Multiplicities ofp-finite modp Galois representations inJ0(Np)

  • Kenneth A. Ribet

DOI: 10.1007/BF01237363

Cite this article as:
Ribet, K.A. Bol. Soc. Bras. Mat (1991) 21: 177. doi:10.1007/BF01237363


LetM≥1 be an integer. LetJ0(M) be the Jacobian Pic0(X0(M)) of the modular curveX0MQ. Let TM be the subring of End (J0(M)) generated by the Hecke operatorsTn withn≥1. Suppose thatp is a maximal ideal of TM. The residue field TM/p is a finite fieldk, whose characteristic will be denotedp. Attached top is a semisimple continuous degree-2 representation ρp of Gal\((\bar Q/Q)\) overk, such that ρp(ϕτ) has characteristic polynomialX2TτX+τ (modp) for each prime numberr prime toMp. We assume that ρp 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 ρp. The number of copies in the direct sum is themultiplicity μp attached top. Results of Mazur and the author show that μp=1 ifp is odd and prime toM, or ifp exactly dividesM and ρp is not finite atp. This article concerns the case wherep exactly dividesM and ρp is finite atp. We prove thatJ0(M)[p] is multiplicity free (in the sense that μp) if and only ifB[p] is multiplicity free, whereB is thep-new subvariety ofJ0(M).

Copyright information

© Sociedade Brasileira de Matemática 1991

Authors and Affiliations

  • Kenneth A. Ribet
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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