Brauer-Siegel theorem for elliptic surfaces
aa r X i v : . [ m a t h . AG ] D ec BRAUER–SIEGEL THEOREM FOR ELLIPTICSURFACES
B. `E. KUNYAVSKI˘I AND M. A. TSFASMAN
Abstract.
We consider higher-dimensional analogues of the clas-sical Brauer-Siegel theorem focusing on the case of abelian varietiesover global function fields. We prove such an analogue in the caseof constant families of elliptic curves and abelian varieties.
To our teachers V.E. Voskresenski˘ı and Yu.I. Maninto their 80th and 70th birthdays, respectively Introduction
The classical Brauer–Siegel theorem, which is one of the milestonesof the number theory of the past century, reflects deep connections be-tween algebraic, arithmetical, analytic, and (in the function field case)geometric properties of global fields. Not only is the theorem a workingtool in a variety of problems concerning number and function fields, butthe underlying ideas have been recently put into much broader contextexpanding far beyond number theory (see, for example, [ST]).Recall that the theorem describes the asymptotic behaviour of theproduct of two important arithmetic invariants of a number field K ,the class number h ( K ) and the regulator R ( K ), as the discriminant d ( K ) tends to infinity. More precisely, it says that the ratio r =log( hR ) / log( p | d | ) tends to 1 provided at least one of the followingconditions is satisfied: 1) the degree n = [ K : Q ] remains the same forall K ’s in the sequence of fields under consideration; 2) n/ log( | d | ) tendsto 0 and all K ’s are normal. Even in this not-so-effective form there This research was supported in part by the French-Israeli grant 3-1354 and theRussian-Israeli grant RFBR 06-01-72004-MSTIa. The first named author was alsosupported in part by the Minerva Foundation through the Emmy Noether Instituteof Mathematics. The second named author was also supported in part by RFBR 02-01-22005, 02-01-01041, 06-01-72550-CNRSa, 07-01-00051, and INTAS 05-96-4634. are many useful applications. Some effective versions of the theoremare known in several particular cases (see [St] and references therein).A natural question whether the statement of the theorem still holdswhen none of conditions 1) and 2) is satisfied, or under some weakerassumptions, remained widely open until recently. In the paper [TV2]there were obtained some asymptotic bounds on r generalizing thestatement of the Brauer–Siegel theorem. These techniques, togetherwith those of an earlier paper [TV1] led to a new concept of infiniteglobal field which is an important object for further investigation. Com-bined with Weil’s “explicit formulae” (see [LT]), they yielded quite afew concrete arithmetic applications, like new estimates for regulators.Note that even more general approach was used in a recent paper [Zy]where the normality assumption on K was weakened.The above mentioned results present the state of the art in the re-search area concentrated around the classical Brauer–Siegel theorem.In the present paper we make an attempt to treat some new problemsarising from these achievements. Namely, one can think about higherdimensional analogues of the Brauer–Siegel theorem. In particular,if E is a commutative algebraic group defined over a global field K ,one can define an analogue of the class number h ( E ) and the regula-tor R ( E ). Moreover, the classical analytical class number formula ofDirichlet admits higher dimensional analogues both for algebraic tori[Shyr] and, conjecturally, for abelian varieties (Birch and Swinnerton-Dyer). This motivates the study of asymptotic behaviour of h ( E ) R ( E )in appropriately chosen families of groups E when the “discriminant” d ( E ) tends to infinity. In the case where E is an abelian variety, recentwork of Hindry and Pacheco contains quite a new approach to thiskind of asymptotic problems, both in the number field case [Hi] and inthe function field case [HP]. This work was an additional motivationfor publishing our results because the approach of Hindry and Pachecois, in a sense, “orthogonal” to ours: loosely speaking, they consider“vertical” families of abelian varieties (say, in the function field case RAUER–SIEGEL THEOREM FOR ELLIPTIC SURFACES 3 the genus of the underlying curve X is fixed and the conductor of theabelian variety grows) while we consider “horizontal” families wherethe genus of X tends to infinity.2. Main theorem
We fix the ground field k = F q and consider a (smooth, projective,geometrically irreducible) curve X/ F q of genus g . Let K = F q ( X ), andlet E/K be a (smooth, connected) commutative algebraic K -group.Our goal is to study asymptotic behaviour of the “class number” h ( E )as g → ∞ . In the present paper we focus on the particular case where E = A is an abelian variety (see, however, Section 3 for the case where E is an algebraic K -torus). Let X := | X ( A ) | be the order of theShafarevich–Tate group of A , and ∆ the determinant of the Mordell–Weil lattice of A (cf. [Mi], [Hi]). In this section we consider the mosttrivial “constant” case, i.e. E ∼ = E × F q K where E is an F q -group;see Section 3 for a more general setting.To state our main result, we recall some notation from [TV1]. Ifthe ground curve X = X varies in a family { X i } , we denote by g i the genus of X i ( g i → ∞ ), by N m ( X i ) the number of F q m -points of X i , and we always assume that for every m ≥ β m := lim i →∞ N m ( X i ) g i . Such families are called asymptotically exact ;any family contains an asympotically exact subfamily; any tower (i.e.a family such that k ( X i ) ⊂ k ( X i +1 ) for every i ) is asymptotically exact;see [Ts], [TV1] for more details. We shall often drop the index i if thisdoes not lead to confusion. Theorem 2.1.
Let E = E × F q K where E a fixed elliptic F q -curve.Let K vary in an asymptotically exact family, and let β m be the corre-sponding limits. Then lim i →∞ g i log q ( X · ∆) = 1 − ∞ X m =1 β m log q N m ( E ) q m , where N m ( E ) = | E ( F q m ) | . B. `E. KUNYAVSKI˘I AND M. A. TSFASMAN
Proof.
Denote by ω j ( j = 1 , . . . , g ) the eigenvalues of Frobenius actingon H ( X ) , and by ψ , ψ the eigenvalues of Frobenius acting on H ( E ) . We have ω j ω j = ψ ψ = q. Put t = q − s and consider the Hasse–Weil L -function of E/K.
Accord-ing to the Birch and Swinnerton-Dyer conjecture (which, under our hy-potheses, is a theorem [Mi], [Oe]), the value of L E ( t ) / (1 − qt ) r at t = q − equals q − g · X · ∆ / [ E ( k )] . Here r is the rank of E ( K ) /E ( K ) tors ;this number is equal to the number of pairs ( i, j ) such that ψ i = ω j (loc. cit.). This gives us Milne’s formula X · ∆ = q g Y ω j = ψ i (cid:18) − ψ i ω j (cid:19) . It is convenient to put ψ i = α i √ q, ω j = γ j √ q, and, taking into accountthat the Frobenius roots can be written as conjugate pairs, to writethe above formula as(1) X · ∆ = q g Y α i =1 /γ j (1 − α i γ j ) . Set α = α, α = α. First consider the case where r = 0 . Then theright-hand side of (1) can be written as q g P X ( α/ √ q ) P X ( α/ √ q ) , where P X ( t ) is the numerator of the zeta-function of X : Z X ( t ) = P X ( t )(1 − t )(1 − qt ) . Hence the right-hand side of (1) equals q g (cid:20)(cid:18) − α √ q (cid:19) (1 − α √ q ) Z X (cid:18) α √ q (cid:19) (cid:18) − α √ q (cid:19) (1 − α √ q ) Z X (cid:18) α √ q (cid:19)(cid:21) . We now write Z X ( t ) = ∞ Q m =1 (1 − t m ) − B m , then we have β m = lim g →∞ B m g (by our assumption, the limit exists), and we getlim g →∞ g log q ( X · ∆) = 1 + log q ∞ Y m =1 (cid:18) − α m q m (cid:19) − β m (cid:18) − α m q m (cid:19) − β m ! = 1 − ∞ X m =1 β m log q (cid:18) q m − α m + α m q m (cid:19) = 1 − ∞ X m =1 β m log q N m q m RAUER–SIEGEL THEOREM FOR ELLIPTIC SURFACES 5 (here N m = | E ( F q m ) | , and the last equality follows from the Weilformula). Note that the series on the right-hand side converges accord-ing to [Ts]. Indeed, we know that the series ∞ P m =1 mβ m q m − converges [Ts,Cor.1]. We have N m q m = 1 + q − m − α m + α m q m . Fix m > m big enough. Put x = α m + α m q m − q − m . Since | α m + α m | ≤ , we have (cid:12)(cid:12)(cid:12)(cid:12) log q N m q m (cid:12)(cid:12)(cid:12)(cid:12) = | log q (1 − x ) | ≤ c ∞ X n =1 (cid:0) q − m (cid:1) n ≤ c ′ q − m ≤ c ′ mq m − . Hence the series P β m log q N m q m converges.Let us now consider the case where r > . Our key observation isthat as g → ∞ , the rank cannot grow as fast as g, i.e., we always havelim g →∞ rg = 0 . Indeed, if lim g →∞ rg = c > , then there is at least one multiple Frobeniusroot ω j = ψ or ψ with multiplicity ≥ cg. Hence the Weil measure (cf.[TV1]) µ Ω = 1 g g X j =1 δ γ j (where δ γ j is the Dirac measure)tends (as g → ∞ ) to a measure that is greater than or equal to cδ γ j . But according to [TV1, Th.2.1], the limit measure µ = lim g →∞ µ Ω musthave a continuous density, contradiction.(As pointed out by the referee, this observation might happen tobe deducible from the “explicit formulae” for elliptic curves, see, e.g.,[Br].)Thus, in the general case where r > , we get the required result asfollows.Let us introduce an auxiliary function δ ( g ) = 1 + ε ( g ) such thatlim g →∞ ε ( g ) = 0 and lim g →∞ (cid:16) r log ε ( g ) g (cid:17) = 0 . Let F ( g ) = q g P X ( δ ( g ) α/ √ q ) P X ( δ ( g ) α/ √ q ) . We have, on the one hand,(2) lim g →∞ log q F ( g ) g = 1 − ∞ X m =1 β m log q (cid:18) N m q m (cid:19) , B. `E. KUNYAVSKI˘I AND M. A. TSFASMAN and, on the other hand,lim g →∞ g log q F ( g ) = lim g →∞ (cid:18) g log q ( X · ∆) (cid:19) . To prove the last equality, we write F ( g ) = q g g Y j =1 (1 − αγ j δ ( g ))(1 − αγ j δ ( g )) = δ ( g ) g q g g Y j =1 „ δ ( g ) − αγ j « „ δ ( g ) − αγ j « = δ ( g ) g q g Y γ j =1 /α „ δ ( g ) − αγ j « „ δ ( g ) − αγ j « · Y γ j =1 /α „ δ ( g ) − αγ j « „ δ ( g ) − αγ j « = δ ( g ) g „ δ ( g ) − « r · q g · Y γ j =1 /α (1 − αγ j δ ( g ))(1 − αγ j δ ( g )) · δ ( g ) g − r = (1 − δ ( g )) r · q g · Y γ j =1 /α (1 − αγ j δ ( g ))(1 − αγ j δ ( g )) . Hencelim g →∞ g log q F ( g ) = lim g →∞ (cid:18) g log q (1 − δ ( g )) r (cid:19) + lim g →∞ (cid:18) g log q ( X · ∆) (cid:19) = lim g →∞ r log q ε ( g ) g + lim g →∞ (cid:18) g log q ( X · ∆) (cid:19) = lim g →∞ (cid:18) g log q ( X · ∆) (cid:19) . Note that the series ∞ X m =1 β m log q (cid:18) q m + ( δ ( g ) α ) m + ( δ ( g ) α ) m q m (cid:19) converges for every fixed δ ( g ) sufficiently close to 1. Hence the passageto the limit in (2) is legitimate. (cid:3) A direct analogue of Theorem 2.1 is true for constant abelian varietiesof arbitrary dimension.
Theorem 2.2.
Let A = A × F q K where A a fixed abelian F q -varietyof dimension d . Let K vary in an asymptotically exact family, and let β m be the corresponding limits. Then lim i →∞ dg i log q ( X · ∆) = 1 − ∞ X m =1 β m log q N m ( A ) /d q m , where N m ( A ) = | A ( F q m ) | .Proof. The proof goes as for elliptic curves, mutatis mutandis . Thevalue of L A ( t ) / (1 − qt ) r at t = q − equals q − dg · X · ∆ / ( A ( k ) · RAUER–SIEGEL THEOREM FOR ELLIPTIC SURFACES 7 A ∨ ( k )), where A ∨ stands for the dual abelian variety. According to[Mi, Th. 3], this leads to a formula similar to (1) X · ∆ = q dg Y α i =1 /γ j (1 − α i γ j ) , where α i ( i = 1 , . . . , d ) are the (normalized) Frobenius roots of A . Therefore the case r A = 0 is treated, word for word, as in the case d = 1 . If r A > , we have to prove that r A g X → g X → ∞ , and then applythe same argument as for elliptic curves. Assume the contrary, i.e.,lim g X →∞ r A g X = c > . Note that the Mordell–Weil group A ( K ) /A ( K ) tors isisomorphic to Hom k ( J X , A ) . This implies that at least one Frobeniusroot of J X (or of X , which is the same) appears with the multiplicityproportional to g. As in the one-dimensional case, we then consider theWeil measure µ Ω and see that its limit as g → ∞ has discontinuousdensity which contradicts [TV1].The theorem is proved. (cid:3) Generalizations
In this section we shall describe some possible generalizations of The-orem 2.1. To make our approach more clear, we shall first restrict our-selves to considering the case where E is an elliptic K -curve. Denote by E the corresponding elliptic surface (this means that there is a properconnected smooth morphism f : E → X with the generic fibre E ). As-sume that f fits into an infinite Galois tower, i.e. into a commutativediagram of the following form:(3) E = E ←−−− E ←−−− . . . ←−−− E j ←−−− . . . y f y y X = X ←−−− X ←−−− . . . ←−−− X j ←−−− . . . , where each lower horizontal arrow is a Galois covering. Let us introducesome notation. For every v ∈ X , let E v = f − ( v ), let r v,i denote thenumber of points of X i lying above v , β v = lim i →∞ r v,i /g i (we supposethe limits exist). Furthermore, denote by f v,i the residue degree of apoint of X i lying above v (the tower being Galois, this does not depend B. `E. KUNYAVSKI˘I AND M. A. TSFASMAN on the point), and let f v = lim i →∞ f v,i . If f v = ∞ , we have β v = 0.If f v is finite, denote by N ( E v , f v ) the number of F q fv -points of E v .Finally, let τ denote the “fudge” factor in the Birch and Swinnerton-Dyer conjecture (see [Ta] for its precise definition). Under this setting,we dare formulate the following Conjecture 3.1.
Assuming the Birch and Swinnerton-Dyer conjecturefor elliptic curves over function fields, we have lim g →∞ g log q ( X · ∆ · τ ) = 1 − X v ∈ X β v log q N ( E v , f v ) q f v . Remark 3.2.
One can check that in the constant case Conjecture 3.1 isconsistent with Theorem 2.1. The first nontrivial case to be consideredis that of an isotrivial elliptic surface.Here are some questions for further investigation.
Question 3.3.
How can one formulate an analogue of Conjecture 3.1for more general towers when diagram (3) does not commute? for moregeneral families when there are no upper horizontal arrows in diagram(3)?With an eye towards even further generalizations of the Brauer–Siegel theorem to arbitrary commutative algebraic groups, the nextextreme case to be considered is that of algebraic tori. In that case theanalogues of the class number and the regulator are known [Ono], [Vo].Moreover, there is an analogue of the analytic class number formulaof Dirichlet established in [Shyr] for tori over number fields. Togetherwith Theorem 2.2, this motivates the following
Conjecture 3.4.
Let T = T × F q K , where T is a fixed F q -torus. Then lim g →∞ g log h ( T ) = lim g →∞ g log p D T − ∞ X m =1 β m log q N m ( T ) q md , where d = dim T, N m ( T ) = | T ( F q m ) | , D T is the “quasi-discriminant”of T ( cf. [Shyr]) , and all other notation is as in the previous sections. RAUER–SIEGEL THEOREM FOR ELLIPTIC SURFACES 9
Acknowledgement.
A substantial part of this work was done duringthe visits of the first named author to the Mediterranean Universityand the Institute of Mathematics of Luminy in 2003 and 2007 andthe visit of the second named author to Bar-Ilan University in 2005.Hospitality and support of these institutions are gratefully appreciated.The authors thank A. Zykin for useful discussions and the referee forhelpful remarks.
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E-mail address : [email protected] Tsfasman: French-Russian Poncelet Laboratory; Institut de Math´e-matiques de Luminy; Independent University of Moscow; and Institutefor Information Transmission Problems
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