# p-adic Galois representations

Here you can find references to the literature.

Lectures: 1 2 2bis 3 4 5 5bis

Last modified: October 18, 2020

## General references

The course follows the outline of the book of Fontaine and Ouyang although I shall diverge from it in many ways. Another useful reference is the notes of Olivier Brinon and Brian Conrad on p-adic Hodge theory.

J.-M. Fontaine, Y. Ouyang. Theory of p-adic Galois representations.

O. Brinon, B. Conrad. CMI summer school notes on p-adic Hodge theory.

## Lecture 1

To familiarize yourself with the language of representation theory have a look at the lecture notes of Kowalski [Chapter 1, Chapter 2, §2.1 – §2.6].

A concise exposition of Galois theory for infinite extensions is contained in Neukirch's book [Chapter IV, §1, §2]. It is important to work out Example 5 in §2 where Neukirch computes absolute Galois groups of finite fields.

The book of Neukirch  [Chapter II, §1] also contains an excellent introduction to p-adic numbers. Hensel's lemma can be found in the same chapter. An alternative reference for Hensel's lemma is Milne's course notes  [Theorem 7.33] (version 3.07).

The classical book of Silverman is one of the best introductions to the theory of elliptic curves. Tate modules are discussed in Chapter III §7. The trace formula of the lecture is Theorem 2.3.1 (b) in Chapter V.

If you have not studied elliptic curves before it would be better to take all the facts I use in the lectures on faith.

## Lectures 2 and 2bis

Serre's book is the basic reference on local fields. Chapters I and II (§1–§3) are especially important.

An alternative reference is Neukirch's book [Chapter II, §3, §4, §8, §9].

Serre's Chapter II also contains a section on Witt vectors (§5 – §6). This construction is of fundamental value to the course, and will be discussed in detail in the coming lectures.

The statements on ramification of cyclotomic characters follow from the theory of cyclotomic extensions. Have a look at Propositions 7.12, 7.13 in Neukirch [Chapter II, §7] and Propositions 16, 17 in Serre [Chapter IV, §4]. Pay attention to the formula for cyclotomic polynomials in the book of Milne [p. 96] (v. 3.07).

Eisenstein polynomials play a fundamental role in the theory of local fields. This is discussed in Serre's book [Chapter I, §6, Proposition 17 and the following Corollary].

It is important to know that the wild inertia subgroup of an absolute Galois group of a local field is a (pro-solvable) pro-p-group: see the monographs of Neukirch [Chapter 2, Proposition 10.2] and Serre [Chapter IV, §1, §2]. Excercise 2 in §2 of Serre is especially important. Extra work is necessary to deduce the claim of the lecture from these facts.

The standard definition of a semi-stable representation is Definition 1.22 from the book of Fontaine and Ouyang [p. 11]. This is equivalent to the definition from the lecture thanks to Lee-Kolchin theorem.

If you have not heard of Lee-Kolchin theorem before, it would be nice to have a look at the exposition in the book of Serre [Chapter V, §3*, p. 35, Theorem]. The proof relies on another important result of representation theory, the density theorem of Burnside. Note that neither of these theorems are necessary for the course. Warning: Wikipedia's version of Lee-Kolchin is not the one discussed here.

The equivalence of the two definitions of semi-stability can be proven without a recourse to Lee-Kolchin.

A proof of Grothendieck's monodromy theorem can be found in the book of Fontaine and Ouyang [Theorem 1.24]. Note that the monodromy theorem of the lecture proves a little bit more than Theorem 1.24. The difference has to do with our definition of a semi-stable representation.

## Lecture 3

The existence of a $\mathbb{Z}_\ell$-lattice in every $\mathbb{Q}_\ell$-representation is an important fact. Fontaine and Ouyang mention this in Section 1.2.2, see the discussion after Definition 1.7. The existence follows since the absolute Galois group is compact and the stabilizer of every lattice is an open subgroup.

Unfortunately I do not have a comprehensive source for the main part of the lecture.

## Lecture 4

The material was borrowed from the book of Serre , although a few arguments were changed. I denoted Teichmüller representatives by the square brackets. This is standard in p-adic Hodge theory, but the notation of Serre is different. The content of the lecture is covered by Chapter II §5 and by Proposition 8 in Chapter II §4 of Serre .

## Lectures 5 and 5bis

I presented a version of the theorem of Katz for perfect fields. Brinon-Conrad and Fontaine-Ouyang treat the case of an arbitrary field of positive characteristic. For this reason they work with Cohen rings rather than rings of Witt vectors.

You can find the theorem of Katz in Part I §3 of Brinon-Conrad . We shall need the following definitions and results: Definition 3.1.1, Definition 3.1.4, Lemma 3.1.7, Theorem 3.1.8, Definition 3.2.2, Lemma 3.2.4, Theorem 3.2.5, Lemmas 3.2.6 and 3.2.7.

An alternative source is Chapter 2, Sections 2.2–2.3 of Fontaine-Ouyang .

The primary source is the monograph of Katz on p-adic modular forms , Proposition 4.1.1. This may be a bit hard to read at this stage.

(Coming soon)

## References

N.M. Katz. p-adic properties of modular schemes and modular forms. Modular functions of one variable, III, pp. 69–190. Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973.

E. Kowalski. Representation theory. Lecture notes, ETH Zürich – Spring Semester 2011.

J.S. Milne. Algebraic number theory. Lecture notes, version 3.07.

J. Neukirch. Algebraic number theory. Grundlehren der mathematischen Wissenschaften, Vol. 322. Springer, Berlin, Heidelberg, 1999.

J.-P. Serre. Local fields. Graduate Texts in Mathematics, Vol. 67. Springer, New York, 1979.

J.-P. Serre. Lie Algebras and Lie Groups. Lecture Notes in Mathematics, Vol. 1500. Springer, Berlin, Heidelberg, 1992.

J.H. Silverman. The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, Vol. 106. Springer, New York, 1986.