Multiplicities ofp-finite modp Galois representations inJ ₀ (Np)

Abstract

LetM≥1 be an integer. LetJ ₀ (M) be the Jacobian Pic⁰(X ₀ (M)) of the modular curveX ₀ M _Q. Let T _M be the subring of End (J ₀ (M)) generated by the Hecke operatorsT _n withn≥1. Suppose thatp is a maximal ideal of T _M . The residue field T _M /p is a finite fieldk, whose characteristic will be denotedp. Attached top is a semisimple continuous degree-2 representation ρ _p of Gal$(\bar Q/Q)$ overk, such that ρ _p (ϕτ) has characteristic polynomialX ²−T _τ X+τ (modp) for each prime numberr prime toM _p. We assume that ρ _p is absolutely irreducible. By a result of Boston, Lenstra, and the author, the representation

$$\mathfrak{p}]: = \left\{ {x \in J_0 (M)(\bar Q)|\lambda x = 0forall \lambda \in \mathfrak{p}} \right\}$$

is a direct sum of copies of ρ _p . The number of copies in the direct sum is themultiplicity μ _p attached top. Results of Mazur and the author show that μ _p =1 ifp is odd and prime toM, or ifp exactly dividesM and ρ _p is not finite atp. This article concerns the case wherep exactly dividesM and ρ _p is finite atp. We prove thatJ ₀ (M)[p] is multiplicity free (in the sense that μ _p ) if and only ifB[p] is multiplicity free, whereB is thep-new subvariety ofJ ₀ (M).