Powers of 2 in modular degrees of modular abelian varieties

Dummigan, Neil; Krishnamoorthy, Srilakshmi

doi:10.1016/j.jnt.2012.08.024

Cited by 3 publications

(2 citation statements)

References 14 publications

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“…By Remark 3.1 W ′ has at most index 2 in W(n). Now, as in Proposition 2.1 in [7] we construct a homomorphism θ : W ′ → E(K) [2]. First of all, we fix a K-rational point x 0 ∈ X 0 (n), then for…”

Section: Watkins' Conjecture For Semi-stable Elliptic Curvesmentioning

confidence: 99%

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Watkins' conjecture for elliptic curves over function fields

Caro¹

2022

Preprint

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Watkins conjectured that given an elliptic curve defined over Q, its Mordell-Weil rank is at most the 2-adic valuation of its modular degree. We consider the analogous problem over function fields of positive characteristic, and we prove it in several cases. More precisely, every modular semi-stable elliptic curve over Fq(T ) after extending constant scalars, and every quadratic twist of a modular elliptic curve over Fq(T ) by a polynomial with sufficiently many prime factors satisfy the analogue of Watkins' conjecture. Furthermore, for a well-known family of elliptic curves with unbounded rank due to Ulmer, we prove the analogue of Watkins' conjecture.

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Section: Watkins' Conjecture For Semi-stable Elliptic Curvesmentioning

confidence: 99%

“…Using Atkin-Lehner involutions we prove a potential version of Watkins' conjecture for semistable elliptic curves over K (see [7] and [3] for other applications of Atkin-Lehner involutions in the context of Watkins' conjecture).…”

Section: Introductionmentioning

confidence: 99%

Watkins' conjecture for elliptic curves over function fields

Caro¹

2022

Preprint

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Watkins’s conjecture for elliptic curves over function fields

Caro

2024

Math. Z.

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Watkins’s conjecture for elliptic curves with non-split multiplicative reduction

Caro

Pasten

2022

Proc. Amer. Math. Soc.

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Let E E be an elliptic curve over the rational numbers. Watkins [Experiment. Math. 11 (2002), pp. 487–502 (2003)] conjectured that the rank of E E is bounded by the 2 2 -adic valuation of the modular degree of E E . We prove this conjecture for semistable elliptic curves having exactly one rational point of order 2 2 , provided that they have an odd number of primes of non-split multiplicative reduction or no primes of split multiplicative reduction.

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Powers of 2 in modular degrees of modular abelian varieties

Cited by 3 publications

References 14 publications

Watkins' conjecture for elliptic curves over function fields

Watkins' conjecture for elliptic curves over function fields

Watkins’s conjecture for elliptic curves over function fields

Watkins’s conjecture for elliptic curves with non-split multiplicative reduction

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