Serre's Conjecture and Base Change for GL(2)

The Fontaine-mazur Conjecture for Gl2

Introduction 641 1. Breuil-Mézard conjecture and the p-adic local Langlands 644 (1.1) The Breuil-Mézard conjecture 644 (1.2) Review of Colmez’s functor 647 (1.3) Hilbert-Samuel multiplicities 650 (1.4) Representations and pseudo-representations 653 (1.5) GL2(Qp)-representations mod p. 657 (1.6) Local patching and multiplicities 661 (1.7) From pseudo-representations to representations 669 2. Mod...

Base Change, Tensor Product and the Birch-swinnerton-dyer Conjecture

We prove the Rankin-Selberg convolution of two cuspidal automorphic representations are automorphic whenever one of them arises from an irreducible representation of an abelian-by-nilpotent Galois extension, which extends the previous result of Arthur-Clozel. Moreover, if one of such representations is of dimension at most 3 and another representation arises from a nearly nilpotent extension or...

Base sizes for simple groups and a conjecture of Cameron

Let G be a permutation group on a finite set Ω. A base for G is a subset B ⊆ Ω with pointwise stabilizer in G that is trivial; we write b(G) for the smallest size of a base for G. In this paper we prove that b(G) ! 6 if G is an almost simple group of exceptional Lie type and Ω is a primitive faithful G-set. An important consequence of this result, when combined with other recent work, is that b...

Base Change for Semiorthogonal Decompositions

Let X be an algebraic variety over a base scheme S and φ : T → S a faithful base change. Given an admissible subcategory A in D(X), the bounded derived category of coherent sheaves on X, we construct an admissible subcategory AT in D b(X×ST ), called the base change of A, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of D(X) is given then the base...

Non - Abelian Base Change For