## 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

### Citations:

Zbl 0271.10033; Zbl 0586.10010
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### References:

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