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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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