# SCIENTIFIC PROGRAMS AND ACTIVITIES

December 12, 2013
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
 Number Theory Seminar 2013-2014 Fields Institute, Stewart Library, Mondays at 3:30 p.m. Organizing Committee: Leo Goldmakher, Jing-Jing Huang
 Upcoming Seminars Talks will continue in Janaury 2014 Past Seminars December 2 Monday Lluis Vena (University of Toronto) The removal lemma for homomorphisms in abelian groups The triangle removal lemma states that if a graph has a subcubic number of triangles, then removing a subquadratic number of edges suffices to make G free of triangles. One of its most famous applications is a simple proof of Roth's theorem, which asserts that any subset of the integers with positive upper density contains a 3-term arithmetic progression. In 2005, Green showed an analogous result for linear equations in finite abelian groups, the so-called removal lemma for groups. In this talk, we will discuss a combinatorial proof of Green's result, as well as a generalization to homomorphism systems in finite abelian groups. In particular, our results imply a multidimensional version of Szemeredi's theorem. November 25 Jonathan Bober (University of Bristol) Conditionally bounding analytic ranks of elliptic curves I'll describe how to use the explicit formula for the L-function of an elliptic curve to compute upper bounds for the analytic rank, assuming GRH. This method works particularly well for elliptic curves of large rank and (relatively) small conductor, and can be used to compute exact upper bounds for the curves of largest known rank, assuming BSD and GRH. November 18 Kevin McGown (Ursinus College) Euclidean Number Fields and Ergodic Theory When does a number field possess a Euclidean algorithm? We will discuss how generalizations of this question lead us to studying the S-Euclidean minimum of an ideal class, which is a real number attached to some arithmetic data. Generalizing a result of Cerri, we show that this number is rational under certain conditions. We also give some corollaries and discuss the relationship with Lenstra's notion of a norm-Euclidean ideal class and the conjecture of Barnes and Swinnerton-Dyer on quadratic forms. The proof involves using techniques of Berend from ergodic theory and topological dynamics on the appropriate compact group. November 11 No seminar October 28 Chantal David (Concordia University) One-level density for zeroes in famlies of elliptic curves Using the ratios conjectures as introduced by Conrey, Farmer and Zirnbauer, we obtain closed formulas for the one-level density for some families of L-functions attached to elliptic curves, and we can then determine the underlying symmetry types of the families. The one-level density for some of those families was studied in the past for test functions with Fourier transforms of small support, but since the Fourier transforms of the three orthogonal distributions (O, SO(even) and SO(odd)) are undistinguishable for small support, it was not possible to identify the distribution with those techniques. This can be done with the ratios conjectures. The results confirm the conjectures of Katz and Sarnak, and shed more light on the phenomenon of "independent" and "non-independent" zeroes, and the repulsion phenomenon. This is joint work with Duc Khiem Huynh and James Parks. We also present some work in progress in collaboration with Sandro Bettin where we obtain general formulas for the one-level density of one-parameter families of elliptic curves in term of the rank over Q(t) and the average root number. October 14 No seminar (Reading Week) October 7 4:30-5:30 **please note time change for this week only Alex Iosevich, University of Rochester Group actions and Erdos type problems in vector spaces over finite fields We shall use group invariances to study the distribution of simplexes in vector spaces over finite fields. It turns out that the most convenient way to study repeated simplexes is via appropriate norms of the natural "measure" on the set $E-gE$, where $E$ is a subset of the ${\Bbb F}_q^d$, $d \ge 2$, and $g$ is an element of the orthogonal group $O_d({\Bbb F}_q)$. September 30 No seminar (Fields Medal Symposium) September 23 Leo Goldmakher (University of Toronto) On the least quadratic nonresidue I will discuss the relationship between bounds on long character sums and bounds on the least quadratic nonresidue. In particular, I will show how small savings on one leads to massive savings in the other. This is joint work with Jonathan Bober. September 16 Giorgis Petridis, University of Rochester Higher sumsets with different summands