Power series expansions of modular forms and p-adic interpolation of the square roots of Rankin–Selberg special values

Computing power series expansions of modular forms

We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra. As applications, we compute Shimura curve parametrizations of elliptic curves over a totally real field, including the image of CM points, and equations for Shimura curves.

p-adic interpolation of square roots of central values of Hecke L-functions

p-adic measures and square roots of special values of triple product L-functions

Introduction Let p be a prime number. In this note, we combine the methods of Hida with the results of [HK1] to define a p-adic analytic function, the squares of whose special values are related to the values of triple product L-functions at their centers of symmetry. More precisely, let f , g, and h be classical normalized cuspidal Hecke eigenforms of level 1 and (even) weights k, l, and m, re...

p-adic interpolation of half-integral weight modular forms

The p-adic interpolation of modular forms on congruence subgroups of SL2(Z) has been succesfully used in the past to interpolate values of L-series. In [12], Serre interpolated the values at negative integers of the ζ-series of a totally real number field (in fact of L-series of powers of the Teichmuller character) by interpolating Eisenstein series, which are holomorphic modular forms, and loo...

Behavior of $R$-groups for $p$-adic inner forms of quasi-split special unitary groups

‎We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups‎. ‎We prove Arthur's conjecture‎, ‎the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group‎, ‎for quasi-split special unitary groups and their inner forms‎. ‎Furthermore‎, ‎we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms‎. ‎This w...