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Factoring Hecke polynomials modulo a prime

J. Brian Conrey·David W. Farmer·Peter Jake Wallace
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Abstract

Let T^{N,chi}_{p,k}(x) be the characteristic polynomial of the Hecke operator T_p acting on the space of cusp forms S_k(N,chi). We describe the factorization of T^{N,chi}_{p,k}(x) mod l as k varies, and we explicitly calculate those factorizations for N=1 and small l. These factorizations are used to deduce the irreducibility of certain T^{1,1}_{q,k}(x) from the irreducibility of T^{1,1}_{2,k}(x).

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J. Brian Conrey, David W. Farmer, Peter Jake Wallace. 1998-11-15. Factoring Hecke polynomials modulo a prime. https://arxiv.org/abs/math/9811179

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