WAMS research school Topics in Analytic and Transcendental Number Theory Institute for Advanced Studies in Basic Sciences (IASBS) Zanjan, Iran - July 1th - July 13th 2017

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Courses

Elementary methods in analytic number theory Lecturers: Mehdi Hassani & Jean-Marc Deshouillers Part I. Elementary and combinatorial number theory Generalities on primes Arithmetical functions Dirichlet convolution Formal Euler products Mobius inversion formulas Chebyshev's upper and lower bounds for primes up to X. Brun's sieve and twin primes Part II. Calculus Mean value of some arithmetical functions Divisors and the hyperbola Sums of two squares and the circle problem Part III. Fourier Upper bounds for trigonometric sums : van der Corput Application to Voronoi's theorem Application to the circle problem Part IV. Heuristics and probabilistic methods (Time permitting) Normal orders of arithmetic functions Distribution modulo 1 REFERENCES: Gerald Tenenbaum Course on analytic and probabilistic number theory (published in English by AMS) is as a very good reference for the above course and related topics.

Analytic Problems for Elliptic curves Lecturers: Amir Akbary & Francesco Pappalardi Part I : Algebraic Number Theory basics of algebraic number theory (Number fields, rings of integers, splitting of primes) splitting of primes in cyclotomic fields and primes in arithmetic progressions Part II : Elliptic Curves basics of elliptic curves (structure of the group of rational points, elliptic curves over finite fields) reduction mod p of elliptic curves splitting of primes in division fields of elliptic curves Part III : Analytic Results Prime number theorem for arithmetic progressions Bombieri-Vinogradov theorem Chebotarev density theorem Part IV : Applications classical Titchmarch divisor problem elliptic Titchmarsh divisor problem Serre's cyclicity problem Artin's conjecture and its elliptic analogue (time permitting) REFERENCES: Lawrence C. Washington, Elliptic Curves: Number Theory and Crptography. Chapman & Hall (CRC) 2003. Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer GTM 2009 Alina Carmen Cojocaru, Questions about the reductions modulo primes of an elliptic curve. Number theory, 61-79, CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI, 2004