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On higher congruences between automorphic forms. (English) Zbl 1367.11050

Summary: We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If \(C\) denotes the congruence module for a fixed automorphic Hecke eigenform \(\pi_0\), we prove an exact relation between the \(p\)-adic valuation of the order of \(C\) and the sum of the exponents of \(p\)-power congruences between the Hecke eigenvalues of \(\pi_0\) and other automorphic forms. We apply this result to several situations including the congruences described by Mazur’s Eisenstein ideal.

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
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