## Arithmetic

### Class field theory

These old notes summarize class field theory. As such they are not particularly well organized, as I realize now. They were created when studying the approach WEIL had chosen in his *Basic Number Theory*: the theory of simple algebras, in favor over group cohomology, which I was used to from Serre. So, in part, these notes follow his suggestion to translate between these two points of view. The emphasis here is on the local situation, as well as on the global function field case. Almost no proofs are given.

### Counting points on curves over finite fields by *Enrico BOMBIERI*

This is the *BOURBAKI* seminar talk n° 430 given by *Enrico BOMBIERI* in June 1973, set in TeX with the appropriate *BOURBAKI* class style. I only took advantage of the time since passed to complete the reference to *STARK*‘s article, published later in 1973, and added a reference to *SCHMIDT*‘s Lecture Notes that appeared in 1976. I corrected a minor typo in formula (2).

### Artin L-functions after Weil, Grothendieck

This paper summarizes some notes taken when comparing Weil’s presentation of Artin’s L-functions with Grothendieck’s.

Dedicated to my brother *Joachim*.

### Local ε-factors

A short note on local L-series, functional equation, ε-factors, Gauss sums and integrals.

Dedicated to my sister *Petra*.

### Numbers of solutions of equations in finite fields by *André WEIL*

I thank the American Mathematical Society for their kind permission to publish my TeX version here.

This is the classical article by André Weil that contain his famous conjectures on the ζ-function of varieties over finite fields. It has since July 2007 again become available at the American Mathematical Society site: Bull. Am. Math. Soc. **55** (1949), 497-508.

### Zeta-Function of *x*^{4} + y^{4} + z^{4} = 0

^{4}+ y

^{4}+ z

^{4}= 0

This note investigates the ζ-function of the Curve *C : x ^{4} + y^{4} + z^{4} = 0*, whose interesting factorization stimulated a closer look at its geometry and arithmetic. It was initiated by the question to find simple examples of curves with no rational points.

The first part develops the ζ-function formalism and its connection to rational points. In particular an isogeny for the Jacobian variety *J* of *C* is postulated: *J ∼ E x E x E*, where *E* is the elliptic curve *y ^{2} = x^{3} + x*. The second part provides the proof, based on methods from the classical paper of André WEIL and a theorem of Gerd FALTINGS.

### Algorithms for computer programs

This is a collection of routines for doing elementary arithmetic on a computer, like generating primes, factoring into primes and calculations on curves over finite fields. They originated from some Algol programs of mine written in 1978-1980 at the Faculty of Mathematics in Bielefeld. They are now implemented in C.

These notes are meant to supplement the program logic, with emphasis put on the conceptual ideas behind those algorithms. These programs are made available under the Apache license. The only change in version 1.7 is to the factorisation routine. In version 1.8 the program listings make use of the package *listings*.

### Zeta-function and Bernoulli numbers

This note gathers the salient features of Dirichlet-series and their convergence, in particular the Riemann ζ-function, some special values, including Bernoulli numbers. We follow Riemann for the proof of its *functional equation*, which was added in version 1.1 in 2011.

The appendix treats the *analytical continuation* of the ζ-function in an elementary way, without using the functional equation.

### My thesis: Kohomologie S-arithmetischer Untergruppen der SL(2) über einem Funktionenkörper und arithmetische Eigenschaften automorpher Formen

- version 1.5, 2015-03-04 : (deutsch)

first published: 1999-07-09

MSC Primary 11F75, Secondary 11F70, 11F67

My thesis was written under the guidance of Günter Harder. The results in part one confirmed a conjecture of his: the vanishing of the rational cohomology H^{i}(Γ,**Q**) = 0 of an S-arithmetic group Γ ⊂ SL(2,F) for a function field F/**F**_{q} in degree i ≠ 0, card(S). This was generalized by Harder to other algebraic groups (published in Inventiones in 1977). The other parts deal with the theory of Eisenstein series and its application to the construction of automorphic cohomology classes.

The new version 1.5 loads the package hyperref with option colorlinks, instead of german.sty the package babel is used. The source is now under git control (git repository).