seminars:arit

## The Arithmetic Seminar

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, etc.

PLACE and TIME: This semester the seminar meets primarily on Tuesdays at 4:15 p.m. on Zoom, with possible special lectures on Mondays at 3:30 or other days. Zoom link

ORGANIZERS:
Regular Faculy: Alexander Borisov, Marcin Mazur, Adrian Vasiu,
Post-Docs: Vaidehee Thatte, Fikreab Solomon Admasu.

Current Ph.D. students: Patrick Carney, Andrew Lamoureux, Micah Loverro, and Sayak Sengupta.

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).

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

Related seminar: The student/postdoc “No Theory Seminar”: https://sites.google.com/view/vaideheethatte/service-outreach/nts-bu

#### Spring 2021

• February 16
Speaker: N/A
Title: Organizational Meeting
Abstract: We will discuss plans for this semester
• February 23
Title: Bhargava's composition law for binary cubic forms
Abstract: The Delone-Fadeev correspondence shows that binary cubic forms with integer coefficients parametrize orders in cubic fields. With this result in mind, Bhargava constructs a binary cubic form from 2x3x3 boxes of integers and proves that there is a natural composition law for the boxes of integers. The group resulting from this law is then shown to be isomorphic to the class group of a corresponding cubic order. This is a cubic analogue of Gauss's theory of composition for binary quadratic forms and its relation to ideal classes of quadratic orders. The talk is based on Bhargava's “Higher composition laws II: On cubic analogues of Gauss composition.”
• March 2
Speaker: Hyuk Jun Kweon (MIT)
Title: Bounds on the Torsion Subgroups of Néron-Severi Group Schemes
Abstract: Let $X \hookrightarrow \mathbb{P}^r$ be a smooth projective variety defined by homogeneous polynomials of degree $\leq d$ over an algebraically closed field $k$. Let $\mathbf{Pic}\, X$ be the Picard scheme of $X$, and $\mathbf{Pic}\, ^0 X$ be the identity component of $\mathbf{Pic}\, X$. The N\'eron–Severi group scheme of $X$ is defined by $\mathbf{NS} X = (\mathbf{Pic}\, X)/(\mathbf{Pic}\, ^0 X)_{\mathrm{red}}$, and the N\'eron–Severi group of $X$ is defined by $\mathrm{NS}\, X = (\mathbf{NS} X)(k)$. We give an explicit upper bound on the order of the finite group $(\mathrm{NS}\, X)_{{\mathrm{tor}}}$ and the finite group scheme $(\mathbf{NS} X)_{{\mathrm{tor}}}$ in terms of $d$ and $r$. As a corollary, we give an upper bound on the order of the torsion subgroup of second cohomology groups of $X$ and the finite group $\pi^1_{et}(X,x_0)^{\mathrm{ab}}_{\mathrm{tor}}$. We also show that $(\mathrm{NS}\, X)_{\mathrm{tor}}$ is generated by $(\deg X -1)(\deg X - 2)$ elements.
• March 9
Speaker: Thomas Morrill (Trine)
Title: TBA
Abstract: TBA
• March 16
Title: Bhargava's composition law for binary cubic forms 2
Abstract: TBA
• March 23
Speaker: TBA
Title: TBA
Abstract: TBA
• March 30
Speaker: TBA
Title: TBA
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• April 6
Speaker: TBA
Title: TBA
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• April 13
Speaker: TBA
Title: TBA
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• April 20
Speaker: TBA
Title: TBA
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• April 27
Speaker: TBA
Title: TBA
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• May 4
Speaker: TBA
Title: TBA
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• May 11
Speaker: TBA
Title: TBA
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• May 18
Speaker: TBA
Title: TBA
Abstract: TBA