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ANALYTIC NUMBER THEORY IN FUNCTION FIELDS
TCC COURSE 2015
JULIO ANDRADE
1. Introduction
Title: Analytic Number Theory in Function Fields
Time slot: Fridays 10am-12pm (No lectures on 08/05, 12/06 and 19/06)
Additional lectures held on: Monday 11th May 10am-12pm; Monday 1st
June 10am-12pm and Thursday 11th June 10am-12pm.
Starting: 1st May 2015
Ending: 11th June 2015
Level: graduate course (in the Taught Course Centre)
Prerequisites: For the first part of the course, students should know some
basic complex analysis and number theory. Previous experience with analytic number theory would be quite helpful, but not strictly necessary since
we will review several topics of analytic number theory in this course. For
later parts of the course, a standard first course on basic algebraic number
theory also would be useful and an introductory course in abstract algebra
covering, in addition to Galois theory, commutative algebra as presented, for
example, in the classic text of Atiyah and MacDonald [4] should be enough
background.
Office hours: to be confirmed
Course webpage: http://julioandrade.weebly.com/analytic-number-theory-in-function-fi
html
2. Synopsis
Elementary number theory is concerned with the arithmetic properties
of the ring of integers Z, and its field of fractions, the rational numbers,
Q. Early on in the development of the subject it was noticed that Z has
many properties in common with Fq [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 non-zero 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 Z have analogues of ring
Fq [t]. This is indeed the case.
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JULIO ANDRADE
This course will explore the analogies between function fields and number fields through the use of analytic methods to tackle number theory
problems over Fq [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 Q, which are called algebraic number fields,
and investigating properties of the ring of algebraic integers OK ⊂ K, defined as the integral closure of Z in K. Similarly, we can consider k = Fq (t),
the quotient field of Fq [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.
3. Syllabus
The following syllabus is tentative and subject to change.
Lecture 1:
• Number fields and Function fields
– Global fields: Basic analogies and contrasts
• Polynomials over Finite Fields
• Primes and the Fundamental Theorem of Arithmetic over Fq [t]
• Zeta function of Fq [t]
• Prime Number Theorem for Polynomials
Lecture 2:
• Arithmetical Functions and Dirichlet Multiplication for Fq [t]
ANALYTIC NUMBER THEORY IN FUNCTION FIELDS
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Averages of Arithmetical Functions
Congruences and the Reciprocity Law
Dirichlet Characters and L-series for Fq (T )
Dirichlet’s Theorem on Primes in Arithmetic Progression in Fq [t]
Lecture 3:
• Foundations of the Theory of Algebraic Function Fields and Global
Function Fields
– Valuations
– Places
– Rational Function Field Fq (T )
– Additive characters and measure
– Characters of Fq (T )
– Fourier transform
– Divisors
– The Riemann-Roch Theorem
– The zeta function ζK (s) of a global function field
– Analytic continuation, functional equation and Riemann hypothesis for ζK (s)
Lecture 4:
• Average Value Theorems in Function Fields
• Some Sieve methods in function fields
Lecture 5:
• Selberg’s Theorem in Function Fields
• An introduction to the Katz-Sarnak philosophy and Random Matrix
Theory
• Traces of high powers of the Frobenius class in the hyperelliptic
ensemble
Lecture 6:
• Moments of L-functions in function fields
• Ratios Conjecture of L-functions over Fq (T )
Lecture 7:
• Revisiting Mean Values of Arithmetic Functions in Fq [t]
• Equidistribution theorems, statistics of arithmetic functions and matrix integrals
Lecture 8:
• Overview of a Proof of the Function Field Riemann Hypothesis
• New directions and problems
(based on a 2 hours time for each lecture)
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JULIO ANDRADE
4. Organisation
I intend to visit all the centres (Bristol, Bath, Warwick and London), so
office hours can be arranged locally according to demand. Please check this
space for times and dates that I will be available, and email me if you would
like to meet.
Bath
Bristol
London
Warwick
tbc
tbc
tbc
tbc
5. Assessment
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.
References
[1] Andrade, J.C., Keating, J.P.: The mean value of L( 21 , χ) in the hyperelliptic ensemble,
J. Number Theory, 132, 2793–2816 (2012).
[2] Andrade, J.C., Keating, J.P: Conjectures for the Integral Moments and Ratios of
L–functions over function fields, J. Number Theory, 142, 102–148 (2014).
[3] T.M. Apostol, Introduction to Analytic Number Theory. Undergraduate Texts in
Mathematics. Springer-Verlag, New York, 1998. 340 pp. ISBN: 978-0387901633.
[4] M.E. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra, AddisonWesley Series in Mathematics, Westview Press, 1994, iv+138 pp. ISBN: 9780201407518.
[5] D. Carmon, The autocorrelation of the M¨
obius function and Chowla’s conjecture for
the rational function field in characteristic 2, preprint.
[6] D. Faifman and Z. Rudnick, Statistics of the zeros of zeta functions in families of
hyperelliptic curves over a finite field, Compositio Mathematica 146(2010), 81–101.
[7] M. van Frankenhuijsen, The Riemann Hypothesis for Function Fields: Frobenius Flow
and Shift Operators. London Mathematical Society Student Texts, 80. Cambridge
University Press, Cambridge, 2014. 162 pp. ISBN: 978-1107685314.
[8] D. Goss, Basic Structures of Function Field Arithmetic. Springer-Verlag, New York,
2013. 424 pp. ISBN: 978-3540635413.
[9] D. Goss, D. R. Hayes and M. Rosen, The Arithmetic of Function Fields. Ohio State
University Mathematical Research Institute Publications, 2. De Gruyter, Columbus,
1992. 494 pp. ISBN: 978-3110131710.
[10] H. Iwaniec and E. Kowalski, Analytic Number Theory. Colloquium Publications 53.
American Mathematical Society, Providence, 2004. 615 pp. ISBN: 978-0821836330.
[11] N.M. Katz and P. Sarnak, Random Matrices, Frobenius eigenvalues, and monodromy,
American Mathematical Society Colloquium Publications, vol. 45, American Mathematical Society, Providence, RI, 1999.
ANALYTIC NUMBER THEORY IN FUNCTION FIELDS
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[12] N.M. Katz and P. Sarnak, Zeroes of zeta functions and symmetry, Bull. Amer. Math.
Soc. (N.S.) 36 (1999), 1–26.
[13] J. P. Keating and N.C. Snaith, Random matrix theory and ζ( 21 + it), Comm. Math.
Phys. 214 (2000), 57–89.
[14] J. P. Keating and N.C. Snaith, Random matrix theory and L-functions at s = 12 ,
Comm. Math. Phys. 214 (2000), 91–110.
[15] P. Kurlberg and Z. Rudnick, The fluctuations in the number of points on a hyperelliptic curve over a finite field, J. of Number Theory, 129(2009), 580–587.
[16] S. Lang, Algebra. Revised third edition. Graduate Texts in Mathematics, 211.
Springer-Verlag, New York, 2002. xvi+914 pp. ISBN: 0-387-95385-X.
[17] H.L. Montgomery and R.C. Vaughan, Multiplicative Number Theory I: Classical Theory. Cambridge Studies in Advanced Mathematics 97. Cambridge University Press,
Cambridge, 2012. 572 pp. ISBN: 978-1107405820.
[18] M. Rosen, Number theory in function fields. Graduate Texts in Mathematics, 210.
Springer-Verlag, New York, 2002. xii+358 pp. ISBN: 0-387-95335-3.
[19] Z. Rudnick, Traces of high powers of the Frobenius class in the hyperelliptic ensemble,
Acta Arith., 143(2010), 81–99.
[20] Zeev Rudnick, Squarefree values of polynomials over the rational function field.
arXiv:1211.6733 [math.NT]
[21] G.D.V. Salvador, Topics in the Theory of Algebraic Function Fields. Mathematics:
Theory and Applications. Birkhuser, Boston, 2006. 652 pp. ISBN: 978-0817644802.
[22] H. Stichtenoth, Algebraic Function Fields and Codes. Graduate Texts in Mathematics,
254. Springer-Verlag, New York, 2008. ix+360 pp. ISBN: 978-3540768777.
[23] D.S. Thakur, Function Field Arithmetic. World Scientific Publishing Co., Singapore,
2004. 404 pp. ISBN: 978-9812388391.
[24] E.C. Titchmarsh, The Theory of the Riemann Zeta-Function. Oxford Science Publications. Oxford University Press, Oxford, 1987. 422 pp. ISBN: 978-0198533696.
Mathematical Institute - University of Oxford
Andrew Wiles Building, Office N1.04
Radcliffe Observatory Quarter
Woodstock Road
Oxford, OX2 6GG
UK
E-mail address: buenodeandra@maths.ox.ac.uk