Number Theory

The Ostrowski Quotient for a finite extension of number fields

Speaker: 
Abbas Maarefparvar
Date: 
Wed, Nov 20, 2024
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
UBC Number Theory Seminar
Abstract: 

For a number field $K$, the P\'olya group of $K$, denoted by $Po(K)$, is the subgroup of the ideal class group of $K$ generated by the classes of the products of maximal ideals of $K$ with the same norm. In this talk, after reviewing some results concerning $Po(K)$, I will generalize this notion to the relative P\'olya group $Po(K/F)$, for $K/F$ a finite extension of number fields. Accordingly, I will generalize some results in the literature about P\'olya groups to the relative case. Then, due to some essential observations, I will explain why we need to modify the notion of the relative P\'olya group to the Ostrowski quotient $Ost(K/F)$ to get a more 'accurate' generalization of $Po(K)$. The talk is based on a joint work with Ali Rajaei (Tarbiat Modares University) and Ehsan Shahoseini (Institute For Research In Fundamental Sciences).

Class: 

Mean values of Hardy's Z-function and weak Gram's laws

Speaker: 
Hung M. Bui
Date: 
Tue, Nov 19, 2024
Location: 
PIMS, University of Northern British Columbia
Zoom
Online
Abstract: 

We establish the fourth moments of the real and imaginary parts of the Riemann zeta-function, as well as the fourth power mean value of Hardy's Z-function at the Gram points. We also study two weak versions of Gram's law. We show that those weak Gram's laws hold a positive proportion of time. This is joint work with Richard Hall.

Class: 

On the vertical distribution of the zeros of the Riemann zeta-function

Speaker: 
Emily Quesada Herrera
Date: 
Fri, Nov 8, 2024
Location: 
PIMS, University of British Columbia
Zoom
Online
Conference: 
UBC Number Theory Seminar
Abstract: 

In 1973, assuming the Riemann hypothesis (RH), Montgomery studied the vertical distribution of zeta zeros, and conjectured that they behave like the eigenvalues of some random matrices. We will discuss some models for zeta zeros starting from the random matrix model but going beyond it and related questions, conjectures and results on statistical information on the zeros. In particular, assuming RH and a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we will discuss our proof of a conjecture of Berry (1988) for the number variance of zeta zeros, in a regime where random matrix models alone do not accurately predict the actual behavior (based on joint work with Meghann Moriah Lugar and Micah B. Milinovich).

Class: 

Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of ℚ

Speaker: 
Omer Avci
Date: 
Thu, Oct 31, 2024
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

Let E be an elliptic curve defined over ℚ. Let p > 3 be a prime such that p - 1 is not divisible by 3, 4, 5, 7, 11. In this article, we classify the groups that can arise as E(ℚ(ζp))tors up to isomorphism. The method illustrates techniques for eliminating possible structures that can appear as a subgroup of E(ℚab)tors.

Class: 

Subconvexity for GL(2) L-functions and Shifted MDS

Speaker: 
Henry Twiss
Date: 
Wed, Oct 30, 2024
Location: 
PIMS, University of Northern British Columbia
Online
Zoom
Abstract: 

Subconvexity problems have maintained extreme interest in analytic number theory for decades. Critical barriers such as the convexity, Burgess, and Weyl bounds hold particular interest because one usually needs to drastically adjust the analytic techniques involved in order to break through them. It has recently come to light that shifted Dirichlet series can be used to obtain subconvexity results. While these Dirichlet series do not admit Euler products, they are amenable to study via spectral methods. In this talk, we construct a shifted multiple Dirichlet series (MDS) and leverage its analytic continuation via spectral decompositions to obtain the Weyl bound in the conductor-aspect for the L-function of a holomorphic cusp form twisted by an arbitrary Dirichlet character. This improves upon the corresponding bound for quadratic characters obtained by Iwaniec-Conrey in 2000. This work is joint with Jeff Hoffstein, Nikos Diamantis, and Min Lee.

Class: 

Orienteering with One Endomorphism

Speaker: 
Renate Scheidler
Date: 
Thu, Oct 24, 2024
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

Given two elliptic curves, the path finding problem asks to find an isogeny (i.e. a group homomorphism) between them, subject to certain degree restrictions. Path finding has uses in number theory as well as applications to cryptography. For supersingular curves, this problem is known to be easy when one small endomorphism or the entire endomorphism ring are known. Unfortunately, computing the endomorphism ring, or even just finding one small endomorphism, is hard. How difficult is path finding in the presence of one (not necessarily small) endomorphism? We use the volcano structure of the oriented supersingular isogeny graph to answer this question. We give a classical algorithm for path finding that is subexponential in the degree of the endomorphism and linear in a certain class number, and a quantum algorithm for finding a smooth isogeny (and hence also a path) that is subexponential in the discriminant of the endomorphism. A crucial tool for navigating supersingular oriented isogeny volcanoes is a certain class group action on oriented elliptic curves which generalizes the well-known class group action in the setting of ordinary elliptic curves.

Class: 

Parametrization of rings of finite rank - a geometric approach and their use in counting number fields

Speaker: 
Gaurav Patil
Date: 
Thu, Oct 17, 2024
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

We describe parametrizations of rings that generalize the notions of monogenic rings and binary rings. We use these parametrizations to give better lower bounds on the number of number fields of degree n and bounded discriminant.

Class: 

Moments of real Dirichlet L-functions and multiple Dirichlet series

Speaker: 
Martin Čech
Date: 
Wed, Oct 23, 2024
Location: 
PIMS, University of Northern British Columbia
Zoom
Online
Abstract: 

There are two ways to compute moments in families of L-functions: one uses the approximation by Dirichlet polynomials, and the other is based on multiple Dirichlet series. We will introduce the two methods to study the family of real Dirichlet L-functions, compare them and show that they lead to the same results. We will then focus on obtaining the meromorphic continuation of the associated multiple Dirichlet series.

Class: 

A Study of Twisted Sums of Arithmetic Functions

Speaker: 
Saloni Sinha
Date: 
Tue, Oct 15, 2024
Location: 
PIMS, University of British Columbia
Online
Zoom
Abstract: 

We study sums of the form $\sum_{n\leq x} f(n) n^{-iy}$, where $f$ is an arithmetic function, and we establish an equivalence between the Riemann Hypothesis and estimates on these sums. In this talk, we will explore examples of such sums with specific arithmetic functions, as well as discuss potential implications and future research directions.

Class: 

Hybrid Statistics of the Maxima of a Random Model of the Zeta Function over Short Intervals

Speaker: 
Christine K. Chang
Date: 
Tue, Oct 8, 2024
Location: 
PIMS, University of British Columbia
Zoom
Online
Abstract: 

We will present a matching upper and lower bound for the right tail
probability of the maximum of a random model of the Riemann zeta function over
short intervals. In particular, we show that the right tail interpolates
between that of log-correlated and IID random variables as the interval varies
in length. We will also discuss a new normalization for the moments over short
intervals. This result follows the recent work of Arguin-Dubach-Hartung and is inspired by a conjecture by Fyodorov-Hiary-Keating on the local maximum over
short intervals.

Class: 

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