# Bucharest Number Theory Day, July 16th, 2013

The talks will take place at IMAR in Amfiteatrul Miron Nicolescu.

## Schedule

- 10:00-11:00
**Alina Cojocaru**, University of Illinois at Chicago and IMAR

*Frobenius fields for elliptic curves*

Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, let pi_p be the p-Weil root of E and Q(pi_p) the associated imaginary quadratic field generated by pi_p. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes p < x for which Q(pi_p) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. The latter is joint work with Henryk Iwaniec and Nathan Jones.
- 11:20-12:20
**Vicentiu Pasol**, IMAR

*On higher moments for quadratic Dirichlet L-functions*

We construct the "improved" Multiple Dirichlet Series for quadratic Dirichlet L- functions for all moments over any number field. This is joint work with Adrian Diaconu (U. Minnesota and IMAR).
- 12:20-14:15 Lunch break
- 14:15-15:15
**Nathan Jones**, Univ. of Mississippi

*A local-global principle for power maps*

Let f be a function from the set of natural numbers to itself. We call f a global power map if f(n) = n^k for some non-negative integer exponent k. For a set S of prime numbers, we call f a local power map at S if for each prime p in S, f induces a well-defined group homomorphism on the multiplicative group (Z/pZ)^*. In this talk, I will motivate the conjecture that if f is a local power map at an infinite set S of primes, then f must be a global power map. I will also discuss some progress towards this conjecture.
- 15:30-16:30
**Alexandru Popa**, IMAR

*Trace of Hecke operators on modular forms for congruence subgroups*

In 1956 Selberg and Eichler gave a formula for the trace of Hecke operators acting on modular forms for the full modular group, in terms of class numbers of imaginary quadratic fields. I will present a simple proof that uses the theory of period polynomials, based on an idea from 1992 of Don Zagier. The proof generalizes to congruence subgroups as well, yielding simple formulas for the trace of Hecke and Atkin-Lehner operators acting on modular forms for Gamma0(N). This is joint work with Don Zagier.

Organizers: Alina Cojocaru and Alexandru Popa

Partially supported by the Marie Curie grant PIRG05-GA-2009-248569.