Analytic Number Theory in Function Fields (TCC)
(No lectures on 08/05, 12/06 and 19/06)
Additional lectures held on:
 Monday 11th May 10am12pm
 Monday 1st June 10am12pm
 Thursday 11th June 10am12pm
Starting: 1st May 2015
Ending: 11th June 2015
Level: graduate course (in the Taught Course Centre)
Prerequisites: some basic complex analysis, number theory and abstract algebra
Office hours: to be confirmed
Complete Description of the Course (.pdf)
Elementary number theory is concerned with the arithmetic properties of the ring of integers $\mathbb{Z}$, and its field of fractions, the rational numbers, $\mathbb{Q}$. Early on in the development of the subject it was noticed that $\mathbb{Z}$ has many properties in common with $\mathbb{F}_{q}[t]$, the ring of polynomials over a finite field. Both rings are principal ideal domains, both have the property that the residue class ring of any nonzero ideal is finite, both rings have infinitely many prime elements, and boh rings have finitely many units. Thus, one is led to suspect that many results which hold for $\mathbb{Z}$ have analogues of ring $\mathbb{F}_{q}[t]$. This is indeed the case.
This course will explore the analogies between function fields and number fields through the use of analytic methods to tackle number theory problems over $\mathbb{F}_{q}[t]$. The techniques range from complex analysis through relatively elementary algebraic arguments. As such it should be suitable for any graduate student with an interest in number theory, analysis and algebraic geometry.
The first part of the course is devoted to illustrating the analogies between function fields and number fields by presenting, for example, analogues of the little theorem of Fermat and Euler, Wilson's theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlet's theorem on primes in arithmetic progression.
In the second part of the course we will study algebraic function fields. Algebraic number theory arises from elementary number theory by considering finite algebraic extensions $K$ of $\mathbb{Q}$, which are called algebraic number fields, and investigating properties of the ring of algebraic integers $\mathcal{O}_{K}\subset K$, defined as the integral closure of $\mathbb{Z}$ in $K$. Similarly, we can consider $k=\mathbb{F}_{q}(t)$, the quotient field of $\mathbb{F}_{q}[t]$ and finite algebraic extensions $L$ of $k$. Fields of this type are called algebraic function fields. More precisely, an algebraic function field with a finite constant field is called a global function field. A global function field is the true analogue of algebraic number field. In this part of the course the basic theory of algebraic function fields, the RIemannRoch theorem and its corollaries and the zeta functions associated to curves over finite fields.
The last part of the course is concerned about several, but interconnected topics, on number theory over function fields. We will study some sieve methods, Selberg's theorem about distritbution of zeros of zeta, the KatzSarnak philosophy, Equidistribution theorems, and moments of $L$functions.
The prerequisites are minimal. We shall need elementary facts on number theory as well as basic concepts from abstract algebra and complex analysis.
Feel free to contact me at j.c.andrade.math@gmail.com with any questions prior to the start of term.
The following syllabus is tentative and subject to change.
Date 
Content 
Recommended exercises 
01/05/2015 
Polynomials over Finite Fields, Primes and the Fundamental Theorem of Arithmetic over Fq[T] 

11/05/2015 
Arithmetical Functions, Average of Arithmetical Functions and Dirichlet Characters in Fq[T] 

15/05/2015 
Foundations of the theory of algebraic function fields and global function fields 

22/05/2015 
Lfunctions and some sieve methods in function fields 

29/05/2015 
Selberg’s Theorem in Function Fields, Random Matrix Theory and the Hyperellitpic Ensemble 

01/06/2015 
Moments of Lfunctions and Ratios Conjecture in function fields 

05/06/2015 
Revisiting mean values of arithmetic functions, equidistribution theorems and RMT 

11/06/2015 
Overview of a Proof of the R.H. for function fields, New directions and problems 

Lecture 1 (slides .pdf)
Lecture 1 (whiteboard .pdf)
Lecture 2 (slides .pdf)
Lecture 2 (whiteboard .pdf)
Lecture 3 (slides .pdf)
Lecture 3 (whiteboard .pdf)
Lecture 4 (slides .pdf)
Lecture 4 (whiteboard .pdf)
Lecture 5 (slides .pdf)
Lecture 5 (whiteboard .pdf)
Lecture 6 (slides .pdf)
Lecture 6 (whiteboard .pdf)
Lecture 7 (slides .pdf)
Lecture 8 (slides .pdf)
Problem Sheet 1 (Recommended exercises: 1; 3; 4; 7; 814 )
Problem Sheet 2 (Recommended exercises: 3; 6; 7; 8)
Problem Sheet 3 (Recommended exercises: 16 )
Problem Sheet 4 (Recommended exercises: 3; 5; 7)
Problem Sheet 5 (Recommended exercises: 16)
Problem Sheet 6 (Recommended exercises: any 3 problems)
Problem Sheet 7 (Recommended exercises: 46)
Problem Sheet 8 (Recommended exercises: 14)
I intend to visit all the centres (Bristol, Bath, Warwick and London), so office hours can be arranged locally according to demand. Pleae check this space for times and dates that I will be available, and email me if you would like to meet.
Bath tbc
Bristol tbc
London tbc
Warwick tbc
At the end of the course, participants will choose from a list of topics/original research articles and should write up an exposition of the chosen result in the function field setting. This exposition should place the result in the context of what has been discussed in the course, and should be detailed for other course participants to be able to follow the main steps of the argument. The completion of the weekly problem sheets is optional but strongly encouraged.
List of Topics for write up an exposition to be use as assessment for this course:
1) Gamma function over function fields.
2) Gauss Sums over function fields.
3) Large sieve over $\mathbb{F}_{q}[T]$.
4) Mean values of arithmetic functions over general global function fields $K/\mathbb{F}_{q}$.
5) SIeve of Eratosthenes over $\mathbb{F}_{q}[T]$.
6) Brun's Sieve over $\mathbb{F}_{q}[T]$.
7) LargeSieve over $\mathbb{F}_{q}[T]$.
8) BombieriVinogradov Theorem over $\mathbb{F}_{q}[T]$.
9) The least prime in arithmetic progressions over $\mathbb{F}_{q}[T]$.
If you have any other topic which you wish to write about in the $\mathbb{F}_{q}[T]$ setting please send me an email and we can discuss.
[1] J.C. Andrade and J.P. Keating: The mean value of $L(\tfrac{1}{2},\chi)$ in the hyperelliptic ensemble, J. Number Theory, 132, 27932816 (2012).
[2] T.M. Apostol: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics. SpringerVerlag, New York, 1998.
[3] D. Faifman and Z. Rudnick: Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field, Compositio Mathematica 146,
81101 (2010).
[4] N.M. Katz and P. Sarnak: Random Matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society, vol. 45 (1999).
[5] M. Rosen: Number Theory in Function Fields. Graduate Texts in Mathematics, 210. SpringerVerlag, New York 2002.