The rank of Mazur’s Eisenstein ideal

Class groups of Kummer extensions via cup products in Galois cohomology

“…Theorem 4.1.1 establishes that H 1 Σ (F p (−i)) is non-zero if and only if S i is a pth power, for regular p and odd i ≡ −1 mod p − 1. A similar relationship was known to Wake-Wang-Erickson in the case i ≡ 1 mod p − 1; see Theorem 12.5.1 of [10].…”

Higher congruences between newforms and Eisenstein series of squarefree level

“…Lastly, we note that work of Wake and Wang-Erickson [17,18], Ribet-Yoo [19], and Ohta [15] independently study similar questions about various Eisenstein ideals.…”

“…, f r of level N fi dividing N and the weight 2 Eisenstein series E 2,N . Since these forms are normalized eigenforms for all Hecke operators T ℓ with ℓ ∤ N prime, this is equivalent to computing congruences between Fourier coefficients, i.e., congruences of the type (17) a…”

Deformation Conditions for Pseudorepresentations

“…We were stimulated to develop this theory for application in the companion paper [WWE17b] (see also [WWE18b]). After this work was complete, the second-named author identified Galois cohomological data that controls the deformation theory of these pseudodeformation rings: see [WE18b], especially [WE18b, Theorem 3.4.1].…”

“…In particular, it allows one to give a kind of upper bound on the size of the pseudodeformation ring in terms of Selmer-type groups, and this is useful in proving modularity-lifting theorems (see Remark 4.0.1). This method is used in our papers [WWE17b,WWE18b]. REMARK 1.2.1.…”