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Abelian varieties over \(\mathbb{Q}\) and modular forms. (English) Zbl 1092.11029

Cremona, John (ed.) et al., Modular curves and Abelian varieties. Based on lectures of the conference, Bellaterra, Barcelona, July 15–18, 2002. Basel: Birkhäuser (ISBN 3-7643-6586-2/hbk). Prog. Math. 224, 241-261 (2004).
This is in fact an old and quite famous paper. It appeared in [Algebra and topology 1992 (Taejŏn), Korea Adv. Inst. Sci. Tech., Taejŏn, 53–79 (1992)], and is reprinted here by permission of the Korean editors. It is simply reprinted: the author has changed nothing, and in this subject rather a lot has changed in the meantime.
The Taniyama conjecture, the paper begins, says that any elliptic curve \(C\) over \({\mathbb Q}\) of conductor \(N\) has a non-constant map \(X_0(N)\to C\) defined over \({\mathbb Q}\). Of course this is now a theorem. Now, if there is a non-constant map \(X_1(M)\to C\) defined over \({\mathbb Q}\), for some \(M>0\) (not necessarily \(M=N\)) then “one knows that” the Taniyama conjecture holds for that curve. Mazur proved that if there is a map \(X_1(M)\to C\) defined over the complex numbers then there is a map \(X_1(M')\to C\) defined over \({\mathbb Q}\), for some \(M'\) (in general equal neither to \(M\) nor to \(N\)). So the existence of such a map implies Taniyama, and conversely Taniyama gives a map \(X_0(N)\to C\) so by picking out a generator of the cyclic subgroup of order \(N\) one gets a map \(X_1(N)\to C\).
Serre asked for a characterisation of elliptic curves over the complex numbers (in fact, necessarily over \(\overline {\mathbb Q}\)) which admit a map from \(X_1(M)\), i.e. which are quotients of \(J_1(M)=\text{Jac\,} X_1(M)\), for some \(M>0\). All CM curves do (a result of Shimura): discarding them, Ribet defines an abelian variety over \({\mathbb Q}\) to be of \(\text{GL}_2\)-type if \({\text{End}}_{\mathbb Q}(A)\otimes{\mathbb Q}\) contains (is, if A is simple over \({\mathbb Q}\)) a number field \(E\) of degree equal to \(\dim A\).
Ribet proves here: if \(C\) is a non-CM elliptic curve over \(\overline {\mathbb Q}\), then it is a quotient of an abelian variety of \(\text{GL}_2\)-type iff it is isogenous to each of its Galois conjugates (such a curve is called a \({\mathbb Q}\)-curve). He also proves that every abelian variety of \(\text{GL}_2\)-type is a quotient of \(J_1(M)\) for some \(M>0\), assuming a conjecture of J.-P. Serre [Duke Math. J. 54, 179–230 (1987; Zbl 0641.10026)] on representations of \({\text{Gal}}(\overline {\mathbb Q}/{\mathbb Q})\). This says very roughly that absolutely irreducible representations of dimension \(2\) should be modular, arising from newforms of prescribed level, weight and character. Ribet applies this to the representations coming from the action of the Galois group on the Tate module \(V_\ell\), for \(\ell\) a prime of good reduction: these are attached to primes \(\lambda\) of \(E\) dividing \(\ell\). Then there is an abelian variety associated with the modular form, and by a theorem of Faltings it is automatically a quotient of \(J_1(M)\), where \(M\) is the level.
Since this paper first appeared, J. S. Ellenberg and C. Skinner [Duke Math. J. 109, No. 1, 97–122 (2001; Zbl 1009.11038)] have shown that every \({\mathbb Q}\)-curve satisfying a technical condition at the prime \(3\) is modular, i.e. is a quotient of some \(J_1(M)\). They do not prove Serre’s conjecture, however; but there is some recent progress in that direction too in a preprint of C. Khare.
Reviewer’s remark: I was assisted in writing this review by Kevin Buzzard, who drew my attention to the paper of Ellenberg and Skinner and to Khare’s preprint.
For the entire collection see [Zbl 1032.11002].

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11F33 Congruences for modular and \(p\)-adic modular forms
14G35 Modular and Shimura varieties
14H52 Elliptic curves
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