Calendar

Posted January 16, 2023
Last modified January 20, 2023

Algebra and Number Theory Seminar Questions or comments?

Xin Wan, Chinese Academy of Sciences
[NOTE UNUSUAL TIME] Iwasawa main conjecture for universal families

We formulate and prove the Iwasawa main conjecture for the universal family for ${\rm GL}_2/\mathbb{Q}$ in the $p$-adic Langlands program. As a consequence we prove the Iwasawa main conjecture and rank 0 BSD formula at bad primes. This is joint work with Olivier Fouquet.

Posted January 27, 2023
Last modified February 5, 2023

Algebra and Number Theory Seminar Questions or comments?

Jeffrey Lagarias, University of Michigan
The Floor Quotient Partial Order

We say that a positive integer $m$ is a floor quotient of $n$ if $m = [n/k]$ for some integer $k$, where $[\, \cdot \,]$ denotes the floor function. We show this relation between $m$ and $n$ defines a partial order on the positive integers. This partial order refines the multiplicative divisor order on the positive integers and is refined by the additive total order. We describe results on the internal structure of this partial order, especially on its initial intervals. We study the (two-variable) Möbius function of this partial order. This is joint work with David Harry Richman (see $\texttt{arXiv:2212.11689}$).

Posted January 16, 2023
Last modified February 24, 2023

Algebra and Number Theory Seminar Questions or comments?

Pranabesh Das, Xavier University of Louisiana
Perfect Powers in Power Sums

Let $k \geq 1$, $n \geq 2$ be integers. A power sum is a sum of the form $x_1^k+x_2^k+\cdots +x_n^k$ where $x_1, x_2, \cdots, x_n$ are all integers. Perfect powers appearing in power sums have been well studied in the literature and are an active field of research. In this talk, we consider the Diophantine equation of the form $$ (x+r)^k + (x+2r)^k + \cdots + (x+nr)^k = y^m \ \ \ \ \ \ \ \ \ (1) $$ where $x, y, r \in \mathbb{Z}$, $n, k \in \mathbb{N}$, and $m \geq 2$. We begin with discussing the literature on the Diophantine equation (1). Then we consider explicit solutions for a particular case of the Diophantine equation (1); more precisely, we consider the Diophantine equation $$ \ \ \ (x-r)^5 + x^5 + (x+r)^5 = y^n, \ \ \ \ n \geq 2, \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2) $$ where $r, x, y \in \mathbb{Z}$ and $r$ is composed of certain fixed primes. The talk is based on a joint work with Dey, Koutsianas, and Tzanakis where we determine the integral solutions of the Diophantine equation (2) as an application of the modular method.

Posted January 27, 2023
Last modified March 5, 2023

Algebra and Number Theory Seminar Questions or comments?

Wijit Yangjit, University of Michigan
On the Montgomery–Vaughan weighted generalization of Hilbert's inequality

Hilbert's inequality states that $$ \left\vert\sum_{m=1}^N\sum_{\substack{n=1\\n\neq m}}^N\frac{z_m\overline{z_n}}{m-n}\right\vert\le C_0\sum_{n=1}^N\left\vert z_n\right\vert^2, $$ where $C_0$ is an absolute constant. In 1911, Schur showed that the optimal value of $C_0$ is $\pi$. In 1974, Montgomery and Vaughan proved a weighted generalization of Hilbert's inequality and used it to estimate mean values of Dirichlet series. This generalized Hilbert inequality is important in the theory of the large sieve. The optimal constant $C$ in this inequality is known to satisfy $\pi\le C \lt \pi+1$. It is widely conjectured that $C=\pi$. In this talk, I will describe the known approaches to obtain an upper bound for $C$, which proceed via a special case of a parametric family of inequalities. We analyze the optimal constants in this family of inequalities. A corollary is that the method in its current form cannot imply an upper bound for $C$ below $3.19$.

Posted January 16, 2023
Last modified March 27, 2023

Algebra and Number Theory Seminar Questions or comments?

Piper H, University of Toronto
Joint Shapes of Quartic Fields and Their Cubic Resolvents

In studying the (equi)distribution of shapes of quartic number fields, one relies heavily on Bhargava’s parametrizations which brings with it a notion of resolvent ring. Maximal rings have unique resolvent rings so it is possible to live a long and healthy life without understanding what they are. The authors have decided, however, to forsake such bliss and look into what ever are these rings and what happens if we consider their shapes along with our initial number fields. What happens is very nice! Until it isn't! We'd have more to say if our respective jobs had treated us humanely during the global pandemic, which coincidentally, is ongoing. (with Christelle Vincent)