TOPICS: Arithmetic in the broadest sense that includes Number Theory (Elementary Arithmetic, Algebraic, Analytic, Combinatorial, etc.), Algebraic Geometry, Representation Theory, Lie Groups and Lie Algebras, Diophantine Geometry, Geometry of Numbers, Tropical Geometry, Arithmetic Dynamics, Arithmetic Topology, etc.

PLACE and TIME: This semester the seminar meets primarily on Tuesdays at 4:00 pm, with possible special lectures at other days and times. The in-house talks will be in-person, while visitors outside of Binghamton area will be by Zoom: Zoom link

ORGANIZERS:
Regular Faculy: Alexander Borisov, Marcin Mazur, Adrian Vasiu.
Post-Docs: Huy Dang

Current Ph.D. students: Hari Asokan, Mithun Padinhare Veettil.

Graduated Ph.D. students (in number theory and related topics): Ilir Snopce (Dec. 2009), Xiao Xiao (May 2011), Jinghao Li (May 2015), Ding Ding (Dec. 2015), Patrick Milano (May 2018), Changwei Zhou (May 2019), Patrick Carney (May 2023), Sarah Lamoureux (Sep. 2023), Sayak Sengupta (May 2024).

SEMINAR ANNOUNCEMENTS: To receive announcements of seminar talks by email, please join our mailing list.

Related seminar: Upstate New York Online Number Theory Colloquium (online, irregular): http://people.math.binghamton.edu/borisov/UpstateNYOnline/Colloquium.html

• Spring 2025 ——– Fall 2024
• Spring 2024 ——– Fall 2023
• Spring 2023 ——– Fall 2022
• Spring 2022 ——– Fall 2021
• Spring 2021 ——– Fall 2020
• Spring 2020 ——– Fall 2019
• Spring 2019 ——– Fall 2018
• Spring 2018 ——– Fall 2017
• Spring 2017 ——– Fall 2016
• Spring 2016 ——– Fall 2015
• Spring 2015 ——– Fall 2014

• August 26
  Speaker: NA
  Title: Organizational Meeting
  Abstract:
  

• September 9
  Speaker: Huy Dang (Binghamton)
  Title: The lifting problem for curves
  Abstract: The lifting problem for curves asks: given a smooth, projective, connected curve 𝐶 over a field 𝑘 of characteristic 𝑝 > 0, which finite Galois coverings of 𝐶 lift to characteristic zero? In this talk, we provide an overview of the central questions and techniques used to study this problem. We will also discuss connections with other areas of research, including deformation theory and ramification theory.
  

• September 16
  Speaker: Huy Dang (Binghamton)
  Title: Lifting abelian isogenies from characteristic $p$ to characteristic $0$
  Abstract: In characteristic $0$, cyclic field extensions are classified by Kummer theory. In characteristic $p$, in addition to Kummer theory, one also needs Artin–Schreier–Witt theory to describe these extensions. Matsuda constructed a formal morphism that connects these two theories, providing a bridge between characteristic $p$ and characteristic $0$. In this talk, we present an algebraization of Matsuda’s construction to study the lifting of abelian isogenies from characteristic $p$ to characteristic $0$. As an application, we show that every lift of an abelian étale cover of a local scheme arises as the pullback of such a lift of an abelian isogeny. This is joint work with Khai-Hoan Nguyen-Dang.
  

• September 30
  Speaker: Enrique Trevino (Lake Forest College)
  Title: On sets whose subsets have integer mean
  Abstract: We say a finite set of positive integers is balanced if all of its subsets have integer mean. For a positive integer $N$, let $M(N)$ be the cardinality of the largest balanced set all of whose elements are less than or equal to $N$, and let $S(N)$ be the cardinality of the largest balanced set with elements less than or equal to $N$ that has maximal sum. For example, for $N = 3000$, the largest balanced set is $\{3000, 2580, 2160, 1740, 1320, 980, 480, 60\}$ so $M(3000) = 8$, while the largest set with maximal sum is $\{3000, 2940, 2880, 2820, 2760, 2700, 2640\}$, so $S(3000)= 7$. In this talk we will study the question of when $M(N) = S(N)$.
  

• October 16 (Thursday, 2:45-3:45 pm, cross-listed from Geometry and Topology Seminar)
  Speaker: John Abou-Rached (Binghamton)
  Title: Integral models for non-Shimura curves and the Eichler-Shimura congruence relation
  Abstract: We construct integral models for an infinite family of algebraic curves that includes noncongruence modular curves, as well as curves whose uniformizers are non-arithmetic Fuchsian groups. Most of these curves are not Shimura curves. We affirm a conjecture of Mukamel that the set of primes of good reduction of such curves have arithmetic significance and obtain an explicit description of this set. We conjecture that a version of Deligne-Rapoport's study of the reduction of modular curves holds in this context, and conjecture that a version of the Eichler-Shimura congruence relation holds in this setting, in resonance with Shimura curves.
  

• October 21
  Speaker: TBA
  Title: TBA
  Abstract: TBA
  

• October 28
  Speaker: TBA
  Title: TBA
  Abstract: TBA
  

• November 4
  Speaker: Bhargavi Parthasarathy (Syracuse University)
  Title: TBA
  Abstract: TBA
  

• November 11
  Speaker: TBA
  Title: TBA
  Abstract: TBA
  

• November 18
  Speaker: TBA
  Title: TBA
  Abstract: TBA
  

• November 25 (Zoom talk only)
  Speaker: TBA
  Title: TBA
  Abstract: TBA
  

• December 2
  Speaker: Bogdan Ion (University of Pittsburgh)
  Title: Bernoulli operators, Dirichlet series, and analytic continuation
  Abstract: Bernoulli operators are distributions with discrete support associated to Dirichlet series (or rather to the corresponding power series). The most basic case, when the power series has a pole singularity at $z=1$ is analyzed in detail. Its main property is that it naturally acts on the vector space of analytic functions in the plane (with possible isolated singularities) that fall in the image of the Laplace-Mellin transform (for the variable in some half-plane). The action of the Bernoulli operator on the function $t^s$, provides the analytic continuation of the associated Dirichlet series and also detailed information about the location of poles, their resides, and special values. Using examples of arithmetic origin, I will attempt to illustrate what is reasonable to expect when the power series has a non-pole singularity at $z=1$, pointing to an extension of this theory to tempered distributions associated to modular forms.