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93d:11056 11F80 11F25 11G18

Mazur, B. [Mazur, Barry C.] ^{(}1-HRV^{)}; Ribet, K. A. ^{(}1-CA^{)} Two-dimensional representations in the arithmetic of modular curves. (English. French summary) Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). Astérisque No. 196-197 (1991), 6, 215–255 (1992). Let M ≥ 1 be an integer. Let T_{M} be the ring generated by the Hecke operators T (n) (n ≥ 1), of level M, viewed either as endomorphisms of the space of cusp forms of weight 2, or as endomorphisms of the Jacobian J_{1}(M) of the modular curve X_{1}(M). Given a prime p, one can attach to a ring homomorphism φ: T_{M} → \overline{Q}_{p} a degreetwo Galois representation ρ_{φ} over \overline{Q}_{p} (Deligne-Shimura). The classical multiplicity one theorem for newforms implies that if φ factors through the new part of T_{M}, ρ_{φ} occurs with multiplicity one in the Tate module of the Albanese variety J_{1}(M) of X_{1}(M): T_{p}J_{1}(M) ⊗_{T,φ} \overline{Q}_{p} ≃ ρ_{φ}. Similarly, to a ring homomorphism φ: T_{M} → \overline{F}_{p}, one can attach a Galois representation ρ_{φ} of degree 2 over \overline{F}_{p} (namely, the reduction modulo p of the Deligne-Shimura representation attached to a lifting φ̃ of φ to \overline{Q}_{p}). In this mod p situation, the question of describing the Galois representation T_{p}J_{1}(M) ⊗_{T,φ} \overline{F}_{p} in terms of ρ_{φ} is not yet answered in general.

This paper addresses this question in an important particular case, namely when the following two conditions are satisfied: (1) ρ_{φ} is absolutely irreducible; (2) p divides M and the lifting φ̃ corresponds to a p-ordinary newform f whose associated automorphic representation π_{f} is special at p. More precisely, under these assumptions, the main theorem states that T_{p}J_{1}(M) ⊗_{T,φ} \overline{F}_{p} is isomorphic to ρ_{φ}.

Two remarks: The first assumption excludes the so-called Eisenstein prime case where the cusp form f happens to be congruent mod p to an Eisenstein series. Moreover, under this irreducibility assumption, it is easy to see that the semisimplification of T_{p}J_{1}(M) ⊗_{T,φ} \overline{F}_{p} is isomorphic to a sum of copies of ρ_{φ}; therefore, the theorem amounts to proving that ρ_{φ} occurs with multiplicity one under assumption 2. Note that thanks to a recent result by N. Boston, H. W. Lenstra, Jr. and Ribet [Duke Math. J., to appear] the semisimplification is superfluous, since T_{p}J_{1}(M) ⊗_{T,φ} \overline{F}_{p} is always semisimple. The assumption that π_{f} is special at p can be translated as: the conductor of f is exactly divisible by p and the conductor of its Nebentypus is prime to p.

In fact, the details of the proof of this theorem are only carried through in the case where the Nebentypus of f is trivial. The general case stated above would require considering the modular curve for the intermediate congruence subgroup Γ_{1}(M) ∩ Γ_{0}(p). However, it is pretty clear that the proof goes through in that case.

Some earlier instances, where such a mod p multiplicity one theorem was known, include the case where (i) p ≠ 2 does not divide M, and where (ii) p divides M, f is p-ordinary and its Nebentypus is pprimitive. Compare with results of Mazur, Ribet, the reviewer and Wiles cited in the extended list of references in the paper.

However, the paper concludes by giving examples where the multiplicity is greater than one. This hints at the subtlety of the relation between T_{p}J_{1}(M) ⊗_{T,φ} \overline{F}_{p} and ρ_{φ}, a fortiori in the Eisenstein prime case, or in the Shimura curve case, where the absence of cusps prevents one from using a geometric multiplicity one theorem (in that case, Ribet found plenty of examples of multiplicity two), to say nothing of the case of a more general Shimura variety for which the multiplicity one theorem is true in the cohomology in characteristic zero!

The main theorem of the paper has been motivated by Ribet’s work on Serre’s conjectures [91g:11066Invent. Math. 100 (1990), no. 2, 431–476; MR 91g:11066], but it has other applications, as in a paper by Mazur and the reviewer [92e:11060 Inst. Hautes Etudes Sci. Publ. Math. No. 71 (1990), 65–103; MR 92e:11060], and it provides examples for R. Greenberg’s new conjectures on the main conjecture and deformations of Galois representations [in Proceedings of the American Mathematical Society Conference on Motives (Seattle), 1993, to appear].

{For the entire collection see MR 92g:11004}.

Jacques Tilouine (F-PARIS11)