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Ordinary primes in Hilbert modular varieties

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  06 February 2020

Junecue Suh
Junecue Suh*
Affiliation:
Department of Mathematics, University of California, 1156 High St, Santa Cruz, CA 95064, USA email [email protected]
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Abstract

A well-known conjecture, often attributed to Serre, asserts that any motive over any number field has infinitely many ordinary reductions (in the sense that the Newton polygon coincides with the Hodge polygon). In the case of Hilbert modular cuspforms $f$ of parallel weight $(2,\ldots ,2)$, we show how to produce more ordinary primes by using the Sato–Tate equidistribution and combining it with the Galois theory of the Hecke field. Under the assumption of stronger forms of Sato–Tate equidistribution, we get stronger (but conditional) results. In the case of higher weights, we formulate the ordinariness conjecture for submotives of the intersection cohomology of proper algebraic varieties with motivic coefficients, and verify it for the motives whose $\ell$-adic Galois realisations are abelian on a finite-index subgroup. We get some results for Hilbert cuspforms of weight $(3,\ldots ,3)$, weaker than those for $(2,\ldots ,2)$.

Keywords

Hilbert modular formordinary reductionSato–Tate equidistribution

MSC classification

Primary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Secondary: 14G35: Modular and Shimura varieties 11F30: Fourier coefficients of automorphic forms 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 4 , April 2020 , pp. 647 - 678
Copyright
© The Author 2020

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