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DEGREES OF CURVES IN ABELIAN VARIETIES

arXiv:alg-geom/9210005v1 13 Oct 1992DEGREES OF CURVES IN ABELIAN VARIETIESOlivier Debarre (*)R´esum´e – Le degr´e d’une courbe C contenue dans une vari´et´e ab´elienne polaris´ee (X, λ)est l’entier d = C · λ .Lorsque C engendre X , on obtient une minoration de d enfonction de n et du degr´e de la polarisation λ . Le plus petit degr´e possible est d = n etn’est atteint que pour une courbe lisse dans sa jacobienne avec sa polarisation principalecanonique (Ran, Collino).

On ´etudie les cas d = n + 1 et d = n + 2 .Lorsque X estsimple, on montre de plus, en utilisant des r´esultats de Smyth sur la trace des entiersalg´ebriques totalement positifs, que si d ≤1, 7719 n , alors C est lisse et X est isomorphe`a sa jacobienne. Nous obtenons aussi une borne sup´erieure pour le genre g´eom´etrique deC en fonction de son degr´e.Abstract – The degree of a curve C in a polarized abelian variety (X, λ) is the integerd = C · λ .

When C generates X , we find a lower bound on d which depends on n andthe degree of the polarization λ . The smallest possible degree is d = n and is obtainedonly for a smooth curve in its Jacobian with its principal polarization (Ran, Collino).

Thecases d = n + 1 and d = n + 2 are studied. Moreover, when X is simple, it is shown, usingresults of Smyth on the trace of totally positive algebraic integers, that if d ≤1.7719 n ,then C is smooth and X is isomorphic to its Jacobian.

We also get an upper bound onthe geometric genus of C in terms of its degree.1. IntroductionAlthough curves in projective spaces have attracted a lot of attention for a longtime, very little is known in comparison about curves in abelian varieties.

We try in thisarticle to partially fill this gap.Let (X, λ) be a principally polarized abelian variety of dimension n defined overan algebraically closed field k . The degree of a curve C contained in X is d = C · λ .The first question we are interested in is to find what numbers can be degreesof curves C in X .

When C generates X , we prove that d ≥n(λn/n! )1/n ≥n .

It isknown ([C], [R]) that d = n if and only if C is smooth and X is isomorphic to itsJacobian JC with its canonical principal polarization. What about the next cases?

Weget partial characterizations for d = n + 1 and d = n + 2 , and we show (example 6.11)that all degrees ≥n + 2 may actually occur (at least when char(k) = 0 ). However, itseems necessary to assume X simple to go further.We prove, using results of Smyth(*) Partially supported by the European Science Project “Geometry of Algebraic Varieties”,Contract no.

SCI-0398-C (A) and by N.S.F. Grant DMS 92-03919.

([S]), that if C is an irreducible curve of degree < 2n if n ≤7 , and ≤1.7719 n if n > 7 ,on a simple principally polarized abelian variety X of dimension n , then C is smooth,has degree 2n −m for some divisor m of n , the abelian variety X is isomorphic to JC(with a non-canonical principal polarization) and C is canonically embedded in X . Weconjecture this result to hold for any n under the assumption that C has degree < 2n .This would be a consequence of our conjecture 6.2, which holds for n ≤7 : the trace ofa totally positive algebraic integer σ of degree n is at least 2n −1 and equality can holdonly if σ has norm 1 .

Smooth curves of genus n and degree 2n −1 in their Jacobianshave been constructed by Mestre for any n in [Me].The second question is the Castelnuovo problem: bound the geometric genus pg(C)of a curve C in a polarized abelian variety X of dimension n in terms of its degreed . We prove, using the original Castelnuovo bound for curves in projective spaces, theinequality pg(C) < (2d−1)22(n−1) , which is far from being sharp (better bounds are obtained forsmall degrees).

This in turn yields a lower bound in O(n3/2) on the degree of a curve ina generic principally polarized abelian variety of dimension n .Part of this work was done at the M.S.R.I. in Berkeley, and the author thanks thisinstitution for its hospitality and support.2.

Endomorphisms and polarizations of abelian varietiesLet X be an abelian variety of dimension n defined over an algebraically closed fieldk and let End(X) be its ring of endomorphisms. The degree deg(u) of an endomorphismu is defined to be 0 if u is not surjective, and the degree of u as a map otherwise.

Forany prime l different from the characteristic of k , the Tate module Tl(X) is a free Zl –module of rank 2n ([Mu], p.171) and the l –adic representation ρl : End(X) →End(Tl(X))is injective. For any endomorphism u of X , the characteristic polynomial of ρl(u) hascoefficients in Z and is independent of l .

It is called the characteristic polynomial of uand is denoted by Pu . It satisfies Pu(t) = deg(t IdX −u) for any integer t ([Mu], theorem4, p.180).

The opposite Tr(u) of the coefficient of t2n−1 is called the trace of u .The N´eron-Severi group of X is the group of algebraic equivalence classes ofinvertible sheaves on X . Any element µ of NS(X) defines a morphism φµ : X →Pic0(X)([Mu], p.60) whose scheme-theoretic kernel is denoted by K(µ) .The Riemann-Rochtheorem gives χ(X, µ) = µn/n!

, a number which will be called the degree of µ .Onehas deg φµ = (deg µ)2 ([Mu], p.150). A polarization λ on X is the algebraic equivalenceclass of an ample invertible sheaf on X ; it is said to be separable if its degree is prime tochar(k) .

In that case, φλ is a separable isogeny and its kernel is isomorphic to a group(Z/δ1Z)2 × · · · × (Z/δnZ)2 , where δ1| · · ·|δn and δ1 · · · δn = deg(λ) . We will say that λ2

is of type (δ1| · · ·|δn) .We will need the following result:Theorem 2.1. (Kempf, Mumford, Ramanujan)– Let X be an abelian variety of dimensionn , and let λ and µ be two elements of NS(X) .

Assume that λ is a polarization. Then:(i) The roots of the polynomial P(t) = (tλ −µ)n are all real.

(ii) If µ is a polarization, the roots of P are all positive. (iii) If P has no negative roots and r positive roots, there exist a polarized abelianvariety (X′, µ′) of dimension r and a surjective morphism f : X →X′ withconnected kernel such that µ = f ∗µ′ .Proof.

The first point is part of [MK], theorem 2, p.98. The second point follows fromthe same theorem and the fact that if M is an ample line bundle on X with class µ ,one has Hi(X, M) = 0 for i > 0 ([Mu], § 16).

For the last point, the same theorem from[MK] yields that the neutral component K of the group K(µ) has dimension n −r . Therestriction of M to K is algebraically equivalent to 0 (loc.cit., lemma 1, p.95) hence,since the restriction Pic0(X) →Pic0(K) is surjective, there exists a line bundle N onX algebraically equivalent to 0 such that the restriction of M ⊗N to K is trivial.

Itfollows from theorem 1, p.95 of loc.cit. that M ⊗N is the pull-back of a line bundle onX′ = X/K .

(2.2)Suppose now that θ is a principal polarization on X , i.e. a polarization ofdegree 1 .

It defines a morphism of Z –modules βθ : NS(X) →End(X) by the formulaβθ(µ) = φ−1θ◦φµ .Its image consists of all endomorphisms invariant under the Rosatiinvolution, which sends an endomorphism u to φ−1θ◦Pic0(u) ◦φθ ([Mu], (3) p.190).Moreover, one has, for any integer t : (tθ −µ)nn! 2= deg(tφθ −φµ) = degt IdX −βθ(µ)= Pβθ(µ)(t) .

(2.3)Let(X, λ) be a polarized abelian variety.For 0 < r ≤n , we setλrmin =λr/(r! δ1 · · ·δr) .

If k = C , the class of λrmin is minimal (i.e. non-divisible) in H2r(X, Z) .If k is any algebraically closed field, and if l is a prime number different from the char-acteristic of k , the group H1´et(X, Zl) is a free Zl –module of rank n ([Mi], theorem 15.1)and the algebras H∗´et(X, Zl) with its cup-product structure and ∧∗H1´et(X, Zl) with itswedge-product structure, are isomorphic ([Mi], remark 15.4).

In particular, the class [λ]lin H2´et(X, Zl) of the polarization λ can be viewed as an alternating form on a free Zl –module, and as such has elementary divisors. If λ is separable, (X, λ) lifts in characteristic0 to a polarized abelian variety of the same type (δ1| · · ·|δn) .

The elementary divisors3

of [λ]l are then the maximal powers of l that divide δ1, . .

. , δn .

Since intersection corre-sponds to cup-product in ´etale cohomology, the class of λrmin is in H2r´et (X, Zl) and is notdivisible by l .Throughout this article, all schemes we consider will be defined over an algebraicallyclosed field k . We will denote numerical equivalence by ∼.

If C is a smooth curve, JCwill be its Jacobian and θC its canonical principal polarization.3. Curves and endomorphismsWe summarize here some results from [Ma] and [Mo].Let C be a curve on apolarized abelian variety (X, λ) and let D be an effective divisor that represents λ .Morikawa proves that the following diagram, where d is the degree of C and S is the summorphism, defines an endomorphism α(C, λ) of X which is independent on the choice ofD :α(C, λ) :X−−→C(d)S−−−−−→Xtranslation−−−−−−→Xx7−→(D + x) ∩C(3.1)Let N be the normalization of C .

The morphism ι : N →X factorizes through amorphism p : JN →X . Set q = ι∗◦φλ : X →JN ; Matsusaka proves that α(C, λ) = p ◦q([Ma], lemma 3).

(3.2)He also proves (loc.cit., theorem 2) that α(C, λ) = α(C′, λ) if and only if C ∼C′ .Since α(λn−1, λ) = (λn/n)IdX , it follows that:α(C, λ) = m IdX ⇐⇒C ∼m(n −1)! deg λλn−1 .If moreover λ is separable of type (δ1| · · ·|δn) and if l is a prime distinct fromchar(k) , the discussion of (2.3) yields that there exists a class ǫ in H2´et(X, Zl) such thatλn−1 · ǫ is (n −1)!δ1 · · · δn−1 times a generator of H2n´et (X, Zl) .

It follows that c = m/δnmust be in Zl . But δn is prime to char(k) , hence c is an integer and C ∼cλn−1min .Let θN be the canonical principal polarization on JN .

One has:(3.3)φq∗θN = Pic0(q) ◦φθN ◦q = φλ ◦p ◦φ−1θN ◦ι∗◦φλ = φλ ◦α(C, λ) .Similarly:φp∗λ = Pic0(p) ◦φλ ◦p = φθN ◦q ◦φλ ◦φ−1λ◦p = φθN ◦q ◦p .4

Note that, if g is the genus of N , one has C · λ = N · p∗λ = θg−1N/(g −1)! · p∗λ .Inparticular, −2(C · λ) is the coefficient of t2g−1 in the polynomial:deg(t θN −p∗λ)2 = deg(t φθN −φp∗λ) = deg(t IdJN −q ◦p) .Since Tr(q ◦p) = Tr(p ◦q) = Trα(C, λ), the following equality, originally proved byMatsusaka ([Ma], corollary, p.8), holds:(3.4)Trα(C, λ)= 2(C · λ) .

(3.5)If the N´eron-Severi group of X has rank 1 (this holds for a generic principallypolarized X by [M], theorem 6.5, hence for a generic X with any polarization by [Mu],corollary 1, p.234), and ample generator l′ , we can write q∗θN = rl′ and l = sl′ with rand s integers. We get rφl′ = sφl′ ◦α(C, l) hence α(C, l)(sx) = rx for all x in X .

Bytaking degrees, one sees that s divides r and α(C, l) = (r/s)IdX . By (3.2), the curve Cis numerically equivalent to a rational multiple of λn−1 and its degree is a multiple of n .If l′ is separable of type (δ1| · · · |δn) , the curve C is numerically equivalent to an integralmultiple of λn−1min and its degree is a multiple of nδn .Lemma 3.6.– Let C be an irreducible curve that generates a polarized abelian variety (X, λ)of dimension n .

Then, the polynomial Pα(C,λ) is the square of a polynomial whose rootsare all real and positive.Proof. Let α = α(C, λ) .

By (3.3), one has φq∗θN = φλ ◦α , hence, for any integer t :Pα(t) deg φλ = deg(t IdX −α) deg φλ= deg(t φλ −φλ ◦α)= deg(t φλ −φq∗θN)= deg(φtλ−q∗θN) = 1n! (tλ −q∗θN)n 2 .The lemma then follows from theorem 2.1.We end this section with a proof of Matsusaka’s celebrated criterion:Theorem 3.7.

(Matsusaka)– Let C be an irreducible curve in a polarized abelian variety(X, λ) of dimension n . Assume that α(C, λ) = IdX .

Then C is smooth and (X, λ) isisomorphic to (JC, θC) .Proof. Let N be the normalization of C .

The morphism α(C, λ) is the identity andfactors as:X −−→N(n) −→Wn(N) −→JN −→X .5

It follows that dim JN = g(N) ≥n . Moreover, the image of X in JN has dimension n ,hence is the entire Wn(N) , which is therefore an abelian variety.

This is possible only ifg(N) ≤n . Hence N has genus n .

It follows that the morphism q : X →JN is an isogeny,which is in fact an isomorphism since p ◦q = α(C, λ) = IdX . By (3.3), the polarizationsq∗θN and λ are equal, hence q induces an isomorphism of the polarizations.4.

Degrees of curvesLet C be a curve that generates a polarized abelian variety (X, λ) of dimension n .We want to study its degree d = C · λ . First, by description (3.1), the dimension of theimage of α(C, λ) is the dimension of the abelian subvariety < C > generated by C .

Thisand the definition of α(C, λ) imply:C · λ ≥n .It was proved by Ran ([R]) for k = C and by Collino ([C]) in general, that if C · λ = n ,the minimal value, then C is smooth and (X, λ) is isomorphic to its Jacobian (JC, θC) .This suggests that there should be a better lower bound on the degree that involves thetype of the polarization λ . The following proposition provides such a bound.Proposition 4.1.– Let C be an irreducible curve that generates a polarized abelian variety(X, λ) of dimension n .

Then:C · λ ≥n(deg λ)1n .If λ is separable, there is equality if and only if C is smooth and (X, λ) is isomorphic to(JC, δθC) , for some integer δ prime to char(k) .Recall that by (3.5), the degree of any curve on a generic polarized abelian variety(X, λ) is a multiple of n . When λ is separable of type (δ1| · · ·|δn) , this degree is even amultiple of nδn .Proof of the proposition.We know by lemma 3.6 that Pα(C,λ) is the square of apolynomial Q whose roots β1, .

. ., βn are real and positive.

We have:C · λ = 12 Tr α(C, λ) = 12(2β1 + · · · + 2βn)≥n (β1 · · · βn)1n = n Q(0)1n = n Pα(C,λ)(0)12n= ndeg α(C, λ)12n ≥n (deg φλ)12n = n (deg λ)1n .This proves the inequality in the proposition. If there is equality, β1, .

. ., βn must beall equal to the same number m , which must be an integer since Pα(C,λ) has integral6

coefficients. It follows from the proof of lemma 3.6 that: 1n!

(tλ −q∗θN)n 2 = Pα(t) deg φλ = (t −m)2n deg φλ .Theorem 2.1. (iii) yields mλ = q∗θN .

It follows from (3.3) that α(C, λ) = m IdX .If λ is separable of type (δ1| · · ·|δn) , by (3.2), the number c = m/δn is an integerand C is numerically equivalent to cλn−1min . We get:cnδn = C · λ = n(deg λ)1n = n(δ1 · · ·δn)1n ≤nδn .This implies c = 1 and δ1 = · · · = δn = δ .

But then λ is δ times a principal polarizationθ ([Mu], theorem 3, p.231) and C ∼θn−1min . The conclusion now follows from Matsusaka’scriterion 3.7.Corollary 4.2.

(Ran, Collino)– Let C be an irreducible curve that generates a polarizedabelian variety (X, λ) of dimension n . Assume that C · λ = n .

Then C is smooth and(X, λ) is isomorphic to its Jacobian (JC, θC) .Proof. Although the converse of the proposition was proved only for λ separable, we stillget from its proof that α(C, λ) is the identity of X and we may then apply Matsusaka’scriterion 2.7.

This is the same proof as Collino’s.More generally, if C · λ = dim < C > , the same reasoning can be applied on < C >with the induced polarization to prove that C is smooth and that (X, λ) is isomorphic tothe product of (JC, θC) with a polarized abelian variety.Corollary 4.3.– Let X be an abelian variety with a separable polarization λ of type(δ1| · · ·|δn) .Let C be an irreducible curve in X and let m be the dimension of theabelian subvariety that it generates. Then:C · λ ≥m(δ1 · · · δm)1m .Proof.

Apply the proposition on the abelian subvariety Y generated by C . All there isto show is that the degree Y · λm/m!

of the restriction λ′ of λ to Y is at least δ1 · · · δm .We will prove that it is actually divisible by δ1 · · · δm . When k = C , this follows fromthe fact that the class λmmin is integral.The following argument for the general casewas kindly communicated to the author by Kempf.

Let ι be the inclusion of Y in X .Then φλ′ = Pic0(ι) ◦φλ ◦ι , hence deg(λ′)2 , which is the order of the kernel of φλ′ , is amultiple of the order of its subgroup K(λ) ∩Y , hence a fortiori a multiple of the order7

of its (r, r) part K′ . In other words, since K′ ≃(Z/δ′1Z)2 × · · · × (Z/δ′mZ)2 for someintegers δ′1|δ′2| .

. .|δ′m prime to char(k) , it is enough to show that δ′1δ′2 · · · δ′m is a multipleof δ1δ2 · · · δm .Let l be a prime number distinct from char(k) and let Fl be the field withl elements.For any integer s , let Xs be the kernel of the multiplication by ls onX .Then Xs/Xs−1 is a Fl –vector space of dimension 2n of which Ys/Ys−1 is asubspace of dimension 2m .Since K(λ) is isomorphic to (Z/δ1Z)2 × · · · × (Z/δnZ)2 ,the rank over Fl ofK(λ) ∩Xs/K(λ) ∩Xs−1is twice the cardinality of the set{i ∈{1, .

. ., n} | ls divides δi} .

The dimension formula yields:rankK(λ) ∩Ys/K(λ) ∩Ys−1≥2 Card {i | ls divides δi} −2n + 2m .But the rank ofK(λ) ∩Ys/K(λ) ∩Ys−1= (K′ ∩Xs)/(K′ ∩Xs−1) is also twice thecardinality of {i ∈{1, . .

., m} | ls divides δ′i} . It follows that:Card {i ∈{1, .

. ., n} ls̸ | δi} ≥Card {i ∈{1, .

. ., m} ls̸ | δ′i} .This implies what we need.Corollary 4.4.– Let C be an irreducible curve that generates a principally polarizedabelian variety (X, θ) of dimension n .

Assume that C is invariant by translation by anelement ǫ of X of order m . Then C · θ ≥n m1−1n .Proof.

Let H be the subgroup scheme generated by ǫ . The abelian variety X′ = X/Hhas a polarization λ of degree mn−1 whose pull-back on X is mθ ([Mu], corollary, p.231).If C′ is the image of C in X′ , the proposition yields C · θ = C′ · λ ≥n m1−1n .Note that in the situation of corollary 4.4, if (X, θ) is a generic principally polarizedabelian variety of dimension n , and m is prime to char(k) , then mn divides C · θ .With the notation of the proof above, this follows from the fact that any curve on X′ isnumerically equivalent to an integral multiple of λn−1min (see (3.5)).5.

Bounds on the genusWe keep the same setting: C is a curve that generates a polarized abelian variety(X, λ) , its normalization is N , and its degree is d = C · λ . The composition:X −−→N(d) −→Wd(N) −→JNis a morphism with finite kernel (since α(C, λ) is an isogeny), hence Wd(N) containsan abelian variety of dimension n .

We can apply the ideas of [AH] to get a bound of8

Castelnuovo type on the genus of N . Note that if C does not generate X , the samebound holds with n replaced by the dimension of < C > .Theorem 5.1.– Let C be an irreducible curve that generates a separably polarized abelianvariety (X, λ) of dimension n > 1 .

Let N be the normalization of C and let d = C · λ .Then:g(N) < (2d −1)22(n −1) .The inequality in the second part of lemma 8 in [AH] would improve this boundwhen char(k) = 0 , but its proof is incorrect.Proof. Let A be the image of X in Wd(N) and let A2 be the image of A × A in W2d(N)under the addition map.

We want to show that the morphism associated with a genericpoint of A2 is generically injective on N . The linear systems corresponding to pointsof A2 are of the form |ON(2Dx)| , where x varies in X , where D is an effective divisorthat represents λ and Dx = D + x .

It is therefore enough to show that the restriction toC −x of the morphism φ2D associated with |2D| is generically injective for x generic. Ifnot, for x generic in X and a generic in C −x , there exists b in C −x with a ̸= b andφ2D(a) = φ2D(b) .

The same holds for a generic in X and x generic in C −a . Since φ2Dis finite ([Mu], p.60), b does not depend on x , hence C −a = C −b .

Since C generatesX , this implies that ǫ = a −b is torsion, hence does not depend on a . Letting a vary,we see that any divisor in |2D| is invariant by translation by ǫ .

The argument in [Mu],p.164, yields a contradiction.It follows that the image of the morphism N →Pr that corresponds to a genericpoint in A2 is a curve of degree a divisor d′ of 2d , with normalization N . Moreover,one has r ≥n ([AH], lemma 1).Castelnuovo’s bound ([ACGH], p.116 and [B] whenchar(k) > 0 ) then gives:g(N) ≤m(d′ −1) −m(m + 1)(r −1)2,where m = d′−1r−1.

Hence:g(N) ≤md′ −1 −(m + 1)(r −1)2< d′ −1r −1d′ −1 −d′ −12≤(d′ −1)22(n −1) ≤(2d −1)22(n −1) .This finishes the proof of the theorem.9

In particular, in a principally polarized abelian variety (X, λ) of dimension n , anysmooth curve numerically equivalent to cθn−1min has genus < (2cn−1)22(n−1) . For curves in genericprincipally polarized abelian varieties of dimension n , I conjecture the stronger inequalityg(C) ≤cn + (c −1)2 .The theorem also gives a lower bound on the degree of any curve in a genericcomplex polarized abelian variety of dimension n , whose only merit is to go to infinitywith n .

A better bound is obtained in [D2].Corollary 5.2.– Let C be a curve in a generic complex polarized abelian variety (X, λ)of dimension n and let c be the integer such that C is numerically equivalent to c λn−1min .Then:c >rn8 −14 .Proof. We may assume that λ is a principal polarization and that n > 12 .

Let N be thenormalization of C . Corollary 5.5 in [AP] yields g(N) > 1 + n(n + 1)/4 , which, combinedwith the proposition, gives what we want.We can get better bounds on the genus when d/n is small.Proposition 5.3.– Let C be an irreducible curve that generates a complex polarized abelianvariety (X, λ) of dimension n .

Let N be the normalization of C and let d = C · λ . Then:(i) If d < 2n , then g(N) ≤d .

(ii) If d = 2n , then g(N) < 3d2 = 3n . (iii) If d ≤3n , then g(N) ≤4d .

(iv) If d ≤4n , then g(N) ≤6d .Proof. We keep the notation of the proof of theorem 5.1.

In particular, Wd(N) containsan abelian variety A of dimension n . If 2n > d , it follows from proposition 3.3 of [DF]that g(N) ≤d .Recall that we proved earlier that the morphisms that correspond togeneric points in A2 are birational onto their image.It follows from corollary 3.6 ofloc.cit.

that g(N) < 3d/2 when d = 2n .This proves (ii).We will do (iv) only, (iii)being analogous. First, we may assume that the embedding of A in Wd(N) satisfies theminimality assumptions made in [A1].

Let Ak be the image of A × · · · × A in Wkd(N)under the addition map and let rk be the maximum integer such that Ak is contained inWrkkd(N) . If g(N) > 6d , we get, as in the proof of proposition 3.8 of [DF], the inequalitiesr6 ≥8n + 2 and n ≤6d −3r6 .

It follows that d ≥(n + 3r6)/6 ≥(25n + 6)/6 > 4n . Thisproves (iv).10

The inequality (ii) should be compared with the inequality g(C) ≤2n + 1 provedby Welters in [W] when char(k) = 0 for any irreducible curve C numerically equivalentto 2θn−1minon a principally polarized abelian variety (X, θ) of dimension n (so thatC · θ = 2n ). Equality is obtained only with the Prym construction.6.

Curves of low degreesLet C be an irreducible curve that generates a principally polarized abelian variety(X, θ) of dimension n . We keep the same notation: N is the normalization of C andq : X →JN is the induced morphism.

From (2.2), we get that the square of the monicpolynomial QC(T) = (Tθ −q∗θN)n/n! has integral coefficients (and is the characteristicpolynomial of α(C, θ) ).

It follows that QC itself has integral coefficients, and we get fromtheorem 2.1 and (3.4):(i) The roots of QC are all real and positive. (ii) The sum of the roots of QC is the degree d = C · θ .

(iii) The product of the roots of QC is the degree of the polarization q∗θN .Smyth obtained in [S] a lower bound on the trace of a totally real algebraic integerin terms of its degree. His results can be partially summarized as follows.Theorem 6.1.

(Smyth)– Let σ be a totally positive algebraic integer of degree m . ThenTr(σ) > 1.7719 m, unless σ belongs to an explicit finite set, in which case Tr(σ) = 2m −1and Nm(σ) = 1 .It is tempting to conjecture:(6.2)Conjecture Cm – Let σ be a totally positive algebraic integer of degree m .

ThenTr(σ) ≥2m −1 . If there is equality, then Nm(σ) = 1 .

(6.3)The inequality in the conjecture follows from Smyth’s theorem for m ≤8 (andholds also for m = 9 according to further calculations). Smyth also worked out a list ofall totally positive algebraic integers σ for which Tr(σ) −deg(σ) ≤6 .

It follows from thislist that the full conjecture holds for m ≤7 .There are infinitely many examples for which the conjectural bound is obtained:if M is an odd prime, the algebraic integer 4 cos2(π/2M) is totally positive, has degree(M −1)/2 , trace M −2 and norm 1 .Proposition 6.4.– Let C be an irreducible curve that generates a principally polarizedabelian variety (X, θ) of dimension n and let QC be the polynomial defined above. Then, if11

|QC(0)| = 1 , the curve C is smooth, X is isomorphic to its Jacobian and C is canonicallyembedded.Proof.By fact (iii) above, the polarization q∗θN is principal.The proposition thenfollows from the following lemma.Lemma 6.5.– Let (JN, θN) be the Jacobian of a smooth curve, let X be a non-zero abelianvariety and let q : X →JN be a morphism. Assume that q∗θN is a principal polarization.Then q is an isomorphism.Proof.

Since q∗θN is a principal polarization, q is an closed immersion. By Mumford’sproof of Poincar´e’s complete reducibility theorem ([Mu], p.173), there exist another abeliansubvariety Y of JN and an isogeny f : X × Y →JN such that f ∗θN is the product ofthe induced polarizations on each factor.As in loc.cit., for any k –scheme S , the set(X ∩Y)(S) is contained in K(q∗θN)(S) , which is trivial.

Hence f is an isomorphism ofpolarized varieties. But a Jacobian with its canonical principal polarization cannot be aproduct, hence Y is 0 and q is an isomorphism.We now give a result on curves on simple abelian varieties.

The part that dependson the validity of conjecture 6.2 holds in particular for n ≤7 .Theorem 6.6.– Let C be an irreducible curve in a simple principally polarized abelianvariety (X, θ) of dimension n . Assume that either C · θ ≤1.7719 n , or that conjectureCm holds for all divisors m of n and C · θ < 2n .

Then, the curve C is smooth, X isisomorphic to its Jacobian and C is canonically embedded.Proof. Since X is simple, the polynomial Pα(C,θ) , hence also its “square root” QC , arepowers of an irreducible polynomial R of degree some divisor m of n .

If the degree ofC , which is equal to the sum of the roots of QC , is ≤1.7719 n , the sum of the roots ofR is also ≤1.7719 m. It follows from theorem 6.1 that |R(0)| = 1 . On the other hand, ifC · θ < 2n , the sum of the roots of R is also < 2m , hence, since Cm is supposed to hold,we also have |R(0)| = 1 .

The theorem then follows in both cases from proposition 6.4.It follows from the proof of the theorem that C has degree 2n −m for some divisorm of n .In particular, for n prime, either C has degree n and θ is the canonicalprincipal polarization, or it has degree 2n −1 .If one wants curves of degree between n and 2n in a simple abelian variety X ,and if one believes in conjecture 6.2, X needs to be a Jacobian with real multiplications(in the sense that the ring End(X) ⊗Q contains a totally real number field differentfrom Q ).Examples have been constructed in [Me] (see also [TTV]). More precisely,12

for any integer M ≥4 , Mestre constructs an explicit 2 –dimensional family of complexhyperelliptic Jacobians JC of dimension [M/2] whose endomorphism rings contain asubring isomorphic to Z[T]/GM(T) , where:GM(T) =Y0

They are simple if 2n + 1 is prime. For n = 2 ,these examples are Humbert surfaces, which contain curves of degree 3 ([vG], p.221).If the assumption X simple is dropped, much less can be said.

If Q is a monicpolynomial with integral coefficients whose roots are all real, we will say that a curve Chas real multiplications by Q if there is an endomorphism of JC whose characteristicpolynomial (see § 2) is Q2 . If k = C , this is the same as asking that the characteristicpolynomial of the endomorphism acting on the space of first-order differentials of C beQ .Proposition 6.7.– Let C be an irreducible curve that generates a principally polarizedabelian variety (X, θ) of dimension n .

Then, if C · θ = n + 1 , the curve C is smooth, Xis isomorphic to its Jacobian and C is canonically embedded. Moreover, the curve C hasreal multiplications by (T −1)n−2(T2 −3T + 1) .Proof.By theorem 6.1 and Smyth’s list in [S], the polynomial QCcan only be(T −1)n−1(T −2) or (T −1)n−2(T2 −3T + 1) .

By proposition 6.4, we need only excludethe first case.By theorem 2.1, there exist a polarized elliptic curve (X′, λ′) and amorphism f ′ : X →X′ such that f ′∗λ′ = q∗θN −θ . Similarly, there exist an (n −1) –dimensional polarized abelian variety (X′′, λ′′) and a morphism f ′′ : X →X′′ such thatf ′′∗λ′′ = 2θ −q∗θN .

The isogeny (f ′, f ′′) : (X, θ) →(X′, λ′) × (X′′, λ′′) is a morphism ofpolarized abelian varieties. Since θ is principal, it is an isomorphism and λ′ and λ′′ areboth principal polarizations.

Then, (X, q∗θN) is isomorphic to (X′, 2λ′) × (X′′, λ′′) . Inparticular, the pull-back of θN by X′′ →JN is a principal polarization.

By lemma 6.5,this cannot occur.In the next case where deg(C) = n + 2 , the same techniques give partial results.Proposition 6.8.– Let C be an irreducible curve that generates a principally polarized13

abelian variety (X, θ) of dimension n > 2 .Assume that char(k) ̸= 2, 3 .Then, ifC · θ = n + 2 , one of the following possibilities occurs:(i) the curve C is smooth of genus n , X is isomorphic to its Jacobian and Cis canonically embedded. Moreover, the curve C has real multiplications by(T −1)n−3(T3 −5T2 + 6T −1) or (T −1)n−4(T2 −3T + 1)2 .

(ii) the curve C is smooth of genus n and bielliptic, i.e. there exists a morphismof degree 2 from C onto an elliptic curve E .

The abelian variety X is thequotient of JC by an element of order 3 that comes from E . (iii) The normalizationNofChas genusnand real multiplications by(T −1)n−2(T2 −4T + 2) .

There is an isogeny JN →X of degree 2 , andeither C is smooth, or it has one node and N is hyperelliptic. (iv) the curve C is smooth and bielliptic of genus n + 1 , and has real multipli-cations by T(T −1)n−2(T2 −4T + 2) .

The abelian variety X is the “Prymvariety” associated with the bielliptic structure.Remark 6.9.Mestre’s construction for N = 7 gives examples of curves of degree 5in principally polarized abelian varieties of dimension 3 , which fit into case (i) of theproposition.Example 6.11 below show that case (ii) does occur.These are the onlyexamples I know of.Proof.By theorem 5.1 and Smyth’s list in [S], the polynomial QCcan only be(T −1)n−2(T −2)2 ,(T −1)n−1(T −3) ,(T −1)n−3(T3 −5T2 + 6T −1) ,(T −1)n−2(T2 −4T + 2) or (T −1)n−4(T2 −3T + 1)2 . If the constant term is ±1 , thesame proof as above yields that we are in case (i).

The first polynomial is excluded as inproposition 6.7 (use n > 2 ).If QC(T) = (T −1)n−1(T −3) , as in the proof of proposition 6.7, there exist apolarized elliptic curve (X′, λ′) and a morphism f ′ : X →X′ with connected kernelX′′ such that f ′∗λ′ = q∗θN −θ or equivalently q∗θN = θ + (deg λ′)[X′′] .The identity1n! (Tθ −q∗θN)n = (T −1)n−1(T −3) yields (deg λ′)(deg θ|X′′) = 2 .If deg λ′ = 2 , onegets a contradiction as in the proof of proposition 6.7.If λ′ is principal, one hasdeg(q∗θN)|X′′= 2 .

We use the following result.Lemma 6.10.– Let (JN, θN) be the Jacobian of a smooth curve, let X be a non-zero abelianvariety and let r : X →JN be a morphism with finite kernel. Assume that deg(r∗θN) is≤dim(X) and prime to char(k) .

Then g(N) < dim(X) + deg(r∗θN) .Proof. Let K be the kernel of r and let ι : X/K →JN be the induced embedding.

ByPoincar´e’s complete reducibility theorem ([Mu], p.173), there exist an abelian subvariety14

X′ of JN and an isogeny f : X/K × X′ →JN such that the pull-back f ∗θN is theproduct of the induced polarizations.Note that deg(ι∗θN) divides deg(r∗θN) .Inparticular, under our assumptions, the polarization ι∗θN is separable and has a non-empty base locus F , of dimension ≥dim(X) −deg(r∗θN) . If Θ is a theta divisor forJN , it follows from the equation of f ∗Θ given in [D1], proposition 9.1, that f(F × X′) iscontained in Θ .

Lemma 5.1 from [DF] (which is valid in any characteristic) then yieldsdim(F × X′) + dim(X′) ≤g(N) −1 , from which the lemma follows.Since char(k) ̸= 2 , it follows from the lemma applied to the inclusion X′′ →JN thatg(N) = n hence that the morphism q : X →JN is an isogeny of degree 3 . It is not difficultto see (using for example [D1] § 9) that since char(k) ̸= 2, 3 , there is a commutativediagram of separable isogenies:X′′ × X′3:1−−−−−→X′′ × E3:1−−−−−→X′′ × X′4:1y4:1y4:1yXq−−−−−→JNp−−−−−→Xwhere E is the quotient of X′ by a subgroup of order 3 .

The middle vertical arrowinduces an injection of E into JN whose image has degree 2 with respect to θN . Byduality, one gets a morphism f : N →E of degree 2 .

In this situation, one checks thatsince n > 2 , for any two points x and y of N , one cannot have x −y ≡f ∗e , for e ̸= 0in E . Thus C , image of N in X by p , is smooth.If QC(T) = (T −1)n−2(T2 −4T + 2) , the polarization q∗θN has degree 2 .Itfollows from lemma 6.10 that:• either g(N) = n and C is the image of N by an isogeny p : JN →X of degree2 .

In particular, either C is smooth or N is hyperelliptic and C is obtained by identifyingtwo Weierstrass points of N (so that, in a sense, C is bielliptic).• or g(N) = n + 1 and q is a closed immersion. The proof of lemma 6.10 yieldsan elliptic curve X′ in JN and an isogeny f : X × X′ →JN of degree 4 .Moreover,deg(θN)|X′ = 2 , hence the morphism N →X′ obtained by duality has degree 2 .Onechecks as above that C is smooth.

The abelian variety X is the Prym variety associatedwith the bielliptic structure, i.e. is isomorphic to the quotient JN/X′ .

It remains to provethe statement about real multiplications. With the notation of (2.2), we calculate thecharacteristic polynomial of the endomorphism βθN(p∗θ) of JN .

If t is any integer, one15

has:degt IdJN −βθN(p∗θ)= deg(t θN −p∗θ)2= 14 deg(t f ∗θN −f ∗p∗θ) 2= 14 deg(t (θN)|X′) deg(t q∗θN −q∗p∗θ) 2= t24 degt φq∗θN −φq∗p∗θ.Set α = α(C, θ) . Using (3.1) and (3.3), we get:degt IdJN −βθN(p∗θ)= t24 degt φθ ◦α −φα∗θ= t24 degt IdPic0(X) −Pic0(α)deg(φθ ◦α)= PPic0(α)(t) t2 = QC(t)2 t2 .It follows that N has real multiplications by T QC(T) = T(T −1)n−2(T2 −4T + 2) .

Thisfinishes the proof of the proposition.Example 6.11 Case (ii) of the proposition does occur as a particular case of the followingconstruction. Let C be a smooth curve of genus n with a morphism of degree r ontoan elliptic curve E .

Assume that r is prime to char(k) and that the induced morphismE →JC is a closed immersion. Let s be an integer prime to char(k) and congruent to1 modulo r , and let q : JC →X be the quotient by a cyclic subgroup H of order s ofE .

There exist an abelian variety Y of dimension n −1 with a polarization λY of type(1| · · ·|1|r) and an isogeny f : E × Y →JC with kernel isomorphic to (Z/rZ)2 , such thatf ∗θC = pr∗1(rλE) ⊗pr∗2λY , where λE is the polarization on E defined by a point. Theisogeny f induces an isogeny g : E/H × Y →X and, because s ≡1 (mod r) , one checksthat there exists a principal polarization θ on X such that g∗θ = pr∗1(rλE/H) ⊗pr∗2λY .It follows that f ∗q∗θ = pr∗1(rsλE) ⊗pr∗2λY .

We claim that the degree of the curve q(C)on X with respect to the principal polarization θ is n + s −1 . In fact, one has:f ∗θn−1C/(n −1)!

∼rλE (pr∗2λY)n−2/(n −2)! + (pr∗2λY)n−1/(n −1)!hencef ∗θn−1C/(n −1)!

· f ∗q∗θ = rs deg λY + r(n −1) deg λY= r2(s + n −1) .It follows that C · q∗θ = n + s −1 , which proves our claim.When char(k) = 0 , this construction yields examples of curves of degree n + t inJacobians of dimension n , for any n ≥2 and t ≥2 .References16

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Classification: 14K05, 14H40.18

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