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Summary: 1
The Modular number, the Congruence number, and Multiplicity One
Abstract. Let N be a positive integer and let f be a new-
form of weight 2 on 0(N). In [ARS07], the authors 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 con-
tains the annihilator ideal of f satisfies multiplicity one, then
the modular number of Af and the congruence number have the
same p-adic valuation. We also discuss a more general setup and
state a result about the structure of the intersection of certain
abelian subvarieties of J0(N) and of the kernel of a congruence
ideal acting on J0(N) as modules over certain quotients of the
Hecke algebra.
1 Introduction and some of the results
Let N be a positive integer and let X0(N) denote the modular curve
over Q associated to the classification of isomorphism classes of elliptic
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