# Fifth Bucharest Number Theory Day July 10-11, 2017

Organizers: Alina Cojocaru, Vicențiu Pașol, Alexandru Popa

The talks will take place at IMAR in Amfiteatrul "Miron Nicolescu" (parter). No registration is necessary.

The workshop is partly supported by BitDefender and by CNCS-UEFISCDI grant TE-2014-4-2077.

## Speakers

• Nicu Beli, IMAR
• Nicolae Bonciocat, IMAR
• Cristian Cobeli, IMAR
• Alina Cojocaru, University of Illinois at Chicago and IMAR
• Adrian Diaconu, University of Minnesotta and IMAR
• Alexandra Florea, Stanford University
• Nathan Jones, University of Illinois at Chicago
• Florin Nicolae, IMAR
• Alexandru Popa, IMAR

## Schedule

Monday, July 10th Tuesday, July 11th Florin Nicolae On the class group of imaginary quadratic fields Radu Gaba A generalization of a congruence of Ramanujan Coffee break Nathan Jones Elliptic curves with non-abelian entanglements Alina Cojocaru Constants in Titchmarsh divisor problems for elliptic curves Adrian Diaconu On averages of families of L-functions associated to moduli of curves over finite fields Nicolae Bonciocat Using prime numbers in attempts to understand polynomials Lunch break Alexandra Florea The fourth moment of quadratic Dirichlet L-functions over function fields Nicu Beli Analogues of the pn-th Hilbert symbol in characteristic p Coffee break Alexandru Popa On the trace formula for Hecke operators Cristian Cobeli On the numbers behind some beautiful pictures

## Abstracts

Nicu Beli: Analogues of the pn-th Hilbert symbol in characteristic p
The p-th Hilbert symbol (_,_) from characteristic different from p has two analogues in characteristic p: [_,_) and the lesser known ((_,_)). The symbol [_,_) generalizes via Witt vectors to an analogue of the pn-th Hilbert symbol, which takes values in the pn torsion of the Brauer group. We prove that ((_,_)) admits similar generalizations. The symbols we introduce are in terms of central simple algebras. Our construction involves Weyl algebras and Witt vectors. With the help of the new symbols we give a representation theorem for the pn torsion of the Brauer group, that extends a known result when n=1.
Nicolae Bonciocat: Using prime numbers in attempts to understand polynomials
We describe several ways in which prime numbers help us understand polynomials, especially factorization properties such as irreducibility and separability. We will analyse this from a number theoretical point of view, then from a geometrical point of view by Newton polygon methods, and finally from an algebraic point of view, in a non-archimedean setting.
Cristian Cobeli: On the numbers behind some beautiful pictures
We discuss a few problems related to the distribution of some nice sequences.
Alina Cojocaru: Constants in Titchmarsh divisor problems for elliptic curves
Inspired by the analogy between the group of units of the finite field $$\mathbb{F}_p$$ and the group of points of an elliptic curve $$E/\mathbb{F}_p$$, we consider elliptic curve analogues of the classical Titchmarsh divisor problem, such as analogues investigated by E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg in the context of reductions modulo primes of an elliptic curve $$E/\mathbb{Q}$$. We pursue a comprehensive study of the constants emerging in these problems and prove upper bounds, explicit formulae, and averages for these constants. This is joint work with Renee Bell (MIT), Clifford Blakestad (Univ. Colorado), Alexander Cowan (Columbia Univ.), Nathan Jones (Univ. Illinois, Chicago), Vlad Matei (Univ. Wisconsin, Madison), Geoffrey Smith (Harvard Univ.), and Isabel Vogt (MIT), based on a research project started at the 2016 Arizona Winter School.
Adrian Diaconu: On averages of families of L-functions associated to moduli of curves over finite fields
In this talk I will discuss some aspects of equivariant Euler characteristics of the moduli space of stable curves over finite fields. These Euler characteristics occur naturally in the study of mean values of the corresponding families of L-functions.
Alexandra Florea: The fourth moment of quadratic Dirichlet L-functions over function fields
I will discuss moments of L-functions over function fields, and I will focus on the fourth moment in the family of quadratic Dirichlet L-functions, obtaining part of an asymptotic formula conjectured by Andrade and Keating.
Radu Gaba: A generalization of a congruence of Ramanujan
In this talk, we present a generalization to modular forms of prime level of Ramanujan's congruence between the Fourier coefficients of a cusp form and those of an Eisenstein series of weight 12 for the modular group. The proof is based on methods introduced by Ihara and used by Ribet and Serre in order to prove congruences between cusp forms of different level.
Nathan Jones: Elliptic curves with non-abelian entanglements
Let $$K$$ be a number field. An elliptic curve $$E/K$$ is said to have a non-abelian entanglement if there are relatively prime positive integers, $$m_1$$ and $$m_2$$, such that $$K(E[m_1])\cap K(E[m_2])$$ is a non-abelian Galois extension of $$K$$. In this talk, we will discuss our ongoing efforts to classify, using explicit methods, all infinite families of elliptic curves $$E/K$$, for a fixed $$K$$, with non-abelian entanglements. This problem is closely related to that of determining when the image of $$\rho_E$$ in $$GL_2(\hat{Z})$$ is maximal, and to the study of correction factors for various conjectural constants for elliptic curves over $$\mathbb{Q}$$. This is based on joint work with Ken McMurdy.
Florin Nicolae: On the class group of imaginary quadratic fields
Which are the imaginary quadratic fields with class group of exponent 4?
Alexandru Popa: On the trace formula for Hecke operators
In joint work with Don Zagier, we gave a new, algebraic proof of the trace formula for Hecke operators on modular forms for the modular group. I will present a generalization for congruence subgroups, which can be stated as a very simple cohomological trace formula.