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
 BombieriVinogradov 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, 6179,
CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI, 2004
