Brian Conrad Professor Department of Mathematics Stanford University Building 380, Sloan Hall Stanford, CA 94305, USA office: 383CC, Sloan Hall Photo Credit: Rod Searcey

notes typed by Sam Lichtenstein, lectures and editing by Brian Conrad February 8, 2020 Please send any errata (typos, math errors, etc.) to [email protected]. The sequel course \Linear Algebraic Groups II" treats the Borel-Tits structure theory of reductive groups over general elds (assuming knowledge of schemes, as in this course). Contents

OLIVIER BRINON AND BRIAN CONRAD Contents Part I. First steps in p-adic Hodge theory 4 1. Motivation 4 1.1. Tate modules 4 1.2. Galois lattices and Galois deformations 6 1.3. Aims of p-adic Hodge theory 7 1.4. Exercises 9 2. Hodge–Tate representations 10 2.1. Basic properties of CK 11 2.2. Theorems of Tate–Sen and Faltings 12 2.3. Hodge ...

BRIAN CONRAD 1. Valuation spectra and Huber/Tate rings 1.1. Introduction. Although we begin the oral lectures with a crash course on some basic high-lights from rigid-analytic geometry in the sense of Tate, some awareness of those ideas is taken as known for the purpose of reading these written notes that accompany those lectures. The intro-

BRIAN CONRAD LECTURE NOTES BY TONY FENG CONTENTS Note to the reader 3 1. Basic theory 4 1.1. Group schemes 4 1.2. Complex tori 5 1.3. Link between complex abelian varieties and complex tori 6 1.4. The Mordell–Weil Theorem 7 1.5. Commutativity 8 1.6. Torsion 11 1.7. Rigidity 12 2. The Picard functor 17 2.1. Overview 17 2.2. Rigidification 17 ...

Professor Brian Conrad MWF, 9--9:50pm, 380D. This is a course on class field theory in its many guises. See the syllabus link for a more detailed discussion.

BRIAN CONRAD 1. Introduction 1.1. Motivation. In [DR], Deligne and Rapoport developed the theory of generalized elliptic curves over arbitrary schemes and they proved that various moduli stacks for (ample) “level-N” structures on generalized elliptic curves over Z[1/N]-schemes are Deligne–Mumford stacks over Z[1/N]. These stacks were ...

BRIAN CONRAD These notes explain the equivalence between certain topological and coherent data on complex-analytic manifolds, and also discuss the phenomenon of “regular singularities” connections in the 1-dimensional case. It is assumed that the reader is familiar with the equivalence of categories between the category of locally constant

Brian Conrad Abstract. Liouville proved that certain integrals, most famously R e x2dx, cannot be expressed in elementary terms. We explain how to give precise meaning to the notion of integration \in elementary terms", and we formu-late Liouville’s theorem that characterizes the possible form of elementary an-tiderivatives.

Brian Conrad 1. Introduction In p-adic Hodge theory there are fully faithful functors from certain categories of p-adic representations of the Galois group G K:= Gal(K/K) of a p-adic field K to certain categories of semi-linear algebra structures on finite-dimensional vector spaces in characteristic 0. For example, semistable representations ...