Euler Products and Eisenstein Series

Shimura, Goro

doi:10.1090/cbms/093

Cited by 126 publications

(221 citation statements)

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“…This Eisenstein series is the Eisenstein series studied by Shimura in [37] and [38]. In particular, one knows that E f converges absolutely and uniformly for (g, s) on compact subsets of G n (A) × {s ∈ C : Re(s) > (n + 1)/2}, it defines an automorphic form on G n (A) and a holomorphic function on {s ∈ C : Re(s) > (n + 1)/2} that has meromorphic continuation to C with at most finitely many poles.…”

Section: Siegel Eisenstein Series -A Choice Of Sectionmentioning

confidence: 87%

On the Bloch–Kato conjecture for elliptic modular forms of square-free level

Agarwal

Brown

2013

Math. Z.

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Abstract. Let κ ≥ 6 be an even integer, M an odd square-free integer, and f ∈ S2κ−2(Γ0(M )) a newform. We prove that under some reasonable assumptions that half of the λ-part of the Bloch-Kato conjecture for the near central critical value L(κ, f ) is true. We do this by bounding the ℓ-valuation of the order of the appropriate Bloch-Kato Selmer group below by the ℓ-valuation of algebraic part of L(κ, f ). We prove this by constructing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel modular form.

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Section: Siegel Eisenstein Series -A Choice Of Sectionmentioning

confidence: 87%

On the Bloch–Kato conjecture for elliptic modular forms of square-free level

Agarwal

Brown

2013

Math. Z.

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“…11.2]); it also follows from [Sh97, Prop. A3.7] (see also the paragraph preceeding Lemma 18.8 of [Sh97]). …”

Section: Siegel Eisenstein Series and Their Pull-backsmentioning

confidence: 99%

The Iwasawa Main Conjectures for GL2

2013

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“…In the following we write <·, ·> for the adelic Petersson inner product (see for example [9] for the definition) and we denote by S k (Q) the space of weight k cusp forms with algebraic Fourier coefficients and of any congruence subgroup. Proof As in [1] we pick a half integer σ 0 so that 3n/2 + 1 < σ 0 < m 0 and m 0 − σ 0 / ∈ 2Z and define μ ∈ Z d by the conditions 0 ≤ μ v ≤ 1 and σ 0 −k v +μ v ∈ 2Z for all v ∈ ∞.…”

Section: Lemma 51 the Hecke Operators T (A) Preservementioning

confidence: 99%