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Serre’s conjectures. (English) Zbl 0848.11019
Murty, V. Kumar (ed.), Seminar on Fermat’s last theorem. The Fields Institute for Research in Mathematical Sciences, 1993-1994, Toronto, Ontario, Canada. Proceedings. Providence, RI: American Mathematical Society (publ. for the Canadian Mathematical Society). CMS Conf. Proc. 17, 135-153 (1995).
Let \(F\) be a finite field of characteristic \(p\) and write \(G_{\mathbb{Q}}= \text{Gal}(\overline{\mathbb{Q}}, \mathbb{Q})\). Serre’s conjectures are concerned with irreducible two-dimensional representations \(\rho: G_{\mathbb{Q}}\to \text{GL}_2(\mathbb{F})\) which are assumed to be odd, i.e. for complex conjugation \(c\), \(\rho(c)\) has eigenvalues \(1\) and \(- 1\). If \(\ell\) is a prime at which \(\rho\) is unramified, the characteristic polynomial of \(\rho(\text{Frob}_\ell)\) of the Frobenius element \(\text{Frob}_\ell\) at \(\ell\) is well defined. With the usual notation, one has the notion of normalized eigenforms for Hecke operators acting on spaces of cusp forms (and on spaces of newforms) \(S_k(N, \varepsilon, R)\) of weight \(k\), level \(N\) and character \(\varepsilon\), for any subring \(R\) of \(\mathbb{C}\). The representation \(\rho\) is said to be modular if there exists a normalized eigenform \(f\) (of some weight \(k\geq 2\), level \(N\), and character \(\varepsilon\)) with Fourier coefficients in \(\mathbb{F}\), such that for all \(\ell\nmid Np\), where \(\rho\) is unramified, \(\rho(\text{Frob}_\ell)\) has characteristic polynomial \(x^2- a_\ell x+ \ell^{k- 1} \varepsilon(\ell)\), where \(a_\ell\) is the \(\ell\)th Fourier coefficient of \(f\).
In this case \(\rho\) and \(f\) are called associated. Serre’s conjectures are stated in two forms: (S1, the vague form) Any odd irreducible \(\rho\) as above is modular; (S2, the precise form) There exists a normalized \(\text{mod } p\) eigenform of level \(N(\rho)\), weight \(k(\rho)\), and \(\mathbb{F}\)-valued character \(\varepsilon(\rho)\) which is associated to \(\rho\). Here the \(N(\rho)\), \(k(\rho)\) and \(\varepsilon(\rho)\) are given by a precise recipe of Serre. (The \(\text{char } 2\) or 3 cases need special treatment.) Of course, with respect to Wiles’ work on the Shimura-Taniyama conjecture and Fermat’s last theorem, a case of particular importance is the one of the Galois representations \(\rho_{E, p}: G_{\mathbb{Q}}\to \operatorname{Aut}(E_p)\simeq \text{GL}_2(\mathbb{F}_2)\) associated to a semi-stable elliptic curve \(E\) defined over \(\mathbb{Q}\) with reduction \(\text{mod } p\), \(E_p\). If \(N_E\) denotes the conductor of \(E\) (i.e. the product of the primes of bad reduction of \(E\)) and \(\Delta_E\) is the minimal discriminant of \(E\), one has: (i) \(N(\rho_{E, p})\) is the product of all primes \(\ell\neq p\) such that \(\text{ord}_p(\Delta_E)\neq 0\pmod p\); (ii) \(k(\rho_{E, p})= 2\), if \(\text{ord}_p(\Delta_E)\equiv 0\pmod p\), and is equal to \(p+ 1\) otherwise; (iii) \(\varepsilon(\rho_{E, p})= 1\). Unfortunately, S2 remains very difficult to attack in most cases, but one may try to prove Serre’s epsilon conjecture: (Assume \(\text{char}(\mathbb{F})> 3\)) S1 implies S2. In this direction one has results due to Mazur and Ribet.
In particular, Ribet’s result is enough to prove that the Shimura-Taniyama conjecture implies Fermat’s last theorem. Several examples of elliptic curves illustrating Mazur’s and Ribet’s results are discussed in some detail. The latest result is due to F. Diamond: Assume \(F\) has odd characteristic. If \(\rho: G_\mathbb{Q}\to \text{GL}_2(\mathbb{F})\) is a representation arising from an eigenform, then \(\rho\) is associated to an eigenform of level \(N(\rho)\), weight \(k(\rho)\), and, if \(\rho\) is not induced from a character of \(\text{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q}(\sqrt{- 3}))\) in case \(\text{char}(\mathbb{F})= 3\), character \(\varepsilon(\rho)\).
There are a few cases where S1 can be shown to be true, e.g. if the image of \(\rho\) is isomorphic to a dihedral group \(D_{2n}\), \((n, p)= 1\). This may then be combined with Diamond’s result. The case \(\mathbb{F}= \mathbb{F}_2\) remains unclear.
One may also try to attack the author’s and Granville’s generalized Fermat conjectures on equations of the form \(x^p+ y^q= z^r\), \({1\over p}+ {1\over q}+ {1\over r}< 1\), \(\text{gcd}(x, y, z)= 1\), \(xyz\neq 0\). Assuming S2 one has some results in this direction. The idea is to associate an appropriate Frey curve (an elliptic curve) to the equation, and use Serre’s conjecture to show that the associated \(\text{ mod } p\) representation does not exist, and, via Ribet’s result, apply the Shimura-Taniyama conjecture for this situation. Frey made the following conjecture: (F) Let \(A\) be an elliptic curve defined over a number field \(K\). There are only finitely many pairs \((E, p)\) consisting of an elliptic curve \(E\) over \(K\), not isogenous to \(A\), and a prime number \(p> 5\), such that \(E_p \simeq A_p\) as \(G_K\)-modules. If both S2 and F hold, then the generalized Fermat equation of the form \(Ax^n+ By^n= Cz^n\), \(n\geq 3\), \(\text{gcd}(x, y, z)= 1\), admits only finitely many integer solutions \((x, y, z, n)\). Apart from the fact that S2 implies the Shimura-Taniyama conjecture, Ribet proved the following: If S2 is true, then an elliptic curve over \(\overline\mathbb{Q}\) is modular (i.e. is the quotient of some \(J_0(N)\)) if and only if \(E\) is a \(\mathbb{Q}\)-curve, i.e. \(E\) is an elliptic curve defined over a number field which is isogenous to all of its Galois conjugates.
The fifth (and last) section relates Wiles’ work to a generalized Shimura-Taniyama conjecture, essentially due to Mazur. This conjecture can be broken into two parts: (i) Serre’s conjecture on \(\text{mod } p\) representations of \(G_{\mathbb{Q}}\); (ii) A lifting conjecture of Serre’s \(\text{mod } p\) representations to so-called admissible \(p\)-adic representations of \(G_{\mathbb{Q}}\). Here is where Wiles’ work has led to substantial inroads. The paper closes with the remark that Serre’s conjecture (in general), as a first step in the direction of a ‘Langlands philosophy \(\text{mod } p\)’, remains wide open and that it will keep number theorists busy in years to come, perhaps long after the Shimura-Taniyama conjecture has been completely proved.
For the entire collection see [Zbl 0828.00033].

11F11 Holomorphic modular forms of integral weight
11F33 Congruences for modular and \(p\)-adic modular forms
11R32 Galois theory
20C11 \(p\)-adic representations of finite groups
11G05 Elliptic curves over global fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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