urv es namely the o urr en e of omplex multipli ation . The other non-Ni k elian (andidate) singularities, (12), (13 ) do not orr esp ond to omplex m ultipliation of ellipti urv es. Assuming that the non Ni k elian singularities obtained in the linear ODE for the in tegrals (8), will b e, at least, inluded in those for the χ ( n ) , v arious lines of though t are p ossible. One ma y imagine that the deomp osition of the suseptibilit y of the Ising mo del in terms of an innite sum of χ ( n ) is quite an artiial one with no deep mathematial meaning, i.e. χ ( n ) are quite arbitrary n -fold in tegrals. In this ase, no in terpretation within the theory of ellipti urv es has to b e lo ok ed for and the o urrene for 1 + 3 w + 4 w 2 = 0 of omplex m ultipliation for ellipti urv es w ould b e just a oinidene. Another option amoun ts to sa ying that one needs to in tro due (motivi) mathematial strutures [36 , 37 , 38 , 39 , 40 ℄ b eyond the the ory of el lipti urves (mo duli spaes, mark ed urv es, ...), and b ey ond the ellipti urv es of the Ising (or Baxter) mo del, to get a mathemati al interpr etation of these singularities . W e tend to fa v our the latter option. A kno wledgmen ts: W e ha v e deriv ed great b enet from disussions on v arious asp ets of this w ork with F. Ch yzak, S. Fis hler, P . Fla jolet, A. J. Guttmann, L. Merel, B. Ni k el, I. Jensen, B. Salvy and J-A. W eil. W e thank A. Bostan for a sear h of linear ODE mo d. prime with one of his Magma programs. W e a kno wledge CNRS/PICS nanial supp ort. One of us (NZ) w ould lik e to a kno wledge the kind hospitalit y at the LPTMC where part of this w ork has b een ompleted. One of us (JMM) thanks MASCOS (Melb ourne) where part of this w ork w as p erformed. Singularities of n -fold inte gr als 24 App endix A. Series expansions of Φ ( n ) H and of single in tegrals Φ ( n ) k W e giv e in this App endix, the series expansion that has b een used for Φ ( n ) H . Expanding the in tegrand of (8) in the v ariables x j , one obtains Φ ( n ) H = 1 n ! · n − 1 Y j =1 Z 2 π 0 dφ j 2 π · ∞ X p =0 (2 − δ p, 0 ) · n Y j =1 y j x p j . (A.1) W e mak e use of the y j x p j F ourier expansion [4 , 5, 8℄ y j x p j = w p · ∞ X k = −∞ w | k | · a ( p, | k | ) · Z k j , Z j = exp( i φ j ) (A.2) where a ( k , p ) is a non-terminating h yp ergeometri funtion that reads (with m = k + p ): a ( k , p ) = m k × (A.3) 4 F 3 1 + m 2 , 1 + m 2 , 2 + m 2 , 2 + m 2 ; 1 + k , 1 + p, 1 + m ; 16 w 2 . W e dene h ρ i b y h ρ i = n Y j =1 Z 2 π 0 dφ j 2 π · 2 π δ n X j =1 φ j · ρ (A.4) where the angular onstrain t is in tro dued through the delta funtion that has the F ourier expansion: 2 π δ n X j =1 φ j = ∞ X k = −∞ ( Z 1 Z 2 · · · Z n ) k (A.5) The in tegrals (A.1) b eome Φ ( n ) H = 1 n ! · ∞ X k = −∞ ∞ X p =0 (2 − δ p, 0 ) · h n Y j =1 y j x p j Z k j i (A.6) where the in tegration is o v er indep enden t angles. Using the F ourier expansion (A.2), one obtains the in tegration rule h y j x p j Z k j i = w p + | k | · a ( p, | k | ) (A.7) and nally: Φ ( n ) H = 1 n ! · ∞ X k =0 ∞ X p =0 (2 − δ k, 0 ) · (2 − δ p, 0 ) · w n ( k + p ) · a n ( k , p ) . (A.8) The deriv ation of the series expansions for the one dimensional in tegrals ( 37 ) pro eeds along similar lines. The in tegrand of the in tegrals (37) is expanded in x 1 1 − x n − k ( φ ) · x k ( φ n ) = ∞ X p =0 x p ( n − k ) ( φ ) x p k ( φ n ) (A.9) with φ n = − n − k k · φ + 2 π j k . (A.10) Singularities of n -fold inte gr als 25 Here, w e use the F ourier expansion x m = w m · ∞ X p =0 (2 − δ p, 0 ) · w p · b ( p, m ) · co s( p φ ) (A.11) where b ( k , p ) is a non-terminating h yp ergeometri funtion that reads (with m = k + p ): b ( k , p ) = m − 1 k × (A.12) 4 F 3 1 + m 2 , 1 + m 2 , 2 + m 2 , m 2 ; 1 + k , 1 + p, 1 + m ; 16 w 2 . The in tegration of the one-dimensional in tegrals (A.9) giv es Φ ( n ) k = h 1 1 − x n − k ( φ ) · x k ( φ n ) i = ∞ X p =0 ∞ X p 1 =0 ∞ X p 2 =0 (2 − δ p 1 , 0 ) · (2 − δ p 2 , 0 ) × w pn + p 1 + p 2 · b ( p 1 , p ( n − k )) · b ( p 2 , p k ) I ( p 1 , p 2 ) (A.13) with I ( p 1 , p 2 ) = 1 2 · (1 + δ p 1 , 0 ) · cos ( c ) , for p 2 · ( n − k ) = k · p 1 , and I ( p 1 , p 2 ) = 1 π b 2 b 2 − p 2 1 · sin( b π ) · cos( b π − c ) , for p 2 · ( n − k ) 6 = k · p 1 , where b = n − k k · p 2 , c = 2 π j k · p 2 . (A.14) App endix B. Linear dieren tial equations of some Φ ( n ) H App endix B.1. Line ar ODE for Φ (3) H The minimal order linear dieren tial equation satised b y Φ (3) H reads 5 X n =0 a n ( w ) · d n dw n F ( w ) = 0 , (B.1) where a 5 ( w ) = (1 − w ) (1 − 4 w ) 4 (1 + 4 w ) 2 (1 + 2 w ) × (1 + 3 w + 4 w 2 ) · w 3 · P 5 ( w ) , (B.2) a 4 ( w ) = (1 − 4 w ) 3 (1 + 4 w ) · w 2 · P 4 ( w ) , a 3 ( w ) = − 2 (1 − 4 w ) 2 · w P 3 ( w ) , a 2 ( w ) = 2 (1 − 4 w ) · P 2 ( w ) , a 1 ( w ) = − 8 P 1 ( w ) , a 0 ( w ) = − 96 P 0 ( w ) , with P 5 ( w ) = − 5 + 21 w + 428 w 2 + 5364 w 3 − 8241 6 w 4 − 2995 04 w 5 + 7149 44 w 6 + 3127 872 w 7 − 8220 672 w 8 − 2585 8048 w 9 − 7077 888 w 10 + 3142 4512 w 11 − 4246 7328 w 12 Singularities of n -fold inte gr als 26 − 3145 7280 w 13 − 4194 304 w 14 + 4194 304 w 15 , P 4 ( w ) = − 40 + 7 w + 523 2 w 2 + 3715 9 w 3 − 4477 78 w 4 − 4947 500 w 5 + 1949 3448 w 6 + 2584 64112 w 7 + 4992 05984 w 8 − 1612 751808 w 9 − 4667 817856 w 10 + 1382 745907 2 w 11 + 6707 841638 4 w 12 + 6239 204147 2 w 13 − 8153 536921 6 w 14 − 1168 354836 48 w 15 + 1246 620549 12 w 16 + 1460 163051 52 w 17 − 1972 581171 20 w 18 − 1316 675911 68 w 19 − 1167 694233 6 w 20 + 1503 238553 6 w 21 , P 3 ( w ) = 35 − 25 w − 868 3 w 2 − 1014 9 w 3 + 6192 46 w 4 + 52 73820 w 5 − 5247 2072 w 6 − 5881 47792 w 7 + 4910 73248 w 8 + 1872 181958 4 w 9 + 4762 277158 4 w 10 − 9745 963059 2 w 11 − 4414 185881 60 w 12 + 6510 035599 36 w 13 + 4694 018588 672 w 14 + 4729 946636 288 w 15 − 7193 770328 064 w 16 − 1181 451970 1504 w 17 + 7399 599505 408 w 18 + 1098 199649 4848 w 19 − 1643 952419 6352 w 20 − 1043 462304 5632 w 21 − 9169 755176 96 w 22 + 1125 281431 552 w 23 , P 2 ( w ) = − 10 + 101 w + 1108 8 w 2 − 4285 5 w 3 − 1117 278 w 4 − 1918 516 w 5 + 7222 1464 w 6 + 4600 80656 w 7 − 4999 186016 w 8 − 3347 442822 4 w 9 + 6744 020032 0 w 10 + 8085 605585 92 w 11 + 5351 666933 76 w 12 − 6771 457933 312 w 13 − 7468 556451 840 w 14 + 4614 351447 6544 w 15 + 9148 886312 5504 w 16 − 7510 773307 8016 w 17 − 2394 386637 78304 w 18 + 3190 472835 0720 w 19 + 2340 588061 98272 w 20 − 2374 461932 17536 w 21 − 1641 815673 40544 w 22 − 1897 516551 3728 w 23 + 1697 371075 3792 w 24 , P 1 ( w ) = − 5 − 11 4 2 w + 8106 w 2 + 2108 46 w 3 − 1070 376 w 4 − 7771 160 w 5 − 2202 9952 w 6 + 8338 94752 w 7 + 3334 510976 w 8 − 3973 644992 0 w 9 − 1561 018593 28 w 10 + 6633 067182 08 w 11 + 2995 615555 584 w 12 − 5033 154314 240 w 13 − 2625 078598 0416 w 14 + 2861 806675 5584 w 15 + 1580 477752 27904 w 16 − 4283 621703 6800 w 17 − 4103 176202 48576 w 18 − 9592 507465 7280 w 19 + 4622 453183 61088 w 20 − 3289 901999 06304 w 21 − 2494 431106 17088 w 22 − 3527 027143 4752 w 23 + 2446 413371 8016 w 24 , P 0 ( w ) = − 5 + 58 w + 323 4 w 2 − 1899 4 w 3 − 2293 30 w 4 + 1516 w 5 + 7017 504 w 6 + 7468 9472 w 7 − 6470 69792 w 8 − 4260 373952 w 9 + 1588 716364 8 w 10 + 9678 961868 8 w 11 − 1361 205084 16 w 12 − 9177 651445 76 w 13 + 8779 966054 40 w 14 + 5646 695006 208 w 15 − 2888 887697 408 w 16 − 1678 515581 7472 w 17 − 5241 017729 024 w 18 + 1795 242642 6368 w 19 − 1305 831119 2576 w 20 − 9329 742708 736 w 21 − 1275 605286 912 w 22 + 8246 337208 32 w 23 . (B.3) Singularities of n -fold inte gr als 27 App endix B.2. Line ar ODE for Φ (4) H The minimal order linear dieren tial equation satised b y Φ (4) H reads (with x = 16 w 2 ) 6 X n =0 a n ( x ) · d n dx n F ( x ) = 0 , (B.4) where a 6 ( x ) = 64 ( x − 4) (1 − x ) 4 x 4 · P 6 ( x ) , a 5 ( x ) = − 128 (1 − x ) 3 x 3 · P 5 ( x ) , a 4 ( x ) = 16 (1 − x ) 2 x 2 · P 4 ( x ) , a 3 ( x ) = − 64 (1 − x ) x · P 3 ( x ) , a 2 ( x ) = − 4 · P 2 ( x ) , a 1 ( x ) = − 8 · P 1 ( x ) , a 0 ( x ) = − 3 (1 − x ) · P 0 ( x ) , with: P 6 ( x ) = 128 + 2 2 33 x − 284 7 x 2 + 3143 x 3 − 3601 x 4 + 144 x 5 − 64 x 6 , P 5 ( x ) = 3 7 12 + 51523 x − 2163 77 x 2 + 2899 18 x 3 − 3128 96 x 4 + 2621 11 x 5 − 6316 7 x 6 + 5512 x 7 − 896 x 8 , P 4 ( x ) = − 1 21856 − 110230 4 x + 1 103828 9 x 2 − 2610 6487 x 3 + 3151 5802 x 4 − 3102 7694 x 5 + 2129 1429 x 6 − 5166 011 x 7 + 4101 60 x 8 − 6777 6 x 9 , P 3 ( x ) = 3 8 144 + 10604 x − 464428 1 x 2 + 2090 9702 x 3 − 3789 0772 x 4 + 4201 1874 x 5 − 3755 2559 x 6 + 2247 4036 x 7 − 5465 572 x 8 + 3925 36 x 9 − 6598 4 x 10 , P 2 ( x ) = 1 6 3840 − 416268 8 x − 181201 52 x 2 + 2771 10610 x 3 − 8800 48289 x 4 + 1357 147519 x 5 − 1395 938590 x 6 + 1141 353668 x 7 − 6213 23833 x 8 + 1508 42795 x 9 − 9676 720 x 10 + 1656 512 x 11 , P 1 ( x ) = − 3 66592 + 311375 2 x + 1 746570 0 x 2 − 1206 58444 x 3 + 2403 21805 x 4 − 2592 77988 x 5 + 2199 51814 x 6 − 1423 14304 x 7 + 4253 4921 x 8 − 2056 040 x 9 + 4352 00 x 10 , P 0 ( x ) = 5 6 1152 − 149640 0 x − 131715 75 x 2 + 3084 0556 x 3 − 2438 1198 x 4 + 2035 2948 x 5 − 1326 8091 x 6 + 3093 60 x 7 − 1200 00 x 8 . App endix B.3. Line ar ODE mo dulo a prime for Φ (5) H The linear dieren tial equation of minimal order sev en teen satised b y Φ (5) H is of the form 17 X n =0 a n ( w ) · d n dw n F ( w ) = 0 , (B.5) with a 17 ( w ) = (1 − 4 w ) 12 (1 + 4 w ) 9 (1 − w ) 2 ( w + 1) (1 + 2 w ) × 1 + 3 w + 4 w 2 2 1 − 3 w + w 2 1 + 2 w − 4 w 2 × (1 + 4 w + 8 w 2 ) 1 − 7 w + 5 w 2 − 4 w 3 × 1 − w − 3 w 2 + 4 w 3 1 + 8 w + 20 w 2 + 15 w 3 + 4 w 4 · w 12 · P 17 ( w ) , Singularities of n -fold inte gr als 28 a 16 ( w ) = w 11 (1 − 4 w ) 11 (1 + 4 w ) 8 (1 − w ) 1 + 3 w + 4 w 2 · P 16 ( w ) , a 15 ( w ) = w 10 (1 − 4 w ) 10 (1 + 4 w ) 7 · P 15 ( w ) , a 14 ( w ) = w 9 (1 − 4 w ) 9 (1 + 4 w ) 6 · P 14 ( w ) , · · · where the 430 ro ots of P 17 ( w ) are app ar ent singularities . The degrees of these p olynomials P n ( w ) are su h that the degrees of a i ( w ) are dereasing as: deg ( a i +1 ( w )) = deg ( a i ( w )) + 1 . In fat, with 2208 terms w e ha v e found the ODE of Φ ((5) H at order q = 28 using the follo wing ansatz for the linear ODE sear h ( D w denotes d/dw ) q X i =0 s ( i ) · p ( i ) · Dw i (B.6) with: s ( i ) = w α ( − 1+ i ) · (1 − 1 6 w 2 ) α ( − 1+ i ) · s α (1+ i − q ) 0 (B.7) where α ( n ) = M in (0 , n ) and s 0 = (1 + w ) · (1 − w ) · (1 + 2 w ) · (1 − 3 w + w 2 ) (1 + 2 w − 4 w 2 ) × (1 + 3 w + 4 w 2 ) · (1 + 4 w + 8 w 2 ) · (1 − 7 w + 5 w 2 − 4 w 3 ) × (1 − w − 3 w 2 + 4 w 3 ) · (1 + 8 w + 20 w 2 + 15 w 3 + 4 w 4 ) the p ( i ) b eing the unkno wn p olynomials. The minimal order ODE is dedued from the set of linear indep endan t ODEs found at order 28. App endix B.4. Line ar ODE mo dulo a prime for Φ (6) H The linear dieren tial equation of minimal order t w en t y-sev en satised b y Φ (6) H reads (with x = w 2 ) 27 X n =0 a n ( x ) · d n dx n F ( x ) = 0 , (B.8) with a 27 ( x ) = (1 − 16 x ) 16 (1 − 4 x ) 3 (1 − x ) (1 − 25 x ) (1 − 9 x ) x 21 × 1 − x + 16 x 2 1 − 10 x + 29 x 2 · P 27 ( x ) , a 26 ( x ) = (1 − 16 x ) 15 (1 − 4 x ) 2 x 20 · P 26 ( x ) , a 25 ( x ) = (1 − 16 x ) 14 (1 − 4 x ) x 19 · P 25 ( x ) , a 24 ( x ) = (1 − 16 x ) 13 x 18 · P 24 ( x ) , · · · (B.9) where the 307 ro ots of P 27 ( x ) are app ar ent singularities . The degrees of the P n ( w ) p olynomials are su h that the degrees of a i ( w ) are dereasing as: deg ( a i +1 ( w )) = deg ( a i ( w )) + 1 In fat, with 1838 terms w e ha v e found the linear ODE of Φ (6) H at order q = 42 using the follo wing ansatz for the linear ODE sear h ( D x denotes d/dx ) q X i =0 s ( i ) · p ( i ) · Dx i (B.10) Singularities of n -fold inte gr als 29 with: s ( i ) = x α ( − 1+ i ) · (1 − 1 6 x ) α ( − 1+ i ) · s α (1+ i − q ) 0 (B.11) where α ( n ) = M in (0 , n ) and s 0 = (1 − 25 x ) · (1 − 9 x ) · (1 − 4 x ) · (1 − x ) × (1 − x + 16 x 2 ) · (1 − 10 x + 2 9 x 2 ) (B.12) the p ( i ) b eing the unkno wn p olynomials. The minimal order ODE is dedued from the set of linear indep endan t ODEs found at order 42. App endix C. Singularities in the linear ODE for Φ (7) H and Φ (8) H F or Φ (7) H , w e generated large series, (1250 o eien ts and 20000 o eien ts mo dulo primes), unfortunately , insuien t to obtain the orresp onding linear ODE. Ho w ev er, b y steadily inreasing the order q of the ODE (and onsequen tly dereasing the degrees n of the p olynomials in fron t of the deriv ativ es), one ma y reognize, in oating p oin t form, the singularities of the ODE as the ro ots of the p olynomial in fron t of the higher deriv ativ e. A ro ot is onsidered as singularit y of the still unkno wn linear ODE, when as q inrease (and onsequen tly dereasing n ), it p ersists with more stabilized digits. Using 1250 terms in the series for Φ (7) H , the follo wing singularities are reognized (1 − 4 w ) 1 − 5 w + 6 w 2 − w 3 1 + 2 w − 8 w 2 − 8 w 3 (1 + 4 w ) · w 1 + 2 w − w 2 − w 3 1 − 3 w + w 2 1 + 2 w − 4 w 2 (1 + w ) 16 w 8 − 32 w 7 − 17 w 6 + 62 w 5 − 5 w 4 − 35 w 3 + 10 w 2 + 3 w − 1 64 w 7 + 96 w 6 + 16 w 5 − 60 w 4 − 21 w 3 + 15 w 2 + 8 w + 1 128 w 5 − 32 w 4 + 48 w 3 + 16 w 2 + 4 w − 1 4 w 5 + 51 w 3 − 21 w 4 − 1 + 10 w − 35 w 2 4 w 3 + 7 w − 5 w 2 − 1 4 w 4 + 1 + 7 w + 26 w 2 + 7 w 3 4 w 4 + 1 + 8 w + 20 w 2 + 15 w 3 1 + 12 w + 54 w 2 + 112 w 3 + 105 w 4 + 35 w 5 + 4 w 6 = 0 . W e will see in the next setion App endix E.3 that w e missed the p olynomials: 1 + 3 w + 4 w 2 1 + 4 w + 8 w 2 (1 − w ) (C.1) (1 + 2 w ) 1 − w − 3 w 2 + 4 w 3 Note that w e ha v e not seen with the preision of these alulations the o urene of the singularities of the Φ (3) H . With similar alulations using 1200 terms for Φ (8) H , the follo wing singularities are reognized: (1 − 2 w ) (1 + 2 w ) 1 − 2 w 2 (1 − 4 w ) 1 − 4 w + 2 w 2 (1 + 4 w ) 1 + 4 w + 2 w 2 8 w 2 − 1 (3 w − 1) (1 − w ) (1 + w ) ( 3 w + 1) w 1138 w 10 − 1685 w 8 + 960 w 6 − 242 w 4 + 26 w 2 − 1 32 w 4 − 10 w 2 + 1 1312 w 6 − 56 w 4 + 30 w 2 − 1 10 w 2 − 6 w + 1 4 w 3 − 8 w 2 + 6 w − 1 Singularities of n -fold inte gr als 30 (5 w − 1 ) 1 + 2 w 2 (5 w + 1) 10 w 2 + 6 w + 1 4 w 3 + 8 w 2 + 6 w + 1 = 0 . W e will see in the next setion App endix E.3 that w e missed the p olynomials: 1 − 3 w + 4 w 2 1 + 3 w + 4 w 2 1 − 10 w 2 + 29 w 4 Note that the stabilized digits in these singularities an b e as lo w as t w o digits. App endix D. Landau onditions and pin h singularities for Φ ( n ) D and in tegrals of Q y j . Similarly to the in tegral represen tation (16 ) of Φ ( n ) H , one has: Φ ( n ) D = Z 2 π 0 dφ 2 π Z 2 π 0 · dψ 2 π p (1 − 2 w cos φ ) 2 − 4 w 2 D ( φ, ψ ) (D.1) × p (1 − 2 w cos(( n − 1 ) φ )) 2 − 4 w 2 D (( n − 1) φ, ( n − 1 ) ψ ) , and n Y i =1 y i = Z 2 π 0 n Y i =1 dφ i 2 π · dψ i 2 π · 1 D ( φ i , ψ i ) · δ n X i =1 φ i . (D.2) F or Φ ( n ) D the singularities of the asso iated ODEs are giv en as solutions of: D ( φ, ψ ) = 0 , D (( n − 1) φ, ( n − 1) ψ ) = 0 , α 1 · sin( φ ) + α 2 · sin(( n − 1) φ ) = 0 , with α 1 , α 2 6 = 0 , α 1 · sin( ψ ) + α 2 · sin(( n − 1) ψ ) = 0 (D.3) whi h are nothing less than the Landau onditions restrited to pin h singularities of the singularit y manifolds D ( φ i , ψ i ) = 0 . F or ♯ Q n i =1 y i the singularities of the asso iated ODEs an b e written as the solutions of: D ( φ i , ψ i ) = 0 , α i · sin( φ i ) − α n · sin( φ n ) = 0 , i = 1 , · · · , n − 1 , with α i 6 = 0 α i · sin( ψ i ) = 0 , i = 1 , · · · , N (D.4) whi h are also Landau onditions restrited to pin h singularities of the singularit y manifolds D ( φ i , ψ i ) = 0 . App endix E. The singularities from Landau onditions In this App endix, w e giv e further details orresp onding to (28), (29 ) obtained from the Landau onditions: 1 − 2 w · (cos( φ j ) + cos( ψ j )) = 0 , j = 1 , · · · , n, (E.1) α j · sin( φ j ) − α n · sin( φ n ) = 0 , j = 1 , · · · , n − 1 , (E.2) α j · sin( ψ j ) − α n · sin( ψ n ) = 0 , j = 1 , · · · , n − 1 . (E.3) ♯ Q y i or Q y 2 i in tegrand are similar as far as lo ation of singularities of the orresp onding ODE is onerned. Singularities of n -fold inte gr als 31 and: n X j =1 φ j = 0 , n X j =1 ψ j = 0 mo d . 2 π (E.4) W e solv e these equations for the v alues (zero or not) of sin( φ n ) and sin( ψ n ) . F or sin( φ n ) = sin( ψ n ) = 0 , the ase is simple and giv es w = ± 1 / 4 . App endix E.1. The ase sin( φ n ) 6 = 0 , sin( ψ n ) = 0 In this ase, there are k angles ψ j = π and the remaining ones are ψ j = 0 . By (17 ), the in teger k should b e ev en, k = 2 p . F rom ( E.1 ), w e obtain and dene ‡ cos( φ + ) = 1 2 w + 1 , cos( φ − ) = 1 2 w − 1 . (E.5) One obtains 2 p angles φ j = ± φ + and n − 2 p angles φ j = ± φ − . The angles φ j are then partitioned in sets of p 1 angles + φ + , (2 p − p 1 ) angles − φ + , ( n − 2 p − p 2 ) angles + φ − and p 2 angles − φ − . By (E.4 ), one gets (2 p − 2 p 1 ) · φ + = ( n − 2 p − 2 p 2 ) · φ − . Note that some manipulations on the indies lead to cos ( | 2 p | · φ + ) = cos ( | n − 2 p − 2 k | · φ − ) and th us | 2 p | · φ + = ±| n − 2 p − 2 k | · φ − , allo wing us to write T 2 p (1 / 2 w + 1) = T n − 2 p − 2 k (1 / 2 w − 1) , (E.6) 0 ≤ p ≤ [ n/ 2] , 0 ≤ k ≤ [ n/ 2] − p, where T n ( x ) is the Cheb yshev p olynomial of the rst kind. One obtains the same results for the ase sin( φ n ) = 0 and sin( ψ n ) 6 = 0 . App endix E.2. The ase sin( φ n ) 6 = 0 , sin( ψ n ) 6 = 0 In this ase, b y (E.2 ), (E.3 ), w e ha v e sin( φ j ) 6 = 0 and sin( ψ j ) 6 = 0 . The equations (E.1 ), (E.3) b eome: cos( ψ j ) = 1 − 2 w · cos( φ j ) , j = 1 , · · · , n, (E.7) sin( ψ j ) = sin( φ j ) · sin( ψ n ) sin( φ n ) , j = 1 , · · · , n. (E.8) Squaring b oth sides of b oth equations and summing, one obtains (cos( φ j ) − cos( φ n )) · (cos( φ j ) − cos( φ 0 )) = 0 , (E.9) where w e ha v e dened cos( φ 0 ) = 4 w − cos( φ n ) 1 − 4 w cos( φ n ) . (E.10) The angles φ j are then partitioned in to four sets ± φ 0 and ± φ n . Note that a similar ondition (E.9 ) o urs for the angles ψ j whi h are partitioned lik ewise. W riting (E.7 ), (E.8) for j = 0 and j = n and with the onditions ( 17), the equations b eome in terms of Cheb yshev p olynomials †† : T n 1 ( z ) − T n 2 4 w − z 1 − 4 w z = 0 , (E.11) ‡ Note that φ + and φ − (whi h orresp ond to ψ j = π and ψ j = 0 resp etiv ely) are not on the same fo oting: indeed, the n um b er of φ + angles m ust b e ev en, while the n um b er of φ − angles dep ends on the parit y of n . †† Note that in equation (E.11 ) one m ust realise that one tak es the n umerator of these rational expressions. Singularities of n -fold inte gr als 32 T n 1 1 2 w − z − T n 2 1 2 w − 4 w − z 1 − 4 w z = 0 , U n 2 − 1 ( z ) · U n 1 − 1 1 2 w − 4 w − z 1 − 4 w z − U n 2 − 1 1 2 w − z · U n 1 − 1 4 w − z 1 − 4 w z = 0 with n 1 = p, n 2 = n − p − 2 k , (E.12) 0 ≤ p ≤ n, 0 ≤ k ≤ [( n − p ) / 2] (E.13) A t this step, some omputational remarks are in order. In the ourse of deriving (E.11), some manipulations su h as dividing b y a term ha v e b een done. Rigorously , the solutions that ome from ( E.11) ha v e to b e he k ed against this p oin t. W e ha v e found, that as they are written, the form ulas are safe from this p ersp etiv e, exept of the follo wing. F or n = p/ 2 (xing k = 0 for on v eniene), th us for n ev en, the form ulas (E.11 ) giv e a ommon urv e whi h reads: w = 1 2 z 1 + z 2 . (E.14) This relation omes from the ondition cos( φ 0 ) = cos( φ n ) in (E.10) whi h mak es (E.9) a p erfet square. W e ha v e he k ed that onsidering this ondition at the outset, i.e. (E.7 ), (E.8) yields no solution. App endix E.3. L andau singularities W e an write the singularities obtained from (E.6 ) as: n = 3 , (1 − 4 w ) (1 − w ) 1 + 3 w + 4 w 2 = 0 , n = 4 , 1 − 16 w 2 1 − 4 w 2 = 0 , n = 5 , (1 − 4 w ) (1 − w ) 1 + 3 w + 4 w 2 1 − 3 w + w 2 × 1 − 7 w + 5 w 2 − 4 w 3 1 + 8 w + 20 w 2 + 15 w 3 + 4 w 4 = 0 , n = 6 , 1 − 16 w 2 1 − 4 w 2 1 − w 2 1 − 25 w 2 1 − 9 w 2 × 1 + 3 w + 4 w 2 1 − 3 w + 4 w 2 = 0 . The solutions of (E.11) inlude some of the solutions of (E.6). W e giv e in the follo wing only those not o urring in ( E.6 ): n = 3 , w · (1 + 4 w ) (1 + 2 w ) = 0 , n = 4 , w = 0 , n = 5 , w · (1 + 4 w ) (1 + w ) (1 + 2 w ) 1 + 2 w − 4 w 2 × 1 + 4 w + 8 w 2 1 − w − 3 w 2 + 4 w 3 = 0 , n = 6 , w · 1 − 10 w 2 + 29 w 4 = 0 . All these singularities an b e iden tied with the singularities o urring in the linear ODE for Φ ( n ) H , ( n = 3 , · · · , 6 ). F or n = 7 and n = 8 , the solutions of (E.6) and (E.11) an b e iden tied with the singularities giv en in App endix C and obtained in oating p oin t form. They also giv e: n = 7 , 1 + 3 w + 4 w 2 1 + 4 w + 8 w 2 (1 − w ) (1 + 2 w ) Singularities of n -fold inte gr als 33 × 1 − w − 3 w 2 + 4 w 3 = 0 , n = 8 , 1 − 3 w + 4 w 2 1 + 3 w + 4 w 2 1 − 10 w 2 + 29 w 4 = 0 , whi h ha v e not b een found in the series with the urren tly a v ailable n um b er of terms. App endix F. Heegner n um b ers and other seleted v alues of the mo dular j -funtion The nine Heegner n um b ers [ 51 ℄ and their asso iated mo dular j -funtion j ( τ ) , yield the follo wing onditions in the v ariable w : j (1 + i ) = (12 ) 3 , 1 − 8 w 2 1 − 16 w 2 − 8 w 4 = 0 , j (1 + i √ 2) = (20 ) 3 , 64 w 4 + 16 w 2 − 1 × 64 w 8 + 1792 w 6 − 368 w 4 + 32 w 2 − 1 = 0 , j 1 + i √ 3 2 = (0) 3 , 1 − 16 w 2 + 16 w 4 = 0 , j 1 + i √ 7 2 = ( − 15) 3 , 1 − 3 1 w 2 + 256 w 4 1 − 1 6 w 2 + w 4 × (1 + 3 w + 4 w 2 ) (1 − 3 w + 4 w 2 ) = 0 , j 1 + i √ 11 2 = ( − 32) 3 , P 3 = 1 − 48 w 2 + 816 w 4 − 5632 w 6 + 4582 4 w 8 − 5365 76 w 10 + 4096 w 12 = 0 , and j 1 + i √ d 2 = ( − m ) 3 , P d = 0 with: P d = P 3 + N · 1 − 16 w 2 · w 8 , with the follo wing v alues for the triplet ( d, m, N ) : (19 , 96 , 85196 8) , (43 , 960 , 88470323 2) , (67 , 5280 , 14719 791923 2 ) , (163 , 640320 , 2625 3 741264 0735232) Bey ond Heegner n um b ers there are man y other seleted quadrati v alues [ 52 , 53 ℄ of j , for instane: j = − 409 6 · 15 + 7 √ 5 3 = j 1 + i √ 35 2 (F.1) Whi h is kno wn [51 ℄ to b e one of the eigh teen n um b ers ha ving lass n um b er h ( − d ) = 2 , and whi h orresp onds to the quadrati relation − 1342 177280 00 + 117964800 j + j 2 = 0 . Realling the expression of the mo dular j -funtion in term of the v ariable w j = 1 − 16 w 2 + 16 w 4 3 (1 − 16 w 2 ) w 8 , (F.2) this quadrati relation in j b eomes a quite in v olv ed p olynomial expression that w e ha v e not seen emerging as singularities of (the linear ODE's of ) our n -fold in tegrals. Singularities of n -fold inte gr als 34 App endix G. Landen transformations and the mo dular j -funtion In this App endix the mo dular j -funtion (53 ) will b e seen, alternativ ely , as a funtion of the mo dulus k , and th us denoted j [ k ] , or as a funtion of the half p erio d ratio τ , and th us denoted j ( τ ) . The mo dular funtion alled the j -funtion when seen as a funtion of the mo dulus k reads: j [ k ] = 25 6 · 1 − k 2 + k 4 3 k 4 · (1 − k 2 ) 2 . (G.1) Inreasing the mo dulus b y (48), the mo dular funtion j ( k ) b eomes: j [ k 1 ] = j 1 [ k ] = 16 · 1 + 14 k 2 + k 4 3 k 2 · (1 − k 2 ) 4 . (G.2) Iterating this pro edure one more one obtains: j 1 [ k 1 ] = j 2 [ k ] = 4 · k 4 + 60 k 3 + 134 k 2 + 60 k + 1 3 k · (1 + k ) 2 (1 − k ) 8 . (G.3) The derease of the mo dulus b y (47 ) giv es: j [ k − 1 ] = j − 1 [ k ] = 16 · k 4 − 16 k 2 + 16 3 k 8 (1 − k 2 ) . (G.4) The next iterations (the ub e of (48 ) and the square of (47 )) giv es algebrai expressions for j [ k ] . It is easy to get a r epr esentation of the L anden tr ansformation on the mo dular j -funtions b y elimination of the mo dulus k b et w een ( 53) and (G.2). One obtains the w ell-kno wn fundamen tal mo dular urve [49 , 50 ℄: Γ 1 ( j, j 1 ) = j 2 · j 2 1 − ( j + j 1 ) · ( j 2 + 14 87 j j 1 + j 2 1 ) + 3 · 1 5 3 · (16 j 2 − 40 27 j j 1 + 16 j 2 1 ) (G.5) − 12 · 30 6 · ( j + j 1 ) + 8 · 3 0 9 = 0 . This algebrai urv e is symmetri in j and j 1 . W e will obtain the same mo dular urv e (G.5) b y elimination of the mo dulus k b et w een (G.2 ) and (G.3), or b et w een (G.1) and (G.4). The t w o mo dular funtions j and j 1 are in v arian t b y the S L (2 , Z ) mo dular group, and, in partiular, transformation τ → 1 /τ . As a onsequene, the transformation τ → 2 · τ , and its in v erse τ → τ / 2 , have to b e on the same fo oting in the mo dular urv e represen tation (G.5) for the Landen and Gauss transformations. Similarly , one an easily nd the (gen us zero) mo dular urv e Γ 2 obtained b y the elimination of the mo dulus k b et w een (G.1) and ( G.3), (or b et w een (G.4) and (G.4)), whi h orresp onds to the transformation τ → 4 · τ and, at the same time , to its in v erse τ → τ / 4 . This last algebrai urv e is, of ourse, also a mo dular urve . App endix G.1. Fixe d p oints of these mo dular r epr esentations in terms of j -funtion T ransformations lik e j → j 1 , or j → j 2 , orresp onding to the previous mo dular urv es, are not (one-to-one) mappings, they are alled orresp ondene b y V eselo v [ 23 , 24 ℄. In order to lo ok at the xed p oin ts of the Landen, Gauss transformations (or their iterates) seen as transformations on omplex variables , within the framew ork of (mo dular) represen tations on the mo dular j -funtions, w e write, resp etiv ely , Γ 1 ( j, j 1 = j ) = 0 and Γ 2 ( j, j 2 = j ) = 0 Singularities of n -fold inte gr als 35 The xed p oin ts Γ 1 ( j, j 1 = j ) = 0 of the (mo dular) orresp ondene (G.5), are j = j 1 = (12) 3 or (20) 3 or ( − 15) 3 . The xed p oin ts Γ 2 ( j, j 2 = j ) = 0 of mo dular urv e orresp onding to the square of the Landen transformation, are j = j 2 = (66) 3 , or 2 · (30) 3 , or ( − 15) 3 or the solutions ‡ of j 2 + 1910 25 · j − 12 1 287375 = 0 , namely: j = − 3 3 · 1 + √ 5 2 2 · (5 + 4 · √ 5) 3 = j τ = 1 + i √ 15 2 (G.6) and its Galois onjugate ( hange √ 5 in to − √ 5 ). App endix G.2. A lternative appr o ah to xe d p oints of the L anden tr ansformation and its iter ates In order to get the xed p oin ts of the Landen transformation, let us imp ose that (G.1) and (G.2) are atually equal, th us j [ k ] = j [ k 1 ] . This yields the ondition (already seen to orresp ond to the χ (3) -singularities 1 + 3 w + 4 w 2 = 0 ): 4 k 2 + 3 k + 1 k 2 + 3 k + 4 = 0 (G.7) together with: 4 k 2 − 3 k + 1 k 2 − 3 k + 4 k 2 + 2 k − 1 k 2 − 2 k − 1 1 + k 2 = 0 . (G.8) The rst t w o p olynomial onditions in (G.8), (4 k 2 − 3 k + 1) ( k 2 − 3 k + 4) = 0 , orresp ond to the Heegner n um b er asso iated with the in teger v alue j = ( − 15) 3 . The next t w o p olynomial onditions in (G.8), k 2 ± 2 k − 1 = 0 , orresp ond to the Heegner n um b er asso iated with the in teger v alue j = (20 ) 3 . The last p olynomial ondition in (G.8), 1 + k 2 = 0 , orresp onds to the Heegner n um b er asso iated with the in teger v alue j = (12) 3 . Similarly , in order to get the xed p oin ts of the square of the Landen transformation, let us require that (G.1) and (G.3) are atually equal: j [ k ] = j [ k 2 ] . This yields the onditions (G.7) (xed p oin ts of the Landen transformation) together with: k 2 − 6 k + 1 1 + 14 k 2 + k 4 = 0 (G.9) k 4 − 6 k 3 + 17 k 2 + 36 k + 16 (G.10) × 16 k 4 + 36 k 3 + 17 k 2 − 6 k + 1 = 0 . In (G.9 ) the ondition 1 + 14 k 2 + k 4 = 0 (or 1 − 16 w 2 + 256 w 4 = 0 ) orresp onds to j = 2 (30) 3 whi h is not a Heegner n um b er but atually orresp onds to omplex m ultipliation. The ondition k 2 − 6 k + 1 = 0 in (G.9) (or 1 − 32 w 2 = 0 ) orresp onds to j = (66) 3 whi h is not a Heegner n um b er either but atually orresp onds to omplex m ultipliation. Note that b oth p olynomials under the Landen transformation (48) giv e resp etiv ely j = (0) 3 and j = (12) 3 , i.e. Heegner n um b ers. The last t w o (self-dual) onditions in (G.10 ), read in w 1 − 9 w + 1 7 w 2 + 24 w 3 + 6 w 4 = 0 , (G.11) 1 + 9 w + 1 7 w 2 − 24 w 3 + 6 w 4 = 0 and yield as seleted v alue [52 , 53 ℄ of j , the quadrati ro ots − 1212 87375 + 19102 5 j + j 2 = 0 , already giv en in ( G.6). ‡ This orresp onds to a v alue of j of lass n um b er h ( − d ) = 2 , see (58) in [51℄. Singularities of n -fold inte gr als 36 One more step an b e p erformed writing the ondition j [ k − 1 ] = j [ k 2 ] . One gets the onditions: k 2 + 3 k + 4 2 4 k 2 − 3 k + 1 k 2 + 2 k − 1 k 2 + 1 = 0 previously obtained and orresp onding to j = ( − 15) 3 , 20 3 , 12 3 , together with: k 6 − 27 k 5 + 363 k 4 + 423 k 3 − 168 k 2 − 144 k + 64 = 0 , k 6 + 17 k 5 + 143 k 4 + 203 k 3 + 52 k 2 + 32 k + 64 = 0 (G.12) orresp onding, resp etiv ely , to the t w o ubi relations on j : 15660 283509 40383 − 586826 38134 j + 39 491307 j 2 + j 3 = 0 , 12771 880859 375 − 515129 6875 j + 3 491750 j 2 + j 3 = 0 . (G.13) These onditions (G.13 ) yield quite in v olv ed p olynomial expressions in the v ariable w that w e ha v e not seen emerging as singularities of (the linear ODE's of ) our n -fold in tegrals (or the Y ( n ) or Φ ( n ) either). App endix H. Linear dieren tial op erators for the Sorokin in tegrals Reall the o urrene of zeta funtions ev aluated at in teger v alues in man y n -fold in tegrals orresp onding to partile ph ysis, eld theory , ... F or instane, the follo wing in tegral [33 , 36 ℄ is asso iated with ζ (3) : I n ( z ) = Z 1 0 du dv dw · (1 − u ) n u n · (1 − v ) n v n · (1 − w ) n w n (1 − u v ) n +1 · ( z − u v w ) n +1 (H.1) F rom the series expansion of this holonomi n -fold in tegral, w e ha v e obtained the orresp onding order four F u hsian linear dieren tial equation. On these linear dieren tial op erators the logarithmi nature of these in tegrals b eomes lear. The fully in tegrated series expansion of the triple in tegral (H.1) is giv en b y (where x denotes 1 /z ): I n ( x ) = ∞ X i =0 x n + i +1 · Γ 2 ( n + 1) · Γ 4 ( n + i + 1) Γ( i + 1) · Γ 3 (2 + 2 n + i ) × 3 F 2 ( n + 1 , n + i + 1 , n + i + 1; 2 n + i + 2 , 2 n + i + 2; 1) . The triple in tegral I n ( x ) is solution of the order four F u hsian linear dieren tial op erator ( D x denotes d/dx ) L n = D x 4 + 2 (3 x − 1 ) ( x − 1 ) x · D x 3 + 7 x 2 + ( n 2 + n − 5) x − 2 n ( n + 1 ) ( x − 1) 2 x 2 · D x 2 + x 2 + 2 n ( n + 1) ( x − 1) 2 x 3 · D x + n ( n + 1) · ( n 2 + n + 1) x + ( n − 1) ( n + 2) ( x − 1) 2 x 4 whi h has the follo wing fatorization L n = D x + d ln( A 1 ) dx · D x + d ln( A 2 ) dx (H.2) × D x + d ln( A 3 ) dx · D x + d ln( A 4 ) dx Singularities of n -fold inte gr als 37 where: A 1 = − ( n − 1) · ln( x ) + 2 · ln( x − 1) + ln( P n ) , A 2 = ( n + 1) · ln( x ) − ( n − 1) · ln( x − 1) − ln( P n ) + ln( Q n ) , A 3 = − n · ln( x ) + ( n + 1) · ln( x − 1) + ln( P n ) − ln( Q n ) , A 4 = n · ln( x ) − ln( P n ) , and where P n and Q n are p olynomials in x of degree n . They are the p olynomial solutions b eha ving as · · · + x n for a system of oupled dieren tial equations ( P ( m ) n (resp. 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