Second Bucharest Number Theory Day, July 22nd, 2014
Conferinta in memoria domnului profesor
Serban Basarab.
The talks will take place at IMAR in Amfiteatrul "Miron Nicolescu" (parter).
Invited speakers
- Florin Boca, University of Illinois at Urbana-Champaign and IMAR
- Alexandru Buium, University of New Mexico
- Alina Cojocaru, University of Illinois at Chicago and IMAR
- Adrian Diaconu, University of Minnesota and IMAR
- Nathan Jones, University of Mississippi
Schedule and abstracts
- 9:30-10:20
- Alexandru Buium
Arithmetic analogue of Painleve VI
Yu. I. Manin discovered in the early nineties that the Painleve VI differential equation admits a geometric description involving a pencil of elliptic curves. The talk explains how one can develop an arithmetic analogue of this construction in which functions are replaced by numbers and derivatives are replaced by Fermat quotient operators. The talk is based on recent joint work with Yu. I. Manin.
- 10:40-11:20
- Alina Cojocaru
Drinfeld modules, Frobenius endomorphisms, and CM-liftings
Given a finite Galois extension L/K of global fields and a conjugacy class
C of Gal(L/K), a fundamental problem is that of describing the (unramified)
primes p of K for which the conjugacy class of the Frobenius at p
is C. The Chebotarev Density Theorem provides the density of these primes, while,
in general, the characterization of the primes themselves is a finer and deeper question.
We focus on unraveling this question for the division fields of a generic Drinfeld module. For Drinfeld modules of rank 2, we obtain an explicit global description
of the Frobenius. We apply this description to derive a criterion for the splitting modulo primes of a class of non-solvable polynomials and to study the frequency with which the reductions of Drinfeld modules have small endomorphism rings. We also generalize some of these results to higher rank Drinfeld modules and prove CM-lifting theorems for Drinfeld modules. This is joint work with Mihran Papikian (Pennsylvania State University, USA).
- 11:30-12:10
- Nathan Jones
Elliptic curves with 2-torsion contained in the 3-torsion field
There is a modular curve X'(6) of level 6 defined over Q whose rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this talk I will characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model of X'(6). The motivation is two-fold: on the one hand, X'(6) belongs to the list of modular curves which parametrize non-Serre curves (and is not well-known), and on the other hand, the set of rational points of X'(6) gives an infinite family of examples of elliptic curves with non-abelian ``entanglement fields,'' which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over Q. This is based on joint work with J. Brau.
- 12:10-14:50 Lunch break
- 14:50-15:40
- Adrian Diaconu
Multiple Dirichlet series
- 16:00-16:50
- Florin Boca
Irregularities in the distribution of lattice angles (Euclidean
vs. hyperbolic)
Spacing statistics provide convenient ways of measuring (and
distinguishing between) the randomness of sequences (or more generally
increasing sequences of finite sets of real numbers) that are uniformly
distributed. Roughly speaking, they measure the distribution of n-tuples
of differences (spacings) between these numbers.
A familiar example of a sequence of sets which is uniformly distributed
mod 1 is given by the directions of vectors joining a fixed point in the
Euclidean plane, with all (or only visible) points of integer coordinates
inside balls of fixed center and radius R going to infinity. However,
these directions are not randomly distributed, and even the study of their
most popular spacing statistics (limiting gap distribution and pair
correlation function) turned out to pose challenges.
This talk will discuss some results on the spacing statistics for this
type of geometric configurations and for their hyperbolic counterpart,
where the Euclidean plane and the lattice Z^2 are replaced by the upper
half plane and respectively by a lattice in PSL(2,R).
Organizers: Alina Cojocaru and Alexandru Popa