The special values of the zeta functions associated with Hilbert modular forms

Shimura, Goro

doi:10.1007/978-1-4612-2060-2_3

Cited by 116 publications

(270 citation statements)

References 15 publications

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“…Here we are using the same shorthand as in [Shi78,§1]. The modularity condition implies that f has a Fourier expansion…”

Section: Introductionmentioning

confidence: 99%

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On the rank one abelian Gross–Stark conjecture

Ventullo¹

2015

Comment. Math. Helv.

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Abstract. Let F be a totally real number field, p a rational prime, and χ a finite order totally odd abelian character of Gal(F /F ) such that χ(p) = 1 for some p|p. Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the p-adic L-function associated to χ at its exceptional zero and the p-adic logarithm of a p-unit in the χ component of F × χ . In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture assuming two conditions: that Leopoldt's conjecture holds for F and p, and that if there is only one prime of F lying above p, a certain relation holds between the L -invariants of χ and χ −1 . The main result of this paper removes both of these conditions, thus giving an unconditional proof of the conjecture.

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“…Here we are using the same shorthand as in [Shi78,§1]. The modularity condition implies that f has a Fourier expansion…”

Section: Introductionmentioning

confidence: 99%

“…We will usually write this as f = (f c ) c . For a ray class character χ of conductor dividing n, we will say f has character χ if S(a)f = χ(a) for almost all prime ideals a of F (see [Shi78,p. 648]).…”

Section: Introductionmentioning

confidence: 99%

On the rank one abelian Gross–Stark conjecture

Ventullo¹

2015

Comment. Math. Helv.

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“…In this direction, let us mention Blasius's result [4], which proves the conjecture for the motives attached to algebraic Hecke characters of CM fields (see also [15]). Let us also mention Shimura's works on critical values of L-functions in the case of modular forms and Hilbert modular forms (see for instance [28] and [29]), based on the classical Rankin-Selberg theory of L-functions and an analysis of the arithmetic properties of ratios of Petersson inner products. Following these works, Harris in [17] treated the case of the polarized regular motives over Q coming from automorphic representations of GL n (A Q ).…”

Section: Then For All Critical Integers K > W + N Of M(χ ) We Have Lmentioning

confidence: 99%

On critical values of L-functions of potentially automorphic motives

Salazar

Guerberoff

2016

manuscripta math.

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“…Finally, for primes q n, denote by S q ∈ T n the spherical operator at q. For the definitions of these operators, see [42]. Denote by T new n/r,r the quotient of T n acting faithfully on S new 2 (n/r, r) and set T new n := T new O F ,n .…”

Section: Hilbert Modular Formsmentioning

confidence: 99%

“…the maps defining it (for more details, see [42]). Following Ribet [40], define a modular form ψ ∈ S old 2 (n, ) by:…”

Section: Congruences Between Modular Formsmentioning

confidence: 99%

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

Longo¹

2006

Ann. inst. Fourier

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Let E/F be a modular elliptic curve defined over a totally real number field F and let f be its associated eigenform. This article presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of E over suitable quadratic imaginary extensions K/F. In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachev, that is, when [F : Q] is even and phi not new at any prime

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The special values of the zeta functions associated with Hilbert modular forms

Cited by 116 publications

References 15 publications

On the rank one abelian Gross–Stark conjecture

On the rank one abelian Gross–Stark conjecture

On critical values of L-functions of potentially automorphic motives

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

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