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

Berger, Lisa

doi:10.1016/j.jnt.2008.03.009

Cited by 14 publications

(44 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3). This sequence corresponds to the special case 'a = 1/2' of Berger's example 1 (6) in §4.3 of [Ber08], further studied in §4.4 loc. cit.…”

Section: The Curves E D and Their Invariantsmentioning

confidence: 92%

“…w d = g(v)). In the notations of [Ber08], the pair (f, g) is of type [1, 1][1, 1][1, 1][2] and we have chosen the value 'a = 1/2' for the parameter in f ; see example (6) in §4.3 and example (2) in §4.4 of [Ber08]. Let us summarise the main results of [Ber08] in this case (see also [Ulm13] for a more detailed account).…”

Section: Berger's Construction and The Bsd Conjecturementioning

confidence: 99%

“…By a direct computation, Berger [Ber08,§3] shows that the smooth projective curve X d over K = k(t) which is a model of the curve given in affine coordinates by…”

Section: Berger's Construction and The Bsd Conjecturementioning

confidence: 99%

“…The most striking feature of Berger's construction is the following: Proof: We only sketch the argument and refer the interested reader to [Ber08,§2] 2), it suffices to prove the Theorem in the case where d = d ′ is coprime to q. Hence we assume that d ≥ 1 is coprime to q and we denote by π : E d → P 1 the minimal regular model of E d /K.…”

Section: Berger's Construction and The Bsd Conjecturementioning

confidence: 99%

“…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%

See 4 more Smart Citations

A New Family of Elliptic Curves with Unbounded Rank

Griffon¹

2020

MMJ

View full text Add to dashboard Cite

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...

show abstract

“…3). This sequence corresponds to the special case 'a = 1/2' of Berger's example 1 (6) in §4.3 of [Ber08], further studied in §4.4 loc. cit.…”

Section: The Curves E D and Their Invariantsmentioning

confidence: 92%

Section: Berger's Construction and The Bsd Conjecturementioning

confidence: 99%

“…By a direct computation, Berger [Ber08,§3] shows that the smooth projective curve X d over K = k(t) which is a model of the curve given in affine coordinates by…”

Section: Berger's Construction and The Bsd Conjecturementioning

confidence: 99%

Section: Berger's Construction and The Bsd Conjecturementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A New Family of Elliptic Curves with Unbounded Rank

Griffon¹

2020

MMJ

View full text Add to dashboard Cite

show abstract

Curves and Jacobians over Function Fields

Ulmer

2014

Advanced Courses in Mathematics - CRM Barcelona

View full text Add to dashboard Cite

These notes originated in a 12-hour course of lectures given at the Centre de Recerca Mathemàtica near Barcelona in February, 2010. The aim of the course was to explain results on curves and their Jacobians over function fields, with emphasis on the group of rational points of the Jacobian, and to explain various constructions of Jacobians with large Mordell-Weil rank.More so than the lectures, these notes emphasize foundational results on the arithmetic of curves and Jacobians over function fields, most importantly the breakthrough works of Tate, Artin, and Milne on the conjectures of Tate, Artin-Tate, and Birch and Swinnerton-Dyer. We also discuss more recent results such as those of Kato and Trihan. Constructions leading to high ranks are only briefly reviewed, because they are discussed in detail in other recent and forthcoming publications.These notes may be viewed as a continuation of my Park City notes [69]. In those notes, the focus was on elliptic curves and finite constant fields, whereas here we discuss curves of high genera and results over more general base fields.It is a pleasure to thank the organizers of the CRM Research Program, especially Francesc Bars, for their efficient organization of an exciting meeting, the audience for their interest and questions, the National Science Foundation for funding to support the travel of junior participants (grant DMS 0968709), and the editors for their patience in the face of many delays. It is also a pleasure to thank Marc Hindry, Nick Katz, and Dino Lorenzini for their help. Finally, thanks to Timo Keller and René Pannekoek for corrections.I welcome all comments, and I plan to maintain a list of corrections and supplements on my web page. Please check there for updates if you find the material in these notes useful.Atlanta, Unless explicitly stated otherwise, all schemes are assumed to be Noetherian and separated and all morphisms of schemes are assumed to be separated and of finite type.A curve over a field F is a scheme over F that is reduced and purely of dimension 1, and a surface is similarly a scheme over F which is reduced and purely of dimension 2. Usually our curves and surfaces will be subject to further hypotheses, like irreducibility, projectivity, or smoothness.We recall that a scheme Z is regular if each of its local rings is regular. This means that for each point z ∈ Z, with local ring O Z,z , maximal ideal m z ⊂ O Z,z , and residue field κ(z) = O Z,z /m z , we haveEquivalently, m z should be generated by dim z Z elements. A morphism f : Z → S is smooth (of relative dimension n) at z ∈ Z if there exist affine open neighborhoods U of z and V of f (z) such that f (U ) ⊂ V and a diagram

show abstract