Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras

Lecouturier, Emmanuel

doi:10.1007/s00222-020-00996-1

Cited by 9 publications

(20 citation statements)

References 24 publications

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“…Note that Corollary A has already been proved by Wake and Wang-Erickson in [22] and by Lecouturier in [14] using different methods. However, Wake and Wang-Erickson use results of [13] to prove that the second part of Corollary A implies the first part.…”

Section: So In Particularmentioning

confidence: 82%

“…In [14], Lecouturier extended Merel's result by combining same circle of ideas with new methods. In particular, he gave a necessary and sufficient condition for the Z p -rank of T 0 m to be greater than 2 in terms of a numerical invariant which is similar to the Merel's invariant mentioned above (see [14,Theorem 1.2]). He also gave an alternate proof of Merel's result in [14].…”

Section: Introductionmentioning

confidence: 98%

“…In particular, he gave a necessary and sufficient condition for the Z p -rank of T 0 m to be greater than 2 in terms of a numerical invariant which is similar to the Merel's invariant mentioned above (see [14,Theorem 1.2]). He also gave an alternate proof of Merel's result in [14].…”

Section: Introductionmentioning

confidence: 99%

“…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%

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

Deo¹

2021

Preprint

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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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Section: So In Particularmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Deo¹

2021

Preprint

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“…(i) Theorem 2.1 is analogous to [3,Theorem 4.3], where de Shalit deforms the filtration (5) (notice that our character • is inverse to the one of de Shalit in this theorem). The main difference here is that we have replaced ( 5) by (8). Another important difference is that the Λ-module W 1 is free, while the analogous statement is false in de Shalit's situation.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

On the Mazur--Tate conjecture for prime conductor and Mazur's Eisenstein ideal

Lecouturier

2019

Preprint

Self Cite

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In 1995, Ehud de Shalit proved an analogue of a conjecture of Mazur-Tate for the modular Jacobian J 0 (p). His main result was valid away from the Eisenstein primes. We complete the work of de Shalit by including the Eisenstein primes, and give some applications such as an elementary combinatorial identity involving discrete logarithms of difference of supersingular j-invariants. An important tool is our recent work on the so called "generalized cuspidal 1-motive".

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THE EISENSTEIN IDEAL OF WEIGHT k AND RANKS OF HECKE ALGEBRAS

Deo

2023

J. Inst. Math. Jussieu

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Let p and $\ell $ be primes such that $p> 3$ and $p \mid \ell -1$ 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 $\Gamma _0(\ell )$ at the maximal Eisenstein ideal containing p. We give a necessary and sufficient condition for the $\mathbb {Z}_p$ -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 $R=\mathbb {T}$ theorems under certain hypotheses.

show abstract

Higher Eisenstein elements, higher Eichler formulas and rank of Hecke algebras

Cited by 9 publications

References 24 publications

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

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

On the Mazur--Tate conjecture for prime conductor and Mazur's Eisenstein ideal

THE EISENSTEIN IDEAL OF WEIGHT k AND RANKS OF HECKE ALGEBRAS

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