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On Galois representations defined by torsion points of modular elliptic curves. (English) Zbl 0770.11029
In his famous paper “Sur les représentations modulaires de degré 2 de $$\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})$$”, Duke Math. J. 54, 179-230 (1987; Zbl 0641.10026) J.-P. Serre formulated a series of conjectures on Galois representations, whose verification would lead to remarkable consequences in number theory.
Let $$\rho:G_ \mathbb{Q}=\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\to\text{GL}_ 2(\overline \mathbb{F}_ p)$$ be an irreducible odd representation, where $$\overline\mathbb{F}_ p$$=algebraic closure of $$\mathbb{F}_ p$$. According to conjecture (3.2.4) of (loc. cit.), $$\rho$$ should arise from a cusp form $$f$$ of type $$(N_ \rho,k_ \rho,\varepsilon_ \rho)$$, eigenform for the Hecke operators, and with coefficients in $$\overline\mathbb{F}_ p$$. Here $$N_ \rho$$ is the conductor, $$k_ \rho$$ the weight, and $$\varepsilon_ \rho:(\mathbb{Z}/N_ \rho\mathbb{Z})^*\to\overline\mathbb{F}_ p^*$$ the Dirichlet character of $$f$$, which can be calculated from $$\rho$$ by an explicit recipe.
In particular, let $$\rho:G_ \mathbb{Q}\to\text{Aut}(E_ p)\cong\text{GL}_ 2(\mathbb{F}_ p)$$ be the representation on the $$p$$-torsion points $$E_ p$$ of an elliptic curve $$E/\mathbb{Q}$$, in which case $$\varepsilon_ \rho$$ is trivial. In section one of the paper, the authors give a complete discussion of $$N_ \rho$$ and $$k_ \rho$$, i.e., they express these invariants through Kodaira symbols of $$E$$ reduced (mod $$p$$) and (mod $$\ell\neq p$$), the class of $$p\pmod{12}$$, and the $$p$$-adic valuation of the invariants $$c_ 4,c_ 6$$ of the minimal model of $$E$$ at $$p$$.
This is used in section 2 to give a proof of Serre’s conjecture in the case where $$E$$ is modular with potentially good ordinary reduction at a prime $$p>7$$. In section 3, a proof is sketched for the case of $$E$$ modular with semi-stable reduction at a prime $$p\geq 5$$.

MSC:
 11G05 Elliptic curves over global fields 11F11 Holomorphic modular forms of integral weight 14H52 Elliptic curves
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References:
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