The Modular number, Congruence number, and Multiplicity One

TL;DR

This paper explores the relationship between modular and congruence numbers of abelian varieties associated with newforms, establishing conditions under which their p-adic valuations coincide, extending prior work on elliptic curves.

Contribution

It proves that under certain multiplicity one conditions, the p-adic valuations of modular and congruence numbers are equal for abelian varieties linked to newforms.

Findings

Modular and congruence numbers share the same p-adic valuation under multiplicity one conditions.

The work generalizes earlier results from elliptic curves to higher-dimensional abelian varieties.

Provides a criterion involving maximal ideals of the Hecke algebra for equality of valuations.

Abstract

Let N be a positive integer and let f be a newform of weight 2 on \Gamma_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 A_f of J_0(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.