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Two-dimensional representations in the arithmetic of modular curves. (English) Zbl 0780.14015

Courbes modulaires et courbes de Shimura, C. R. Sémin., Orsay/Fr. 1987-88, Astérisque 196-197, 215-255 (1991).
[For the entire collection see Zbl 0745.00052.]
Let \(N\) be a positive integer, \(p\) a prime not dividing \(N\), and \(M:=pN\). Let \(J\) be the Jacobian of the modular curve \(X_ 0(N)\) and \(T_ M\) the subring of \(\text{End}(J)\) generated by the Hecke operators. Then with a maximal ideal \(m\) of \(T_ M\) of residue characteristic \(p\), there are associated two natural representations in characteristic \(p\) of the Galois group \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\): first the canonical two- dimensional “modular” representation \(\rho_ m\) of level \(M\) satisfying \(\text{trace}(\rho_ m(\text{Frob}_ l))\equiv T_ l \text{mod} m\) and \(\text{det}(\rho_ m(\text{Frob}_ l))\equiv l \text{mod} m\) for all primes \(l\) not dividing \(M\); and secondly the representation on \(J[m]\), the intersection of the kernels of all elements of \(m\) acting on \(J(\overline\mathbb{Q})\). The main theorem of the paper states that the two representations coincide (i.e. \(\rho_ m\) occurs with multiplicity one in \(J[m])\) if \(\rho_ m\) is absolutely irreducible and moreover not modular of level \(N\). The main part of the proof consists in some subtle algebro-geometric properties of the models of \(X_ 0(M)\) and \(J_ 0(M)\) over \(\mathbb{Z}_ p\) (formulated in a somewhat more general setting). At the end of the paper some examples are given where \(\rho_ m\) occurs with higher multiplicity in \(J[m]\).
The investigation was motivated by the authors’ results on Serre’s conjecture [cf. K. A. Ribet, Invent. Math. 100, No. 2, 431-476 (1990; Zbl 0773.11039)].

MSC:

14H25 Arithmetic ground fields for curves
14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
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