A note on the -adic Galois representations attached to Hilbert modular forms.
Abelian surfaces of GL₂-type as Jacobians of curves
Adelic equidistribution of tori and equidistribution of CM points. (Équidistribution adélique des tores et équidistribution des points CM.)
An alternative description of the Drinfeld -adic half-plane
We show that the Deligne formal model of the Drinfeld -adic half-plane relative to a local field represents a moduli problem of polarized -modules with an action of the ring of integers in a quadratic extension of . The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of and for a two-dimensional split hermitian space for .
An annihilator for the -Selmer group by means of Heegner points
Let be a modular elliptic curve, and let be an imaginary quadratic field. We show that the -Selmer group of over certain finite anticyclotomic extensions of , modulo the universal norms, is annihilated by the «characteristic ideal» of the universal norms modulo the Heegner points. We also extend this result to the anticyclotomic -extension of . This refines in the current contest a result of [1].
An arithmetic of modular function fields of degree two
An effective bound of p for algebraic points on Shimura curves of Γ₀(p)-type
In previous articles, we showed that for number fields in a certain large class, there are at most elliptic points on a Shimura curve of Γ₀(p)-type for every sufficiently large prime number p. In this article, we obtain an effective bound for such p.
An effective result of André-Oort type II
We prove some new effective results of André-Oort type. In particular, we state certain uniform improvements of the main result in [L. Kühne, Ann. of Math. 176 (2012), 651-671]. We also show that the equation X + Y = 1 has no solution in singular moduli. As a by-product, we indicate a simple trick rendering André's proof of the André-Oort conjecture effective. A significantly new aspect is the usage of both the Siegel-Tatuzawa theorem and the weak effective lower bound on the class number of an...
Anticyclotomic main conjectures.
Auto-intersection du dualisant relatif des courbes modulaires Xo (N).
Automorphism groups of Shimura varieties.
Automorphy for some l-adic lifts of automorphic mod l Galois representations. II
We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spirit to the methods of [TW] than to those of [W], which relied on Ihara’s lemma.
Automorphy for some l-adic lifts of automorphic mod l Galois representations
We extend the methods of Wiles and of Taylor and Wiles from GL2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge–Tate numbers), minimally ramified, l-adic lifts of certain automorphic mod l Galois representations of any dimension. We also make a conjecture about the structure of mod l automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for GL2. Following Wiles’ method we show that this...