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CONGRUENCES OF SAITO–KUROKAWA LIFTS AND DENOMINATORS OF CENTRAL SPINOR L-VALUES

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  14 October 2021

NEIL DUMMIGAN Open the ORCID record for NEIL DUMMIGAN [Opens in a new window]
NEIL DUMMIGAN*
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, UK e-mail: [email protected]
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Abstract

Following Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.

MSC classification

Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F33: Congruences for modular and $p$-adic modular forms 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 2 , May 2022 , pp. 504 - 525
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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