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On the modular representations of degree two of \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\). (Sur les représentations modulaires de degré 2 de \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\).) (French) Zbl 0641.10026
In the present paper the author precises a very interesting conjecture stated in 1973: If \(\rho\) is an irreducible odd representation of \(G_{{\mathbb Q}}\) to \(\mathrm{GL}(2,{\overline {\mathbb F}}_ p)\) then it should be modular in the following sense: There exists a parabolic cusp form \(f\) of level \(N\), weight \(k\) and character \(\varepsilon\) with coefficients in \({\overline {\mathbb F}}_ p\) [cf. N. M. Katz, \(p\)-adic properties of modular schemes and modular forms, Lect. Notes Math. 350, 69–190 (1973; Zbl 0271.10033)] such that the trace of the image of Frobenius elements \(\pi_{\ell}\) of primes \(\ell \nmid N\) is equal to the \(\ell\)th Fourier coefficient of \(f\) and \(\det\;\rho\) \(=\varepsilon \cdot \chi^{k-1}\) (\(\chi\) cyclotomic character mod \(p\)).
Moreover there is a precise recipe how to determine \(N\), \(k\) and \(\varepsilon\): \(N\) is the Artin conductor of \(\rho\) away from \(p\), and \(k\) is determined by the restriction of \(\rho\) to an inertia group \(I_ p\) at \(p\). For instance: \(k\) is equal to 2 iff \(\det\;\rho| I_ p=\chi\) and the group scheme corresponding to \(\rho | G_{{\mathbb Q}_ p}\) has an extension to a flat finite group scheme over \({\mathbb Z}_ p\) (i.e. \(\rho\) is finite at \(p\)).
This conjecture has very strong and interesting consequences which can be found in § 4. For example: Take \(E/{\mathbb Q}\) as elliptic curve which is semi-stable at \(p\), and take \(\rho =\rho_{E_ p}\) as the representation induced by the action of \(G_{{\mathbb Q}}\) on the points of order \(p\) of \(E\). \(\rho_{E_ p}\) is finite at \(p\) (and hence \(k=2)\) iff \(\mathrm{Min}\{0,v_ p(j_ E)\}\equiv 0 \pmod p.\) For given \(E\) this is so for infinitely many \(p\), and so one gets surprisingly that \(E\) is a quotient of the Jacobian of \(X_ 0(N_ E)\) with \(N_ E\) the conductor of \(E\). Hence Serre’s conjecture implies Taniyama’s conjecture claiming that every elliptic curve over \({\mathbb Q}\) is modular. (For an extension to abelian varieties with real multiplications cf. 4.7.)
On the other side a beautiful result of K. Ribet [On modular representations of \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\) arising from modular forms; Preprint # 06420-87; Math. Res. Inst. Berkeley, CA (1987)] states that if one assumes that \(E/{\mathbb Q}\) is modular and some reasonable conditions are satisfied then Serre’s conjecture is true for \(\rho_{E_ p}\). This can be applied to Fermat’s Last Theorem: A nontrivial solution of \(Z^p_ 1-Z^p_ 2=Z^p_ 3\) \((p\geq 3)\) can be used to define an elliptic curve over \({\mathbb Q}\) which, due to Ribet’s theorem, cannot be modular [cf. the reviewer, Links between stable elliptic curves and certain Diophantine equations, Ann. Univ. Sarav., Ser. Math. 1, No. 1 (1986; Zbl 0586.10010)]. Hence Taniyama’s conjecture (and so Serre’s conjecture) implies Fermat’s Last Theorem.
In the last section of the paper one finds interesting examples (for \(p=2, 3, 7\)) for which Serre’s conjecture is verified (at least partly) mostly by using computer programs implemented by J.-F. Mestre.
Reviewer: G. Frey

MSC:
11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11G05 Elliptic curves over global fields
11R32 Galois theory
11R39 Langlands-Weil conjectures, nonabelian class field theory
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