The Eisenstein ideal with squarefree level

Wake, Preston; Wang-Erickson, Carl

doi:10.1016/j.aim.2020.107543

Cited by 16 publications

(15 citation statements)

References 24 publications

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“…Now, let N be a non-squarefree integer, i.e., N 2 > 1. As noticed by many mathematicians (for example, see [22,Sec. 1.4.3]), when ℓ N, the Atkin-Lehner operator w ℓ is more convenient for studying Eisenstein ideals than the Hecke operator T ℓ .…”

Section: 3mentioning

confidence: 85%

The rational torsion subgroup of $J_0(N)$

Yoo¹

2021

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HWAJONG YOO A. The main result of this paper is to determine the structure of the rational torsion subgroup of the modular Jacobian variety J 0 (N ) for any positive integer N up to finitely many primes. More precisely, we prove that the prime-to-2n part of the rational torsion subgroup of J 0 (N ) is equal to that of the rational cuspidal divisor class group of X 0 (N ), where n is the largest perfect square dividing 3N . As the rational cuspidal divisor class group of X 0 (N ) is already computed in [28], it determines the structure of the rational torsion subgroup of J 0 (N ) up to primes dividing 2n. CConjecture 1.3. For any positive integer N, we haveBefore proceeding, we recall several results on the conjectures above. For simplicity, let C(N) (n) and J 0 (N)(Q)(n) tors denote the prime-to-n parts of C(N) and J 0 (N)(Q) tors , respectively. Also, letdenote the p-primary subgroups of C(N) and J 0 (N)(Q) tors , respectively.(1) For a prime p ≥ 5 such that p ≡ 11 (mod 12) and r ≥ 2, we haveby Lorenzini (1995) [11, Th. 4.6].(2) For any prime p and r ≥ 3, we haveby Ling (1997) [10, Th. 4].(3) For a squarefree integer N, we havewhere n = gcd(3, N) by Ohta (2014) [16, Th.].1 By a rational cuspidal divisor, we mean a divisor supported only on the cusps and fixed under the action of Gal(Q/Q).

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Section: 3mentioning

confidence: 85%

The rational torsion subgroup of $J_0(N)$

Yoo¹

2021

Preprint

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“…For level Γ 0 (M ) with M prime, Mazur [Maz77] gave a necessary and sufficient condition for an Eisenstein congruence to exist. For squarefree M , partial necessary and sufficient conditions have been found by Ribet [Rib10] (see also [Yoo19a]) using geometry of modular Jacobians, and by the second author and Wang-Erickson [WWE21] using Galois deformation theory. For some nonsquarefree M , sufficient conditions have been proven by Martin [Mar17] using Jaquet-Langlands theory and necessary conditions by Yoo [Yoo19b] using geometry of modular Jacobians.…”

Section: Theorem B Assume That N ≡ −1 Mod Pmentioning

confidence: 99%

“…Calegari explains how to use cup products to prove Kobayashi's conjecture in full generality [Cal]. We believe a modular approach is also possible by combining the methods of the current paper with those of the second author and Wang-Erickson in [WWE21], but we have elected not to do so in this paper for simplicity. △ 1.5.…”

Section: Theorem B Assume That N ≡ −1 Mod Pmentioning

confidence: 99%

A modular construction of unramified $p$-extensions of $\mathbb{Q}(N^{1/p})$

Lang¹,

Wake²

2021

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We show that for primes N, p ≥ 5 with N ≡ −1 mod p, the class number of Q(N 1/p ) is divisible by p. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when N ≡ −1 mod p, there is always a cusp form of weight 2 and level Γ 0 (N 2 ) whose ℓ-th Fourier coefficient is congruent to ℓ + 1 modulo a prime above p, for all primes ℓ. We use the Galois representation of such a cusp form to explicitly construct an unramified degree p extension of Q(N 1/p ).

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“…As f is of level Γ 0 (ℓ), we know that ρ f | G Q ℓ is either unramified or Steinberg. Hence, the pseudo-representation attached to ρ f is Steinbergor-unramified at ℓ (see [23,Observation 1.9.2]). This proves the desired properties of the pseudo-representation (τ ℓ , δ ℓ ).…”

Section: Lemma 41 There Exists a Pseudo-representationmentioning

confidence: 99%

“…The key step in both these works is a suitable R = T theorem. In [23], Wake and Wang-Erickson studied this question in the case of squarefree level. We refer the reader to the well-written introductions of [17], [6], [22] [23] and [14] for a summary of the known results, nice exposition of various approaches to the problem and their comparison.…”

Section: Introductionmentioning

confidence: 99%

The Eisenstein ideal of weight $k$ and ranks of Hecke algebras

Deo¹

2021

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Let p and ℓ be primes such that p > 3 and p | ℓ − 1 and k be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight k and level Γ0(ℓ) at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the Zp-rank of this Hecke algebra to be greater than 1 in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for k = 2 using our methods. In addition, we prove some R = T theorems under certain hypothesis.

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The Eisenstein ideal with squarefree level

Cited by 16 publications

References 24 publications

The rational torsion subgroup of $J_0(N)$

The rational torsion subgroup of $J_0(N)$

A modular construction of unramified $p$-extensions of $\mathbb{Q}(N^{1/p})$

The Eisenstein ideal of weight $k$ and ranks of Hecke algebras

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