L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields

Ulmer, Douglas

doi:10.1007/s00222-006-0018-x

Cited by 21 publications

(29 citation statements)

References 19 publications

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“…We again apply [Ulm07], and the members of the 1-parameter family of elliptic curves E (a,a) /F q (t 1/d ) obtain arbitrarily large rank as d → ∞, which completes the proof of our theorem.…”

Section: Large Ranksmentioning

confidence: 68%

“…We believe that curves of arbitrary genus may be constructed using our methods, producing abelian varieties of any dimension that satisfy BSD. By a further analysis of the conductors of the Galois representations that arise, we expect to apply results from [Ulm07] to show that some of these varieties obtain large rank.…”

Section: Bidegree (M N) Curvesmentioning

confidence: 98%

“…Ulmer [Ulm07] proves that certain L-series vanish to high order at s = 1 in towers of function fields. [Ulm07].)…”

Section: Analytic Ranksmentioning

confidence: 99%

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Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields

Berger

2008

Journal of Number Theory

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With the goal of producing elliptic curves and higher-dimensional abelian varieties of large rank over function fields, we provide a geometric construction of towers of surfaces dominated by products of curves; in the case where the surface is defined over a finite field our construction yields families of smooth, projective curves whose Jacobians satisfy the conjecture of Birch and Swinnerton-Dyer. As an immediate application of our work we employ known results on analytic ranks of abelian varieties defined in towers of function field extensions, producing a one-parameter family of elliptic curves over F q (t 1/d ) whose members obtain arbitrarily large rank as d → ∞.

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Section: Large Ranksmentioning

confidence: 68%

Section: Bidegree (M N) Curvesmentioning

confidence: 98%

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Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields

Berger

2008

Journal of Number Theory

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“…The first instance of a sequence of nonisotrivial elliptic curves with unbounded ranks is due to Ulmer in [Ulm02]. Ulmer subsequently proved a rather general theorem (see [Ulm07]) which ensures, under a parity condition on the conductor, that certain so-called Kummer families of elliptic curves over K have unbounded analytic ranks. In parallel, Berger proposed in [Ber08] a construction of Kummer families of elliptic curves for which the BSD conjecture holds for each curve in the family: in these cases, Ulmer's 'unbounded analytic rank' result mentioned above can be translated into a 'unbounded algebraic rank' theorem.…”

Section: Introductionmentioning

confidence: 99%

A New Family of Elliptic Curves with Unbounded Rank

Griffon¹

2020

MMJ

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Let F q be a finite field of odd characteristic and K = F q (t). For any integer d ≥ 2 coprime to q, consider the elliptic curve E d over K defined by y 2 = x · x 2 + t 2d · x − 4t 2d . We show that the rank of the Mordell-Weil group E d (K) is unbounded as d varies. The curve E d satisfies the BSD conjecture, so that its rank equals the order of vanishing of its L-function at the central point. We provide an explicit expression for the L-function of E d , and use it to study this order of vanishing in terms of d.Of course, the second assertion implies the first one, and our treatment might appear redundant. However, the two proofs shed different lights on the behaviour of the sequence d → rank E d (K) . Indeed, for the explicitly constructed (d n ) n≥1 in Theorem 6.4, the corresponding curves E dn have 'very large' ranks (as large as allowed by Brumer's bound, cf.[Bru92] and our Remark 6.6(b)). However, these integers d n are very far apart from each other, and they cannot be deemed representative of the 'typical' size of rank E d (K) . The average result fills in that gap by showing that the rank of E d is 'usually large'; but one then loses the explicit and precise character of the first construction.The second proof also reveals that the sequence {E d } d at hand is quite special: indeed, Brumer has proved that the average rank of elliptic curves over F q (t) is bounded (see [Bru92]).Let us now explain the strategy of the proof of Theorem A as we give the plan of the paper. In section 1, we start by introducing the elliptic curves E d and by computing their relevant invariants. We then describe the torsion subgroup E d (K) tors (Theorem 1.6), and provide a point of infinite order in E d (K) (Corollary 1.7). We also explain why the results in [Ulm07] cannot be used here.Our first step towards the proof of Theorem A will be to give an explicit formula for the L-function of E d /K. To avoid introducing too many notations here, let us only state the following special case of our result (see Theorem 3.1 for the general version):where, for all n ∈ {1, . . . , 2d − 1} with n = d/2, 3d/2, we letThe relevant objects are introduced in section 2 and the proof of Theorem 3.1 is given in section 3. It is based on direct manipulations of character sums related to 'counting points' on the various reductions of E d . Given that there are very few elliptic curves over K for which the L-function is explicitly known, this Theorem may be of independent interest. The expression of L(E d /K, T ) in Theorem B is sufficiently explicit that one can study its order of vanishing at T = q −1 (see Corollary 3.3). Our main interest in doing so is that the BSD conjecture is known to hold for E d /K i.e., one has:This fact has been proved by Berger in [Ber08], and we sketch her proof in section 4. To this effect, we also briefly recall there how the curves E d are constructed.The expression for ord T =q −1 L(E d /K, T ) obtained in Corollary 3.3 becomes more tractable for certain values of d: specifically, for those d ≥ 1 such that 2d is su...

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“…We hope that the explicit expression for L(E d /K, T ) can be of use for several applications. For example, using Theorem 3.1, one could reprove a result of Ulmer stating that as d ≥ 2 ranges through integers coprime to q, the ranks of the Mordell-Weil groups E d (K) are unbounded (see [Ulm07,[2][3][4]).…”

Section: Introductionmentioning

confidence: 99%

Explicit L-functions and a Brauer–Siegel theorem for Hessian elliptic curves

Griffon¹

2018

J. Théor. Nombres Bordeaux

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For a finite field F q of characteristic p ≥ 5 and K = F q (t), we consider the family of elliptic curves E d over K given by y 2 + xy − t d y = x 3 for all integers d coprime to q. We provide an explicit expression for the L-functions of these curves. Moreover, we deduce from this calculation that the curves E d satisfy an analogue of the Brauer-Siegel theorem. Precisely, we show that, for d → ∞ ranging over the integers coprime with q, one haswhere H(E d /K) denotes the exponential differential height of E d , X(E d /K) its Tate-Shafarevich group and Reg(E d /K) its Néron-Tate regulator. Keywords: Elliptic curves over function fields, Explicit computation of L-functions, Special values of Lfunctions and BSD conjecture, Estimates of special values, Analogue of the Brauer-Siegel theorem.

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L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields

Cited by 21 publications

References 19 publications

Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields

Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields

A New Family of Elliptic Curves with Unbounded Rank

Explicit L-functions and a Brauer–Siegel theorem for Hessian elliptic curves

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