The Modular number, Congruence number, and Multiplicity One
Abstract
Let N be a positive integer and let f be a newform of weight 2 on 0(N). In earlier joint work with K. Ribet and W. Stein, we introduced the notions of the modular number and the congruence number of the quotient abelian variety Af of J0(N) associated to the newform f. These invariants are analogs of the notions of the modular degree and congruence primes respectively associated to elliptic curves. We show that if p is a prime such that every maximal ideal of the Hecke algebra of characteristic p that contains the annihilator ideal of f satisfies multiplicity one, then the modular number and the congruence number have the same p-adic valuation.
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