Computing Galois representations and equations for modular curves $X_H(\ell)$

Derickx, Maarten; van Hoeij, Mark; Zeng, Jinxiang

doi:10.48550/arxiv.1312.6819

Cited by 2 publications

(3 citation statements)

References 9 publications

(20 reference statements)

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“…For future use, it will be of value for us to record some basic arithmetic facts about τ (n); these are taken from Swinnerton-Dyer's article [37]. Here σ v (n) denotes the sum of the v-th powers of the divisors of n. 23) for other p = 23 (16) τ (n) ≡ σ 11 (n) (mod 691).…”

Section: Congruences For the τ Functionmentioning

confidence: 99%

“…The restriction that n has no prime divisors p for which τ (p) = 0 is, in fact, necessary if one wishes to obtain a lower bound upon P (τ (n)) that tends to ∞ with n. Indeed, one may observe that, if τ (p) = 0, then (see (18) below) P τ (p 2k ) = P (−1) k p 11k = p is bounded independently of k. While Lehmer's conjecture remains unproven, we do know that if there is a prime p for which τ (p) = 0, then (6) p > 816212624008487344127999, by work of Derickx, van Hoeij and Zeng [16]. Theorem 2 is an easy consequence of the following result.…”

Section: Introductionmentioning

confidence: 96%

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Odd values of the Ramanujan tau function

Bennett¹,

Gherga²,

Patel³

et al. 2021

Preprint

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We prove a number of results regarding odd values of the Ramanujan τ -function. For example, we prove the existence of an effectively computable positive constant κ such that if τ (n) is odd and n ≥ 25 then eitherlog log log n log log log log n or there exists a prime p | n with τ (p) = 0. Here P (m) denotes the largest prime factor of m. We also solve the equation τ (n) = ±3 b 1 5 b 2 7 b 3 11 b 4 and the equations τ (n) = ±q b where 3 ≤ q < 100 is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue-Mahler equations due to Bugeaud and Győry, and the modular approach via Galois representations of Frey-Hellegouarch elliptic curves.

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Section: Congruences For the τ Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Odd values of the Ramanujan tau function

Bennett¹,

Gherga²,

Patel³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…We also refer to the Corollary 1.2 of the unpublished article [4], which claims that τ (n) = 0 for all n ≤ 816212624008487344127999 ≈ 8 • 10 23 .…”

Section: Introductionmentioning

confidence: 99%

Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms

Bilu¹,

Deshouillérs²,

Gun³

et al. 2018

Compositio Math.

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Let τ (·) be the classical Ramanujan τ -function and let k be a positive integer such that τ (n) = 0 for 1 ≤ n ≤ k/2. (This is known to be true for k < 10 23 , and, conjecturally, for all k.) Further, let σ be a permutation of the set {1, ..., k}. We show that there exist infinitely many positive integers m such that |τ (m + σ(1))| < |τ (m + σ(2))| < ... < |τ (m + σ(k))|.We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.

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Computing Galois representations and equations for modular curves $X_H(\ell)$

Abstract: We construct plane models of the modular curve X H (ℓ) and describe the moduli interpretation on these plane models. We use these explicit plane models to compute Galois representations associated to modular forms for values of ℓ that are significantly higher than in prior works.

Cited by 2 publications

References 9 publications

Odd values of the Ramanujan tau function

Odd values of the Ramanujan tau function

Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms

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