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. 1 2 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 [email protected] 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 • • • • 3 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) 4 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 5 [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: [email protected]

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