# Third Bucharest Number Theory Day, August 3rd, 2015

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

## Invited speakers

• Constantin-Nicolae Beli, IMAR
• Florin Boca, University of Illinois at Urbana-Champaign and IMAR
• Alina Cojocaru, University of Illinois at Chicago and IMAR
• Adrian Diaconu, University of Minnesota and IMAR
• Nathan Jones, University of Illinois at Chicago
• Maria Nastasescu, Caltech
• Alexandru Zaharescu, University of Illinois at Urbana-Champaign

## Schedule and abstracts

9:15-10:00
Florin Boca
Some statistical properties of digits in continued fractions
Specifically, we will discuss results on generalized Gauss-Kuzmin statistics and on the distribution of partial sums of digits of a random irrational number.

10:00-10:15 Coffee break

10:15-11:00

11:10-11:40
Alina Cojocaru
The number of non-S_5 quintic extensions of bounded discriminant
A central problem in arithmetic statistics is that of understanding the asymptotic number of number fields with a given Galois group and bounded discriminant.
For example, the study of quintic number fields was pursued by M. Bhargava who showed that the number of such fields with discriminant bounded by x is asymptotically equal to cx for an explicit positive constant c. Moreover, he showed that 100% of these fields have associated Galois group S_5.
I will report on joint work with M. Bhargava and F. Thorne concerning the number of quintic fields of bounded discriminant and Galois group strictly less than S_5. This relates to a conjecture of G. Malle and improves on prior results by A. Shankar and J. Tsimerman.

11:45-12:15
Constantin-Nicolae Beli
Properties of the symbols ( , ), [ , ) and (( , ))

12:15-14:30 Lunch break

14:30-15:15
Nathan Jones
Elliptic curves with non-abelian entanglement fields
It is a classical fact that cyclotomic fields of co-prime level are linearly disjoint. The analogous statement for division fields of an elliptic curve E is not true, and we call the intersection of two such division fields of co-prime level an entanglement field of E. In light of applications to computing the image in GL_2 of the Galois representation on the torsion of E, it is of interest to classify elliptic curves with entanglement fields which are non-abelian over the base field. In this talk I will speak about recent progress on this problem, which represents ongoing joint work with K. McMurdy.

15:15-15:30 Coffee break

15:30-16:15
Maria Nastasescu
Determination of elliptic curves by their adjoint p-adic L-functions
Fix $$p$$ an odd prime. Let $$E$$ be an elliptic curve over $$\mathbb{Q}$$ with semistable reduction at $$p$$. We show that the adjoint $$p$$-adic L-function of $$E$$ evaluated at infinitely many integers prime to $$p$$ completely determines up to a quadratic twist the isogeny class of $$E$$. To do this, we prove a result on the determination of isobaric representations of $$GL(3,\mathbb{A}_\mathbb{Q})$$ by certain L-values of $$p$$-power twists.

16:25-17:10
Alexandru Zaharescu
Zeros of the Riemann zeta-function on the critical line

Organizers: Alina Cojocaru, Vicentiu Pasol, Alexandru Popa