Department of Mathematics and Computer Science Number Theory and Combinatorics Seminar Spring 2018 Talks are at noon on Monday in various rooms in University Hall For more information, or to receive an email announcement of each week's seminar, contact Nathan Ng < ng AT cs DOT uleth DOT ca > or Dave Morris .
 The next talk: May 7 at noon in C630 Alia Hamieh (University of Northern British Columbia) Non-vanishing of $L$-functions of Hilbert Modular Forms inside the Critical Strip In this talk, I will discuss recent joint work with Wissam Raji. We show that, on average, the $L$-functions of cuspidal Hilbert modular forms with sufficiently large weight $k$ do not vanish on the line segments $\Im(s)=t_{0}$, $\Re(s)\in(\frac{k-1}{2},\frac{k}{2}-\epsilon)\cup(\frac{k}{2}+\epsilon,\frac{k+1}{2})$. The proof follows from computing the Fourier expansion of a certain kernel function associated with Hilbert modular forms and estimating its first Fourier coefficient. This result is analogous to the case of classical modular forms which was proved by W. Kohnen in 1997.
 Talks in the series this semester: (Click on any title for more info, including the abstract. Then click on it again to hide the info.)

 Date Speaker Title Jan 15 everyone Open problem session at noon in B543 Please bring your favourite (math) problems. Anyone with a problem to share will be given about 5 minutes to present it. We will also choose most of the speakers for the rest of the semester. Jan 29 Nathan Ng Mean values of long Dirichlet polynomials at noon in B543 (University of Lethbridge) A Dirichlet polynomial is a function of the form $A(t)=\sum_{n \le N} a_n n^{-it}$ where $a_n$ is a complex sequence, $N \in \mathbb{N}$, and $t \in \mathbb{R}$. For $T \ge 1$, the mean values $$\int_{0}^{T} |A(t)|^2 \, dt$$ play an important role in the theory of L-functions. I will discuss work of Goldston and Gonek on how to evaluate these integrals in the case that $T < N < T^2$. This will depend on the correlation sums $\sum_{n \le x} a_n a_{n+h} \text{ for } h \in \mathbb{N}.$ If time permits, I will discuss a conjecture of Conrey and Keating in the case that $a_n$ corresponds to a generalized divisor function and $N > T$. Feb 12 Ha Tran Reduced Ideals from the Reduction Algorithm at noon in B543 (University of Calgary) Reduced ideals of a number field $F$ have inverses of small norms and they form a finite and regularly distributed set in the infrastructure of $F$. Therefore, they can be used to compute the regulator and the class number of a number field [5, 3, 2, 1, 4]. One usually applies the reduction algorithm (see Algorithm 10.3 in [4]) to find them. Ideals obtained from this algorithm are called 1-reduced. There exist reduced ideals that are not 1-reduced. We will show that these ideals have inverses of larger norms among reduced ones. Especially, we represent a sufficient and necessary condition for reduced ideals of real quadratic fields to be obtained from the reduction algorithm. References Johannes Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. In Séminaire de Théorie des Nombres, Paris 1988-1989, volume 91 of Progr. Math., pages 27-41. Birkhäuser Boston, Boston, MA, 1990. Johannes Buchmann and H. C. Williams. On the infrastructure of the principal ideal class of an algebraic number field of unit rank one. Math. Comp., 50(182):569-579, 1988. H. W. Lenstra, Jr. On the calculation of regulators and class numbers of quadratic fields. In Number theory days, 1980 (Exeter, 1980), volume 56 of London Math. Soc. Lecture Note Ser., pages 123-150. Cambridge Univ. Press, Cambridge, 1982. René Schoof. Computing Arakelov class groups. In Algorithmic number theory: lattices, number fields, curves and cryptography, volume 44 of Math. Sci. Res. Inst. Publ., pages 447-495. Cambridge Univ. Press, Cambridge, 2008. Daniel Shanks. The infrastructure of a real quadratic field and its applications. In Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972), pages 217-224. Univ. Colorado, Boulder, Colo., 1972. Feb 26 Andrew Fiori A Geometric Description of Arthur Packets at noon in B543 (University of Lethbridge) In this talk I will discuss joint work with Clifton Cunningham, Ahamed Moussaui, James Mracek and Bin Xu. I will begin by giving a brief overview of the (conjectural) Langlands Correspondence, focusing in particular on Vogan's geometric reformulation of the local Langlands Correspondence. We will then discuss some geometric objects that arise as part of several conjectures which give geometric interpretations to Arthur packets and their associated stable distributions under the LLC. More specifically we shall discuss equivariant perverse sheaves and their vanishing cycles. Mar 5 Amir Akbary On the size of the gcd of $\boldsymbol{a^n-1}$ and $\boldsymbol{b^n-1}$ at noon in B543 (University of Lethbridge) We review some results, from the last twenty years, on the problem of bounding $${\rm gcd}(a^n-1, b^n-1),$$ as $n$ varies. Here either $a$ and $b$ are integers or $a$ and $b$ are polynomials with coefficients in certain fields. In spite of elementary nature of the problem, the results are depended on tools from Diophantine approximation and Diophantine geometry. Mar 12 Steve Wilson The BGCG Construction at noon in B543 (Northern Arizona University) Well, it's not really a construction yet — it's more like a template for constructions. It's a way to take many copies of one tetravalent graph B, the 'base graph', and identify each edge-midpoint with one other according to another graph C, the 'connection graph' to produce a bipartite tetravalent graph. If the identifying is done with caution, wisdom and, um, insouciance, the resulting graph will have lots of symmetry. The cunning of the identifications is related to edge-colorings of the base graph which are themselves nicely symmetric, and we will give several examples where the symmetry can actually be achieved. Mar 19 Peng-Jie Wong On Generalisations of the Titchmarsh divisor problem at noon in B543 (University of Lethbridge) The study of the asymptotic behaviour of the summatory function of the number of divisors of shifted primes was initiated by Titchmarsh, who showed that under the generalised Riemann hypothesis, one has $\sum_{p \le x} \tau(p-a) = x \prod_{p\nmid a} \left( 1+ \frac{1}{p(p-1)}\right)\prod_{p |a} \left( 1- \frac{1}{p}\right) + O\left(\frac{x \log \log x}{\log x}\right),$ where $\tau$ denotes the divisor function. The above formula was first proved unconditionally by Linnik via the dispersion method. Moreover, applying the celebrated Bombieri-Vinogradov theorem, Halberstam and Rodriguez independently gave another proof. In this talk, we shall study the Titchmarsh divisor problem in arithmetic progressions by considering the sum $\sum_{\substack{ p \le x \\ p\equiv b \ (\mathrm{mod}\, r)}} \tau(p-a).$ Also, we will try to explain how to obtain an asymptotic formula for the same, uniform in a certain range of the modulus $r$. If time allows, we will discuss a number field analogue of this problem by considering the above sum over primes satisfying Chebotarev conditions. (This is joint work with Akshaa Vatwani.) Mar 26 Joy Morris Cayley index and Most Rigid Representations (MRRs) at noon in B543 (University of Lethbridge) For any finite group $G$, a natural question to ask is the order of the smallest possible automorphism group for a Cayley graph on $G$. A particular Cayley graph whose automorphism group has this order is referred to as an MRR (Most Rigid Representation), and its Cayley index is the index of the regular representation of $G$ in its automorphism group. Study of GRRs (Graphical Regular Representations, where the full automorphism group is the regular representation of $G$) showed that with the exception of two infinite families and ten individual groups, every group admits a Cayley graph whose MRRs are GRRs, so that the Cayley index is 1. I will present results that complete the determination of the Cayley index for those groups whose Cayley index is greater than 1. This is based on joint work with Josh Tymburski, who was an undergraduate student here at the time. Apr 9 Jean-Marc Deshouillers Values of arithmetic functions at consecutive arguments at noon in B543 (University of Bordeaux) We shall place in a general context the following result recently (*) obtained jointly with Yuri Bilu (Bordeaux), Sanoli Gun (Chennai) and Florian Luca (Johannesburg). Theorem. Let $\tau(\cdot)$ be the classical Ramanujan $\tau$-function and let $k$ be a positive integer such that $\tau(n) \neq 0$ for $1 \le n \le k/2$. (This is known to be true for $k < 10^{23}$, and, conjecturally, for all $k$.) Further, let $\sigma$ be a permutation of the set $\{1, \ldots, k\}$. We show that there exist infinitely many positive integers $m$ such that $$\bigl|\tau \bigl( m + \sigma(1) \bigr)\bigr| < \bigl|\tau \bigl(m + \sigma(2) \bigr)\bigr| < \cdots < \bigl| \tau \bigl( m + \sigma(k) \bigr)\bigr| .$$ The proof uses sieve method, Sato-Tate conjecture, recurrence relations for the values of $\tau$ at prime power values. (*) Hopefully to appear in 2018. May 7 Alia Hamieh Non-vanishing of $L$-functions of Hilbert Modular Forms inside the Critical Strip at noon in C630 (University of Northern British Columbia) In this talk, I will discuss recent joint work with Wissam Raji. We show that, on average, the $L$-functions of cuspidal Hilbert modular forms with sufficiently large weight $k$ do not vanish on the line segments $\Im(s)=t_{0}$, $\Re(s)\in(\frac{k-1}{2},\frac{k}{2}-\epsilon)\cup(\frac{k}{2}+\epsilon,\frac{k+1}{2})$. The proof follows from computing the Fourier expansion of a certain kernel function associated with Hilbert modular forms and estimating its first Fourier coefficient. This result is analogous to the case of classical modular forms which was proved by W. Kohnen in 1997.
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