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POINTS OF LOW DEGREE ON SMOOTH PLANE CURVES

arXiv:alg-geom/9210004v1 13 Oct 1992POINTS OF LOW DEGREE ON SMOOTH PLANE CURVESOlivier Debarre (*) and Matthew J. Klassen1. IntroductionThe purpose of this note is to provide some applications of a theorem of Faltings([Fa1]) to smooth plane curves, using ideas from [A] and [AH].Let C be a smooth projective plane curve defined by an equation of degree dwith rational coefficients.

We show:Theorem 1.– If d ≥7 , the curve C has only finitely many points whose field ofdefinition has degree ≤d −2 over Q .The result still holds for d < 7 , provided that the complex curve C has nomorphisms of degree ≤d −2 onto an elliptic curve, an assumption which we showautomatically holds for d ≥7 .This result is sharp in the sense that if C has arational point, there exist infinitely many points on C with field of definition of degree≤d −1 . These points come from the intersection of C with a rational line through arational point.

We show further:Theorem 2.– If d ≥8 , all but finitely many points of C whose field of definition hasdegree ≤d −1 over Q arise as the intersection of C with a rational line through arational point of C .In particular, if C has no rational points, there are only finitely many pointswhose field of definition has degree ≤d −1 over Q .Again, the result still holds for d = 6 or 7 , provided that C has no morphismsof degree ≤d −1 onto an elliptic curve, and for d = 5 , provided that C has nomorphisms onto an elliptic curve.Both results remain valid if Q is replaced by any number field.These theorems apply in particular to the Fermat curves Fd with equationXd + Yd = Zd , which is the case we had in mind when we started this investigation.Moreover, we can extend the results to all d ≥3 in this case, with the one exceptiond ̸= 6 (see § 6).The first author would like to thank the M.S.R.I., where this work was done, forits support and hospitality. (*) Partially supported by the European Science Project “Geometry of Algebraic Vari-eties”, Contract no.

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

2. NotationFor a projective curve C , we denote by C(d) the symmetric product varietiesof C .

We also denote by J(C) the Jacobian variety of C and by Wd(C) the image ofC(d) under the Abel-Jacobi map to J(C) defined with respect to a chosen base pointon C . It corresponds to isomorphism classes of line bundles on C of degree d whichhave a non-zero section.3.

Faltings’ theoremWe first remark that theorem 1 is equivalent to the statement that the set ofQ –rational points on the symmetric product C(n) is finite for any n ≤d −2 . Thisis obtained by simply observing that any point on C(K) with [K : Q] = n , togetherwith its conjugates, forms a divisor of degree n invariant by Gal (Q/Q) , and hencedefines a Q –rational point on C(n) .

Furthermore, since C is a smooth plane curve ofdegree d , it has no pencils of degree ≤d −2 ([ACGH] p. 56, exercise 18. (i)), henceC(n) maps isomorphically onto Wn(C) .

Thus the proof of theorem 1 is reduced toshowing that Wn(C)(Q) is finite for all n ≤d −2 . We use the following beautifulresult of Faltings:Theorem (Faltings, [Fa1]) – Let A be an abelian variety defined over a number fieldK .If X is a subvariety of A which does not contain any translate of a positive-dimensional abelian subvariety of A , then X contains only finitely many K –rationalpoints.It is therefore enough to show that Wn(C) does not contain any non-zero abelianvariety.The situation in theorem 2 is a bit more complicated since the morphism:ψ : C(d−1) −→Wd−1(C)is no longer an isomorphism: each pencil of degree d −1 on C corresponds to arational curve in C(d−1) which is contracted by ψ .

By [ACGH] p. 56, exercise 18. (ii),all such pencils are given by the lines through a fixed point x of C .

Let Rx be thecorresponding rational curve in C(d−1) . Since ψ induces an isomorphism outside ofthe union of all Rx , any rational point of C(d−1) corresponds either to a rational pointof Wd−1(C) , or to a rational point of some Rx .

Now let x be a point of C suchthat Rx(Q) is non-empty and let D be the divisor on C that corresponds to a pointof Rx(Q) . The points of D are then on a unique line l (which passes through x )and, since D is invariant under the action of Gal (Q/Q) , so is l , which is thereforerational.

It follows that the divisor l · C is rational, hence so is x = l · C −D . Thisreduces the proof of theorem 2 to showing that Wd−1(C)(Q) is finite.

As above, by2

Faltings’ theorem, it is enough to show that Wd−1(C) does not contain any positive-dimensional abelian variety.4. Linear systems on smooth plane curvesBefore proceeding to the proof of the theorems, we gather here some elementaryfacts about linear systems on smooth plane curves, which we will deduce from thefollowing result of Coppens and Kato.

Let H be a hyperplane section on a smoothplane curve C of degree d and let D be an effective divisor on C which belongs to abase-point-free pencil. Then we have:Theorem (Coppens-Kato, [CK]) – If n < k(d −k) for some integer k , the linearsystem |(k −1)H −D| is non-empty.We assume now that d ≥5 .

Here are the consequences that we need:(4.1) If deg(D) ≤2d −5 , then either D ≡H or D ≡H −x for some point x on C . (4.2) If deg(D) = 2d −4 , then D ≡2H −x1 −x2 −x3 −x4 for some points x1 , x2 ,x3 and x4 on C , no three of them collinear.This follows from the theorem with k = 3 , except for d = 5 .In the latter case,Riemmann-Roch says that the 6 points of D are on a conic, which is what we need.

(4.3) If deg(D) = 2d −3 and d ≥7 , then D ≡2H −x1 −x2 −x3 for some points x1 ,x2 and x3 on C , not collinear. (4.4) If deg(D) = 2d −2 and d ≥8 , then D ≡2H −x1 −x2 for some points x1 andx2 on C .

(4.5) If deg(D) = 2d −2 , dim |D| ≥2 and d ≥6 , then D ≡2H −x1 −x2 for somepoints x1 and x2 on C . In particular dim |D| = 3 .For d ≥7 , the linear system |2H −(D −x)| is non-empty for any point x on C by(4.3).

By Riemann-Roch, this implies that |2H −D| is non-empty. For d = 6 , byRiemann-Roch, the 10 points of D are on 3 linearly independent cubics, which mustbe reducible.

Since 7 points of D cannot be on a line (because d = 6 ), all points ofD are on a conic. (4.6) If deg(D) = 2d −2 and dim |D| ≥3 , then D ≡2H −x1 −x2 for some pointsx1 and x2 on C .By (4.2), the linear system |2H −(D −x −y)| is non-empty for any points x and yon C .

By Riemann-Roch, this implies that |2H −D| is non-empty.5. Proof of the theoremsWe only need to prove that, under the hypotheses of theorem 1 and theorem3

2, Wd−2(C) and Wd−1(C) respectively do not contain any non-zero abelian varieties.This will follow from the following proposition.Proposition 1. – Let C be a smooth plane curve of degree d ≥4 .

Then:(i) If d ≥5 , the variety Wd−1(C) does not contain any abelian variety of dimension≥2 . (ii) If the variety Wd−2(C) contains an elliptic curve E , the inclusion is inducedby a morphism C →E of degree d −2 and d ≤6 .

(iii) If d ≥6 , and if the variety Wd−1(C) contains an elliptic curve E , then d ≤7and the inclusion is induced by a morphism C →E of degree d −1 or d −2 .Proof. If d = 4 , property (ii) follows from [A], theorem 11.2.We may therefore assume d ≥5 .

Let 1 ≤e ≤d −1 and assume that We(C)contains an abelian variety A of dimension h > 0 . Let A2 be the image of A × Aunder the addition map We(C) × We(C) →W2e(C) and let r be the dimension of thelinear system on C which corresponds to a generic point of A2 .

It follows from [A],lemma 8 that r ≥h . We may assume that A is not contained in x + We−1(C) forany point x in C .

In this case, the linear system on C that corresponds to a genericpoint of A2 is base-point-free.Assume first h ≥2 .Since r ≥h , we get a family of base-point-free linearsystems of degree ≤2d −2 and dimension ≥2 parametrized by A , which is an abelianvariety. By (4.6), this is possible only if d = 5 , e = d −1 = 4 and h = r = 2 .

By [A]lemma 14, the morphisms C →P2 which correspond to points of A2 factor througha fixed morphism p : C →B of degree n > 1 onto a curve B of genus ≥h = 2 .The induced birational morphims B →P2 then have degree 8/n , hence n = 2 andg(B) = 2 . Let σ be the involution associated with the double cover p , and let Hbe a hyperplane section of C .

Since the embedding of C as a smooth plane curve isunique (this follows for example from (4.1)), one has σ∗(H) ≡H hence σ is inducedby a projective automorphism τ of P2 . By Riemann-Hurwitz, σ has 6 fixed points,hence τ is the symmetry with respect to a line.

But then, the fixed points of σ arethe intersection of this line with C , hence there cannot be 6 of them since C hasdegree 5 . Therefore, this case does not occur.This takes care of the case h ≥2 and we now assume h = 1 .If r ≥3 , we get from (4.6) a non-constant map from the elliptic curve A intoC(2) .

By [A] theorem 11.2, C is bi-elliptic, hence has an elliptic curve of pencils ofdegree 4 . This contradicts (4.1).If r = 2 , we get linear systems of dimension 2 and degree 2e ≤2d −2 .

We4

get e = d −1 from (4.2). By (4.6), if moreover d ≥6 , there are no linear systems ofdegree 2d −2 and dimension exactly 2 , which is a contradiction.The only remaining case is h = r = 1 (except maybe if d = 5 and e = 4 ).

Theembedding of the elliptic curve A in We(C) is then induced by a morphism C →Aof degree e ([A], lemma 13). The pencils of degree 2 on A pull back to an ellipticcurve of base-point-free pencils on C of degree 2e .

Fact (4.1) implies e = d −2 ord −1 .If e = d −2 , fact (4.2) yields an embedding of A into W4(C) . If d > 6 , onehas g(C) ≥8 , and theorem 11 of [A] implies that C has a morphism of degree ≤4onto an elliptic curve E .

But then, the pencils of degree 2 on E pull back to anelliptic curve of base-point-free pencils on C of degree ≤8 , which contradicts (4.1)(since 8 ≤2d −5 . ).

Therefore, d ≤6 .If e = d −1 and d ≥8 , fact (4.4) yields an embedding of A into W4(C) , whichwe just saw cannot exist. Therefore, d ≤7 .This finishes the proof of the proposition.6.

Fermat curvesBoth theorems apply in particular to the Fermat curves Fd defined by theequation Xd + Yd = Zd , at least for d ≥8 . For small d , the situation is the following:• for d = 3 , 4 , 5 or 7 , it is known that J(Fd)(Q) , hence also its subvarietiesWe(Fd)(Q) for all e , are finite ([F1], [F2]).

This of course implies both theorems. Ford = 4 , Faddeev also shows that in addition to its four rational points, F4 has exactlytwelve points defined over quadratic fields, and that the lines through each of the fourrational points of F4 account for all points of F4 in all cubic fields.• for d = 6 , there is a morphism of degree 4 from F6 onto the elliptic curveF3 .

In particular, W4(F6) does contain an elliptic curve and our whole method ofproof collapses. However, since J(F3)(Q) is finite, this does not say anything aboutthe finiteness of W4(F6)(Q) .

On the other hand, J(F6)(Q) is known to be infinite([F2]). One may use here a stronger theorem of Faltings ([Fa2]), which says that ifX is a subvariety of an abelian variety A , both defined over a number field K , thenthe set X(K) lies inside a finite union of K –rationally defined translates of abeliansubvarieties of A contained in X .

Consequently, if any morphism of degree 4 fromF6 onto an elliptic curve has image F3 , theorem 1 will hold for F6 . If, in addition,there are no morphisms of degree 5 from F6 onto an elliptic curve, theorem 2 willhold for F6 .We mentioned in the introduction that our two theorems remained valid over any5

number field K . This applies in particular to Fermat curves for d ≥7 (for theorem 1)and d ≥8 (for theorem 2).

Both theorems remain valid for F5 : the absolutely simplefactors of its Jacobian are surfaces ([KR] theorem 2), and W4(F5) cannot containan abelian surface by proposition 1.(i). However, theorem 1 fails trivially for F3 , forF4 (this curve has a morphism of degree 2 onto the elliptic curve E with equationU2W2 + V4 = W4 , and as soon as E(K) becomes infinite, so will F(2)4 (K) ) and for F6(for the same reason, since there is a morphism of degree 4 from F6 onto the ellipticcurve F3 ).

As far as F7 is concerned, theorem 1 holds, and theorem 2 holds if andonly if there are no morphisms of degree 6 from F7 onto an elliptic curve.We cannot resist giving a different proof of theorem 1 for Fermat curves when dis an odd prime number p which satisfies p ≡2 (mod 3) as a nice application of thefollowing result of [DF] (proposition 3.3): for a complex projective curve C of genus g ,the variety Wd(C) cannot contain an abelian variety of dimension > d/2 for d < g .In fact, it is known in this case that the absolutely simple factors of J(Fp) are all ofdimensionp−12([KR], theorem 2), so that Wp−2(Fp) cannot contain any non-zeroabelian variety.It would be very interesting to know specifically which points constitute thefinite sets of algebraic points in the theorems for the Fermat curves. Of course, thisextends the already difficult question, posed by Fermat, of showing that there are onlythree rational points if d is odd and four if d is even.Assume again that d is an odd prime number p .

The easiest way to producealgebraic points of degree ≤p −3 is to take the other points of intersection of Fp withthe line through the three known rational points. In the affine patch where Z = 1 , thisis just the line y = 1 −x , and Fp is defined by xp + yp = 1 .

The x -coordinates ofthese other points of intersection are then just the roots ofxp + (1 −x)p −1x(x −1)= 0 .One sees, by considering the equation xp + (1 −x)p −1 = 0 and its derivative, thatthe factor x2 −x + 1 always occurs with multiplicity one or two depending on whetherp is 5 or 1 (mod 6) respectively, so we obtain x -coordinates η , and η−1 , where η isa primitive sixth root of unity. The other factor, of degree p −5 or p −7 , is irreducibleover Q for p ≤101 (checked using MAPLE), but the authors do not know if this isalways the case.

Also, the authors do not know of any other points on Fp of degree≤p −2 , i.e. which do not lie on the line y = 1 −x .

Gross and Rohrlich show thatthis line accounts for all the points of degree ≤(p −1)/2 on Fp for the primes p = 3 ,5 , 7 and 11 (see [GR]. )It is interesting to note that the linear equivalence class described above producesthe only known points of infinite order on the Mordell-Weil group J(Fp)(Q) .

More6

specifically, Gross and Rohrlich take the conjugate quadratic points P = (η, η−1, 1) ,and P = (η−1, η, 1) , and form the divisor P + P −2∞on Div0(Fp) .Then, theyshow that for p > 7 , the linear equivalence class in J(Fp) of the divisor P + P −2∞represents a point of infinite order.Finally, one would like to have, if not a complete description, at least an upperbound on the cardinality of the finite sets of algebraic points in theorem 1. It is naturalto begin by trying to bound the number of Q –rational points on Fp .

The greatestsuccess in this regard is Kummer’s proof of Fermat’s Last Theorem for p a regularprime (see [W].) For general p , all bounds depend on the rank of J(Fp)(Q) .

Oneapproach is to come up with bounds which are exponential in the Mordell-Weil rank asin Bombieri’s version of Faltings’ Theorem ([B].) Another approach is to use Coleman’s“Effective Chabauty,” applying his theory of p -adic abelian integrals ([C1], [C2].) Inthis case, one needs to know that the rank of the Mordell-Weil group J(Fp)(Q) is lessthan its dimension (i.e.

the genus of the curve.) This is known to hold in the case whenp is regular ([F3].) However, McCallum has shown that this would hold for all p ifone has a certain bound on the ideal class group Cl of Q(ζ) , where ζ is a primitivepth root of unity.

In particular, if Cl[p] denotes the subgroup of Cl of elements killedby p , then he shows thatrankZ Js(Q) ≤p −74+ 2 rankZ/pZ(Cl[p]).He then goes on to show that if rankZ/pZ(Cl[p]) < p+58, then the number of Q –rational points on Fp is ≤2p −3 ([Mc]. )The second author has begun to applyColeman’s theory to the symmetric products of curves in his Ph.D. thesis (to appear.

)References[A] Abramovich, D., Subvarieties of abelian varieties and of Jacobians of curves. Ph.D.Thesis, Harvard University, 1991.

[ACGH] Arbarello, E., Cornalba, M., Griffiths, P., Harris, J., Geometry of algebraiccurves. I. Grundlehren 267, Springer-Verlag, New York, 1985.

[AH] Abramovich, D., Harris, J., Abelian varieties and curves on Wd(C) , Comp. Math.78 (1991), 227–238.

[B] Bombieri, E., The Mordell Conjecture Revisited, Preprint, 1991. [C1] Coleman, R., Torsion points on curves and p -adic abelian integrals, Ann.ofMath.

121 (1985), 111–168.7

[C2] Coleman, R., Effective Chabauty , Duke Math. J.

52 (1985), 765–770. [CK] Coppens, M., Kato, T., The gonality of smooth curves with plane models,Manuscripta Math.

70 (1990), 5–25, and Correction to: “the gonality of smoothcurves with plane models”, Manuscripta Math. 71 (1991), 337–338.

[DF] Debarre, O., Fahlaoui, R., Abelian Varieties In Wrd(C) And Points Of BoundedDegrees On Algebraic Curves. To appear in Comp.

Math. [F1] Faddeev, D.K., Group of divisor classes on the curve defined by the equationx4 + y4 = 1 , Dokl.

Akad. Nauk.

SSSR 134 (1960), 776–778 and Sov. Math.Dokl.

1 (1960), 1149–1151. [F2] Faddeev, D.K., On the divisor class group of some algebraic curves, Dokl.

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2 (1961), 67–69. [F3] Faddeev, D.K., Invariants of divisor classes for the curves xk(1 −x) = yl in anl -adic cyclotomic field (in Russian), Trudy Math.Inst.Steklov 64 (1961),284–293.

[Fa1] Faltings, G., Diophantine approximation on abelian varieties, Ann. of Math.

133(1991), 549–576. [Fa2] Faltings, G., The general case of S. Lang’s conjecture, to appear.

[GR] Gross, B.H., Rohrlich, D.E., Some results on the Mordell-Weil group of theJacobian of a quotient of the Fermat curve, Invent. Math.

93 (1978), 637–666. [KR] Koblitz, N., Rohrlich, D.E., Simple Factors in the Jacobian of a Fermat Curve,Canadian J.

Math. 30 (1978), 1183–1205.

[Mc] McCallum, W.G., The Arithmetic of Fermat Curves, to appear in Math. Ann.

[W] Washington, L.C., Introduction to Cyclotomic Fields, Springer Verlag, New York,1982.8

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