The Modular number, Congruence number, and Multiplicity One
Amod Agashe
TL;DR
The paper addresses the relationship between the modular number $n_{A_f}$ and the congruence number $r_{A_f}$ for newform quotients $A_f$ of the Jacobian $J_0(N)$. It places these invariants in a general framework with abelian subvarieties $A$ and $B$ of a Jacobian, defining modular/congruence numbers $n_A$ and $r_A$, and proves that for a prime $p$ such that all maximal ideals of the Hecke algebra with residue characteristic $p$ containing $I_A+I_B$ satisfy multiplicity one, one has $ ext{ord}_p(n_A) = ext{ord}_p(r_A^2)$; this extends prior exponent-level analogues to the level of numbers. The main result is established via a homology-based analysis of $Aigcap B$, Mazur-style multiplicity-one for differentials, and a key lemma linking the annihilators $I_A$ and $I_B$. The findings clarify when divisibility $n_A mid r_A^2$ can fail and illuminate the role of multiplicity-one in the arithmetic of congruences and visible factors in modular abelian varieties.
Abstract
Let N be a positive integer and let f be a newform of weight 2 on Γ_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.