Department of Mathematics and Statistics

Department of Mathematics and Statistics
Department of Mathematics and Statistics
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Number Theory Seminar

Number Theory - Neha Prabhu (Queen's University)

Wednesday, April 4th, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: Neha Prabhu (Queen's University)

Title: Moments of the error term in the Sato-Tate law for elliptic curves.

Abstract: The Sato-Tate theorem for elliptic curves was proved by L. Clozel, M. Harris, N. Shepherd-Barron and R. Taylor in a series of papers from 2008-2010. Since the Sato-Tate law is an asymptotic statement, one is naturally interested in studying the nature of the error terms. In this talk, I shall describe some results relating to moments of the error term when we consider averages over certain families of elliptic curves. This is joint work with Stephan Baier.

Number Theory - Siddhi Pathak (Queen's University)

Wednesday, March 28th, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: Siddhi Pathak (Queen's University)

Title: On the primitivity of Dirichlet characters.

Abstract: A Dirichlet character modulo q is said to be imprimitive if it is induced from a lower level. A characterization of the primitivity of characters is the separability of the Gauss sum ( Fourier transform of $\chi$ ), i.e., $G_q(n, \bar{chi}) = \chi(n) G_q(1, \bar{ \chi } )$ for all n. In this talk, we discuss a paper of R. Daielda and N. Jones in which they introduce another way of extending primitive Dirichlet characters so that the above separability property holds even for imprimitive characters.

Number Theory - Francois Seguin (Queen's University)

Wednesday, March 21st, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: François Séguin (Queen's University)

Title: Frequency of primes dividing n and $\Phi_n$.

Abstract: In a previous talk, we have seen that at most one prime Pn, can divide both an integer n and the nth cyclotomic polynomial evaluated at some integer a. During this talk, we will use methods in Dirichlet series to investigate how often Pn actually divides the nth cyclotomic polynomial $\Phi_n$.

Number Theory - Arpita Kar (Queen's University)

Wednesday, March 14th, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: Arpita Kar (Queen's University)

Title: On the normal number of prime factors of Ramanujan Tau function.

Abstract: We will discuss various results concerning $\omega(\tau(p))$, $omega(\tau(n))$, $\omega(\tau(p+1))$ where $\tau$ denotes Ramanujan Tau function and $\omega(n)$ denotes the number of prime factors of $n$ counted without multiplicity. This is work in progress with Prof. Ram Murty.

Number Theory - M. Ram Murty (Queen's University)

Wednesday, March 7th, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: M. Ram Murty (Queen's University)

Title: The Central Limit Theorem in algebra and Number Theory.

Abstract: The central limit theorem is certainly one of the pinnacles of 20th century mathematics that has transformed human civilization extending its influence outside mathematics and now touching every other scientific discipline and beyond. I will give a short historical survey and then highlight how the central limit theorem has inspired the development of probabilistic number theory and probabilistic group theory. At the end, I will report on some old work with Kumar Murty and recent joint work with Arpita Kar and Neha Prabhu regarding arithmetical aspects of Fourier coefficients of modular forms.

Number Theory - Francois Seguin (Queen's University)

Wednesday, February 28th, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: François Séguin (Queen's University)

Title: Prime divisors of sparse values of cyclotomic polynomials.

Abstract: We will be presenting a result about the largest prime divisors of cyclotomic polynomials evaluated at a specific integer. We will also see how this ties in to problems we previously encountered.

Number Theory - Vaidehee Thatte (Queen's University)

Wednesday, January 31st, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: Vaidehee Thatte (Queen's University)

Title: Defect Extensions – I

Abstract: Let $K$ be a valued field of characteristic $p > 0$ with henselian valuation ring $A$. Let $L$ be a non-trivial Artin-Schreier extension of $K$ with $B$ as the integral closure of $A$ in $L$. In the classical theory of complete discrete valuation rings, $B$ is generated as an $A$-algebra by a single element. This in particular, is not true in the defect case. We will discuss a result that allows us to write $B$ as a "filtered union over $A$", when there is defect. Similar results can be obtained in the mixed characteristic case.

Number Theory - François Séguin (Queen's University)

Wednesday, January 24th, 2018

Time: 2:15 p.m.  Place: Jeffery Hall 319

Speaker: François Séguin (Queen's University)

Title: Heights of elliptic curves and the elliptic analogue of the two-variable Artin conjecture

Abstract: Similar to the way Lang and Trotter adapted Artin's primitive root conjecture in the case of elliptic curves, we consider this natural adaptation for the two-variable Artin Conjecture. In light of our recent results for the two-variable setting, we present similar, unconditional lower bounds for this elliptic analogue.

Number Theory - Jung-Jo Lee (Queen's University)

Wednesday, January 17th, 2017

Time: 1:30 p.m.  Place: Jeffery Hall 319

Speaker: Jung-Jo Lee

Title: The p-adic zeta function and Iwasawa’s main conjecture

Abstract: I would explain the role of the p-adic zeta function in describing the structure of certain Iwasawa module.

Note: I consider a series of talks, each "hopefully" self-contained.

Talk 2 : Euler system of cyclotomic units and Iwasawa's main conjecture

Talk 3 : Euler system of Heegner points and Birch and Swinnerton-Dyer conjecture

 

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