Analogue of the Brauer–Siegel theorem for Legendre elliptic curves

Griffon, Richard

doi:10.1016/j.jnt.2018.05.006

Cited by 8 publications

(6 citation statements)

References 12 publications

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“…as shown in [51,Lemma 3.7]. Recent works of Hindry [51], Hindry-Pacheco [52] and Griffon [39,40,41,42] on the analogue of the Brauer-Siegel estimate for abelian varieties show that the numerator of ( 16) is also expected to be comparable (in some cases) to H(A). Hence it is necessary to gain further evidence in order to be able to decide if a Northcott property for the special value L * (A, 1) associated to abelian varieties holds in some cases.…”

Section: Special Values Inside the Critical Strip: Abelian Varietiesmentioning

confidence: 69%

On the Northcott property for special values of L-functions

Pazuki¹,

Pengo²

2020

Preprint

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We propose an investigation on the Northcott, Bogomolov and Lehmer properties for special values of L-functions. We first introduce an axiomatic approach to these three properties. We then focus on the Northcott property for special values of L-functions. We prove that such a property holds for the special value at zero of Dedekind zeta functions of number fields. In the case of L-functions of pure motives, we prove a Northcott property for special values located at the left of the critical strip, assuming the validity of the functional equation.

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Section: Special Values Inside the Critical Strip: Abelian Varietiesmentioning

confidence: 69%

On the Northcott property for special values of L-functions

Pazuki¹,

Pengo²

2020

Preprint

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“…We finish by remarking that Griffon has shown [Gri15] that if E d is the twist of a constant ordinary E/K by the quadratic extension F q (t, √ t d + 1), then as d runs through "supersingular" integers, i.e., those that divide p f + 1 for some f , the limit of BS(E d ) is 1. In conjunction with Theorem 12.4, this shows that the Brauer-Siegel ratio of an elliptic curve E ′ may be large even when the dimension of X(E ′ ) is zero.…”

Section: 3mentioning

confidence: 95%

On the Brauer–Siegel ratio for abelian varieties over function fields

Ulmer

2019

Alg. Number Th.

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A. Hindry has proposed an analogue of the classical Brauer-Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell-Weil group and the order of the Tate-Shafarevich group should have size comparable to the exponential differential height. Hindry-Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate-Shafarevich group and the regulator. We recover the results of Hindry-Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.

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“…We note that there are several sequences of elliptic curves for which similar behaviour has been described. See [HP16,Gri16,Gri18,Gri19,GU20] For a fixed pair (a, b), the genus g of C = C a,b,q is constant as q varies. Hence the term log r g / log H(J) is o(1) as q → ∞.…”

Section: Analogue Of the Brauer-siegel Theoremmentioning

confidence: 99%

On the arithmetic of a family of superelliptic curves

Arpin¹,

Griffon²,

Taylor³

et al. 2021

Preprint

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Let p be a prime, let r and q be powers of p, and let a and b be relatively prime integers not divisible by p. Let C/F r (t) be the superelliptic curve with affine equation y b + x a = t q − t. Let J be the Jacobian of C. By work of Pries-Ulmer [PU16], J satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon-Ulmer [GU20], we compute the L-function of J in terms of certain Gauss sums. In addition, we estimate several arithmetic invariants of J appearing in BSD, including the rank of the Mordell-Weil group J(F r (t)), the Faltings height of J, and the Tamagawa numbers of J in terms of the parameters a, b, q. For any p and r, we show that for certain a and b depending only on p and r, these Jacobians provide new examples of families of simple abelian varieties of fixed dimension and with unbounded analytic and algebraic rank as q varies through powers of p. Under a different set of criteria on a and b, we prove that the order of the Tate-Shafarevich group X(J) is "large" as q → ∞.

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Analogue of the Brauer–Siegel theorem for Legendre elliptic curves

Cited by 8 publications

References 12 publications

On the Northcott property for special values of L-functions

On the Northcott property for special values of L-functions

On the Brauer–Siegel ratio for abelian varieties over function fields

On the arithmetic of a family of superelliptic curves

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