# SCIENTIFIC PROGRAMS AND ACTIVITIES

December 21, 2014
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 TBA Past Seminars April 7 Damien Roy (University of Ottawa) On Schmidt and Summerer parametric geometry of numbers In a series of recent papers, W.M. Schmidt and L. Summerer develop a remarkable theory of parametric geometry of numbers which enables them to recover many results about simultaneous rational approximation to families of Q-linearly independent real numbers, or about the dual problem of forming small linear integer combinations of such numbers. They recover classical results of Khintchine and Jarnik as well as more recent results by Bugeaud and Laurent. They also find many new results of Diophantine approximation. Their theory provides constraints on the behavior of the successive minima of a natural family of one parameter convex bodies attached to a given n-tuple of real numbers, in terms of this varying parameter. In this talk, we are interested in the converse problem of constructing n-tuples of numbers for which the corresponding successive minima obey given behavior. We will present the general theory of Schmidt and Summerer, mention some applications, and report on recent progress concerning the above problem. Mar. 24 Wentang Kuo (University of Waterloo) On Erd\H{o}s-Pomerance conjecture for rank one Drinfeld modules (tentative abstract) Let $\phi$ be a sgn-normalized rank one Drinfeld $A$-module defined over $\mathcal{O}$, the integral closure of $A$ in the Hilbert class field of $A$. We prove an analogue of a conjecture of Erd\H{o}s and Pomerance for $\varphi$. Given any $0 \neq \alpha \in \mathcal{O}$ and an ideal $\frak{M}$ in $\mathcal{O}$, let $f_{\alpha}\left(\frak{M}\right) = \left\{f \in A \mid \phi_{f}\left(\alpha\right) \equiv 0 \pmod{\frak{M}} \right\}$ be the ideal in $A$. We denote by $\omega\big(f_\alpha\left(\frak{M}\right)\big)$ the number of distinct prime ideal divisors of $f_\alpha\left(\frak{M}\right)$. If $q \neq 2$, we prove that there exists a normal distribution for the quantity $$\frac{\omega\big(f_\alpha\left(\frak{M}\right)\big)-\frac{1}{2} \left(\log\deg\frak{M}\right)^2}{\frac{1}{\sqrt{3}} \left(\log\deg\frak{M}\right)^{3/2}}.$$ This is the jointed work with Yen-Liang Kuan and Wei-Chen Yao Mar. 17 Kevin Hare (University of Waterloo) Base $d$ expansions with digits $0$ to $q-1$ Let $d$ and $q$ be positive integers, and consider representing a positive integer $n$ with base $d$ and digits $0, 1, \cdots, q-1$. If $q < d$, then not all positive integers can be represented. If $q = d$, every positive integer can be represented in exactly one way. If $q > d$, then there may be multiple ways of representing the integer $n$. Let $f_{d,q}(n)$ be the number of representations of $n$ with base $d$ and digits $0, 1, \cdots, q-1$. For example, if $d = 2$ and $q = 7$ we might represent 6 as $(110)_2 = 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0$ as well as $(102)_2 = 1 \cdot 2^2 + 0 \cdot 2^1 + 2 \cdot 2^0$. In fact, there are six representations in this case $(110)_2, (102)_2, (30)_2, (22)_2, (14)_2$ and $(6)_2$, hence $f_{2,7}(6) = 6$. In this talk we will discuss the asymptotics of $f_{d,q}(n)$ as $n\to \infty$. This depends in a rather strange way on the Generalized Thue-Morse sequence. While many results are computationally/experimentally true, only partial results are known. Mar. 10 Julian Rosen (University of Waterloo) Multiple zeta values and their truncations The multiple zeta values are real numbers generalizing the values of the Riemann zeta function at positive integers. They are known to satisfy certain algebraic relations, but there are many conjectured transcendence results that have proven to be quite difficult. Truncations of the defining series are called multiple harmonic sums. These rational numbers have interesting arithmetic properties, and are viewed as a finite analogue of the multiple zeta values. We will discuss the parallels between the two theories, as well as some recent results concerning multiple harmonic sums. Mar. 3 Yu-Ru Liu (University of Waterloo) Equidistribution of polynomial sequences in function fields We prove a function field analog of Weyl's classical theorem on equidistribution of polynomial sequences. Our result covers the case when the degree of the polynomial is greater than or equal to the characteristic of the field, which is a natural barrier when one tries to apply the Weyl differencing process to function fields. We also discuss applications to Sakozy's theorem in function fields. This is a joint work with Thai Hoang Le. February 10 Yuanlin Li (Brock University) On The Davenport Constant Let G be a finite abelian group. The Davenport constant D(G) of G is defined to be the smallest positive integer d such that every sequence of d elements in G contains a nonempty subsequence with the product of all its elements equal to 1 the identity of G. The problem of finding D(G) was proposed by H. Davenport in 1966, and it was pointed out that D(G) is connected to the algebraic number theory in the following way. Let K be an algebraic number field and G be its class group. Then D(G) is the maximal number of the prime ideals (counting multiplicity) that can occur in the decomposition of an irreducibleinteger in K. In this talk, we will review some known results regarding the Davenport constant of abelian groups and discuss a few methods which can be used to find the exact value of $D(G). Some recent new results will also be presented. February 3 Stanley Xiao (University of Waterloo) Powerfree values of polynomials In this talk I will give an overview of the progress made on the power-free values of polynomial problem. In particular, I intend to discuss the determinant method of Heath-Brown and Salberger, which so far is the most promising technique on this problem. January 27 Jing-Jing Huang (University of Toronto) Rational points near manifold and Diophantine approximation We will discuss the two topics mentioned in the title. 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