Special values of anticyclotomic L-functions for modular forms

Special Values of Anticyclotomic L-functions

The purpose of the paper is to extend and refine earlier results of the author on nonvanishing of the L-functions associated to modular forms in the anticyclotomic tower of conductor p over an imaginary quadratic field. While the author’s previous work proved that such L-functions are generically nonzero at the center of the critical strip, provided that the sign in the functional equation is +...

Congruences for Fourier Coefficients of Half-integral Weight Modular Forms and Special Values of L−functions

Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different type. Let f(z) = P∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coefficients. If ` is prime, then we shall be interested in congruences of the form

ON ANTICYCLOTOMIC ì-INVARIANTS OF MODULAR FORMS

On Anticyclotomic Μ-invariants of Modular Forms

We prove the μ-part of the main conjecture for modular forms along the anticyclotomic Zp-extension of a quadratic imaginary field. Our proof consists of first giving an explicit formula for the algebraic μ-invariant, and then using results of Ribet and Takahashi showing that our formula agrees with Vatsal’s formula for the analytic μ-invariant.

Special Values of the Standard Zeta Functions for Elliptic Modular Forms

Let M and k be positive integers and φ a Dirichlet character modulo M. For a normalized cuspidal Hecke eigenform f of weight k and Nebentypus φ with respect to Γ0(M), and a Dirichlet character χ modulo N, let L(f, s, χ) be the standard zeta function of f twisted by χ. (For the precise definition of the standard zeta function, see the paragraph immediately preceding Theorem 3.3.) The twisted sta...