# Algebraic Number Theory 3 (Formal Groups) (WS 2016/17)

## Algebraic Number Theory 3 / Formal Groups (WS 2016/17)

In this course we will study the theory of formal group laws, formal groups and their deformation theory.

In a sense, this course will be a continuation of my Algebraic Number Theory 2 course on (local) class field theory. On the other hand, attending the course will not require many prerequisites from the previous terms. Of course a sound knowledge of algebra is required, as is some knowledge about the field of $p$-adic numbers. Further knowledge about algebraic number theory and/or algebraic geometry will be useful, but is not strictly required.

Dates: Tuesday, 14-16, S-U-3.02; Wednesday, 10-12, S-U-3.01.

Exercise session: Monday, 12-2pm, O-3.46.

Contact: ulrich.goertz@uni-due.de

## Problem sheets

 Due Sheet 1 pdf November 2, 2016 Sheet 2 pdf November 9, 2016 Sheet 3 pdf November 16, 2016 Sheet 4 pdf November 23, 2016 Sheet 5 pdf November 30, 2016 Sheet 6 pdf December 7, 2016 Sheet 7 pdf December 14, 2016 Sheet 8 pdf December 21, 2016 Sheet 9 pdf January 11, 2017 Sheet 10 (updated Jan 25, 2017) pdf January 18, 2017 Sheet 11 pdf January 25, 2017 Sheet 12 pdf February 1, 2017

## References

V. Drinfeld, Elliptic modules, Mat. USSR Sbornik 23 (1974)no. 4, 561-592.

M. Hazewinkel, Formal groups and applications, Academic Press

M. Hazewinkel, Three lectures on formal groups, Canad. Math. Soc. Conf. Proc. (1986)

J. Lubin, J. Tate, Formal complex multiplication in local fields, Ann. of Math., 2nd series, 81 (1965), 380-387.

J. Lubin, J. Tate, Formal moduli for one parameter formal Lie groups, Bull. SMF 94 (1966), 49-60.

J. Neukirch, Klassenkörpertheorie /Class field theory (Chapter II, §7)

J. P. Serre, Local class field theory, in: Cassels, Fröhlich (eds.), Algebraic Number Theory

M. Strauch, Deformation spaces of one-dimensional formal modules and their cohomology, Adv. Math. 217 (2008), 889-951.

E. Viehmann, K. Ziegler, Formal moduli of formal $\mathcal O_K$-modules, in: ARGOS Seminar on Intersections of Modular Correspondences, Astérisque 312 (2007)

J. Weinstein, The geometry of Lubin-Tate spaces

T. Yoshida, Local class field theory via Lubin-Tate theory, Annales de la Faculte des Sciences de Toulouse, Ser. 6, 17-2 (2008), 411-438. math.NT/0606108

Th. Zink, Cartiertheorie kommutativer formaler Gruppen, Teubner (German; English translation)

The following references take a point of view closer to algebraic topology:

M. Hopkins, Complex oriented cohomology theories and the language of stacks (Course notes)

J. Lurie, Course on chromatic homotopy theory

N. Strickland, Formal groups (Course notes)

## Overview of the course

• Formal groups: Definitions, invariant derivations and differential forms, the tangent space, the $\mathbb Q$-theorem.
• Lubin-Tate theory (the relationship to local class field theory, see Neukirch’s book, the original account in the 1965 of Lubin and Tate and/or Yoshida’s paper)
• Lazard’s Theorem (see Fargues’s notes and the book by Hazewinkel)
• Deformations of formal groups (see the 1966 paper of Lubin and Tate; Drinfeld’s paper, Fargues’s notes, the papers by Viehmann and Ziegler and by Strauch): The deformation functor of a formal group of finite height is pro-representable. Level structures. The étale tower in the generic fiber.