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EXCEPTIONAL ZEROES OF P-ADIC L-FUNCTIONS OVER NON-ABELIAN FIELD EXTENSIONS

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  21 July 2015

DANIEL DELBOURGO
DANIEL DELBOURGO*
Affiliation:
Department of Mathematics, University of Waikato, Hamilton 3240, New Zealand e-mail: delbourg@waikato.ac.nz
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Abstract

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Suppose E is an elliptic curve over $\Bbb Q$, and p>3 is a split multiplicative prime for E. Let q ≠ p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields $K\subset \Bbb Q\big(\mu_{{q^{\infty}}},\;\!^{q^{\infty}\!\!\!\!}\sqrt{m}\big)$ such that p remains inert in $K\cap\Bbb Q(\mu_{{q^{\infty}}})^+$. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Secondary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 2 , May 2016 , pp. 385 - 432
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

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