a(p) = 2p~ @ ) cos 0(p). Since we know the truth of the Ramanujan-Petersson conjecture, it follows that the 0(p)'s are real. Inspired by the Sato-Tate conjecture for elliptic curves, Serre [14] conjectured that the 0(p)'s are uniformly distributed in the interval [0, rc] with respect to the 1 measure -sin2OdO. Following Serre, we shall refer to this as the Sato-Tate r~ conjecture, there being n...

We study the motivic Serre invariant of a smoothly bounded algebraic or rigid variety X over a complete discretely valued field K with perfect residue field k. If K has characteristic zero, we extend the definition to arbitrary K-varieties using Bittner’s presentation of the Grothendieck ring and a process of Néron smoothening of pairs of varieties. The motivic Serre invariant can be considered...

To motivate the study of Siegel modular varieties and Borel-Serre compactifications let me recall Scholze’s theorem on torsion classes. We start from an Hecke eigenclass h ∈ H (Xg/Γ,Fp) where Xg is the locally symmetric domain for GLg, and we want to attach to it a continuous semisimple Galois representation ρh : GQ → GLg(Fp) such that the characteristic polynomials of the Frobenius classes at ...

— We construct explicitly some analytic families of étale (φ,Γ)-modules, which give rise to analytic families of 2-dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and extend some results (of Deligne, Edixhoven, Fontaine and Serre) on the representations arising from modular ...

In 1987 Serre conjectured that any mod ` two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where ` is unramified. The hard work is in formulating an analogue of the “weight” part...

In 1987 Serre conjectured that any mod l two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where l is unramified. The hard work is in formulating an analogue of the “weight” part...

For a transversal pair of closed Lagrangian submanifolds L, L of a symplectic manifold M so that π1(L) = π1(L′) = 0 = c1|π2(M) = ω|π2(M) and a generic almost complex structure J we construct an invariant with a high homotopical content which consists in the pages of order ≥ 2 of a spectral sequence whose differentials provide an algebraic measure of the highdimensional moduli spaces of pseudo-h...

Let X be a nonsingular algebraic variety. Suppose Z ⊆ X is a closed subscheme of X, with ideal sheaf IZ . When Z has codimension one in X, everything is as nice as it could be: IZ is a locally free sheaf, in fact a line bundle, and Z can locally be defined by a single equation. But starting in codimension two, all these pleasant things are usually false. To begin with, not every closed subschem...

Motivated by a question of A. Rapinchuk concerning general reductive groups, we are investigating the following question: Given a finitely generated integral domain R with field of fractions F , is there a finitely generated subgroup Γ of SL2(F ) containing SL2(R)? We shall show in this paper that the answer to this question is negative for any polynomial ring R of the form R = R0[s, t], where ...

In this short note, we observe that the Serre functor on residual category of a complete intersection can be easily described in framework hybrid models. Using description, recover some recent results Kuznetsov and Perry.