Values of \(p\)-adic \(L\)-functions and a \(p\)-adic Poisson kernel
type: Item
instance of: scholarly article
title: Values of \(p\)-adic \(L\)-functions and a \(p\)-adic Poisson kernel
zbMATH Open document ID: 0731.11065
DOI: 10.1007/BF01231508
author: Jeremy Teitelbaum
published in: Inventiones Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143809
review text: Let \(B\) be a definite quaternion algebra over \({\mathbb Q}\) of discriminant \(N\) and let \(p\) be a prime which splits \(B\). Let \({\mathcal O}\) be a maximal \({\mathbb Z}[1/p]\)-order in \(B\) and let \(\Gamma\) be a congruence subgroup of \({\mathcal O}^*\). The author denotes by \(S_ k(\Gamma)\) the space of rigid modular forms on the \(p\)-adic upper half plane \({\mathcal H}_ p\), which is viewed as a rigid analytic subspace of \({\mathbb P}^ 1\). Let \({\mathbb T}\) be the Hecke algebra for \(\Gamma\). As a \({\mathbb T}\)-module, the space \(S_ k(\Gamma)\) is isomorphic to a space of modular forms for the indefinite quaternion algebra with discriminant \(N_ p\). Associated to a modular form \(f\in S_ k(\Gamma)\) is a harmonic cocycle \(c_ f\) which, in fact, determines \(f\). This is a type of harmonic function defined on the tree \({\mathcal T}\) of \(SL_ 2({\mathbb Q}_ p).\) Let \(P_ k\) be the \(k+1\) dimensional vector space of degree \(k\) polynomials in \(T\) over the field \({\mathbb C}_ p\). The author considers two maps \(\psi\),\(\lambda: S_ k(\Gamma)\to H^ 1(\Gamma,P_{k-2})\), defined from the harmonic cocycles and from the Coleman \(p\)-adic integral, respectively. If \(\psi^{\text{new}}\) and \(\lambda^{\text{new}}\) denote the restriction of \(\psi\) and \(\lambda\) to the new part of \(S_ k(\Gamma)\), then he proves that there exists an element \({\mathcal L}\) in \({\mathbb T}\otimes {\mathbb Q}_ p\) such that \(\lambda^{\text{new}}={\mathcal L}\psi^{\text{new}}\). This invariant \({\mathcal L}\) can be thought of as a ``period'' associated to \(f\) and it generalizes the \({\mathcal L}\) invariant for weight two modular forms constructed by \textit{B. Mazur}, \textit{J. Tate} and the author [(MTT) Invent. Math. 84, 1--48 (1986; Zbl 0699.14028)]. In particular, when \(f\) is a form of weight two, the invariant \({\mathcal L}\) can be derived from the \(p\)-adic period matrix of the Jacobian of \({\mathbb H}_ p/\Gamma.\) The author intro…
reviewed by: Pilar Bayer
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 4208202
zbMATH Keywords: quaternion algebra
zbMATH Keywords: p-adic L function
zbMATH Keywords: periods
zbMATH Keywords: rigid analytic modular forms
zbMATH Keywords: harmonic cocycles
zbMATH Keywords: Coleman integrals
zbMATH Keywords: Shimura curves
zbMATH Keywords: special values of p-adic L-functions
zbMATH Keywords: modular forms of even weight
zbMATH Keywords: modular symbols
MaRDI profile type: Publication
cites work: UNIFORMIZATION OF ALGEBRAIC CURVES BY DISCRETE ARITHMETIC SUBGROUPS OF $ PGL_2(k_w)$ WITH COMPACT QUOTIENTS
cites work: Dilogarithms, regulators and \(p\)-adic \(L\)-functions
cites work: ELLIPTIC MODULES
cites work: Q5183633
cites work: Eichler cohomology and periods of modular forms on p-adic Schottky groups.
cites work: Schottky groups and Mumford curves
cites work: Automorphic forms on GL (2)
cites work: p-adic automorphic functions
cites work: Refined conjectures of the ``Birch and Swinnerton-Dyer type''
cites work: On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer
cites work: Q3884232
cites work: p-adic periods of genus two Mumford-Schottky curves.
description: scientific article; zbMATH DE number 4208202