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p-adic L-functions over the false Tate curve extensions

Published online by Cambridge University Press:  22 July 2013

DOHYEONG KIM
DOHYEONG KIM*
Affiliation:
Department of Mathematics, POSTECH, Republic of Korea, 790-784. e-mail: dohyeongkim@ymail.com
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Abstract

Let f be a primitive modular form of CM type of weight k and level Γ0(N). Let p be an odd prime which does not divide N, and for which f is ordinary. Our aim is to p-adically interpolate suitably normalized versions of the critical values L(f, ρχ,n), where n=1,2,. . .,k − 1, ρ is a fixed self-dual Artin representation of M∞ defined by (1.1) below, and χ runs over the irreducible Artin representations of the Galois group of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, if k ≥ 4, we will show that there are only finitely many χ such that L(f, ρχ,k/2)=0, generalizing a result of David Rohrlich. Also, we conditionally establish a congruence predicted by non-commutative Iwasawa theory and give numerical evidence for it.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 3 , November 2013 , pp. 483 - 498
Copyright
Copyright © Cambridge Philosophical Society 2013 

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