# Search Research Symposium Abstracts

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## Local heights computations for quadratic Chabauty

The method of quadratic Chabauty was a groundbreaking development on the problem of explicitly determining the set of rational points on a curve. A crucial step of this method is the computation of values of local height functions at primes of bad reduction. In this talk, I will discuss algorithms and provide practical examples of computing these local [Read More...]

**Presenter:**Juanita Duque-Rosero, Boston University

**Authors:**Alexander Betts, Juanita Duque-Rosero, Sachi Hashimoto, and Pim Spelier.

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**September 30, 2023; 9:45 am

## Size of isogeny classes of certain abelian varieties over a finite field

In this talk, we will discuss the size of the isogeny classes of certain abelian varieties over a finite field. In particular we will talk about an expected bound for a certain isogeny class with commutative endomorphism algebra, in the non-ordinary Newton [Read More...]

**Presenter:**Tejasi Bhatnagar, University of Wisconsin- Madison

**Authors:**Tejasi Bhatnagar

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**September 30, 2023; 10:10 am

## Relative Oriented Class Groups of Central Quadratic Extensions

In 2019 Zemková defined relative oriented class groups associated to quadratic extensions of number fields L/K, extending work of Bhargava concerning composition laws for binary quadratic forms over number fields of higher degree. Indeed, this work generalized the classical correspondence between the ideal class group of a quadratic number field and classes [Read More...]

**Presenter:**Kelly O'Connor, Colorado State University

**Authors:**Kelly O'Connor

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**September 30, 2023; 10:35 am

## Green’s functions in the arithmetic intersection of cycles over Shimura curves

Green’s functions are crucial in arithmetic intersection theory, particularly at Archimedean places. In this talk, I will introduce a combinatorial version of Green’s functions that arises in the context of intersection pairing at some non-Archimedean places. I will first motivate the discussion with a generalization problem of the Gross-Zagier formula [Read More...]

**Presenter:**Boya Wen, University of Wisconsin-Madison

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**September 30, 2023; 11:00 am

## Hyperelliptic Curves mapping to Abelian Surfaces and applications to Beilinson's Conjecture for zero-cycles

For an abelian surface A over an algebraically closed field k we describe a rich collection of rational equivalences in the kernel of the Albanese map of A arising from hyperelliptic curves. When the field k has characteristic zero and the abelian surface A is isogenous to a product of elliptic curves, we produce a very large collection of hyperelliptic [Read More...]

**Presenter:**Evangelia Gazaki, University of Virginia

**Authors:**Evangelia Gazaki, Jonathan Love

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**September 30, 2023; 2:00 pm

## Isogeny Classes of Drinfeld Modules over Finite Fields via Frobenius Distributions

Classically, the size of an isogeny class of an elliptic curve - or more generally, a principally polarized abelian variety - over a finite field is given by a suitable class number. Gekeler expressed the size of an isogeny class of an elliptic curve over a prime field in terms of a product over all primes of local density functions using a random matrix [Read More...]

**Presenter:**Amie Bray, Colorado State University

**Authors:**Amie Bray

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**September 30, 2023; 2:25 pm

## Torsion for CM Elliptic Curves Defined Over Number Fields of Degree 2p

Let E be an elliptic curve defined over a number field F. By the Mordell-Weil theorem we know that the points of E with coordinates in F can be given the structure of a finitely generated abelian group. We will focus on the subgroups of points with finite order. For a given prime p > 3 and an elliptic curve E defined over a number field of degree 2p, we [Read More...]

**Presenter:**Holly Paige Chaos, University of Vermont

**Authors:**Abbey Bourdon and Holly Paige Chaos

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**September 30, 2023; 2:50 pm

## Panel discussion: career development resources

We will feature women panelists who have successfully leveraged research and organizing opportunities to advance their careers and the careers of other women. For example, one timely opportunity in this area is the Arizona Winter School in 2023 on Abelian Varieties that the organizers have been involved in in various capacities, and we plan to touch on this [Read More...]

**Presenter:**Padmavathi Srinivasan, Boston University

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**September 30, 2023; 3:15 pm

## Some dynatomic modular curves in positive characteristic

In arithmetic dynamics, the smoothness and irreducibility of the dynatomic modular curves $Y_1(n)$ and $Y_0(n)$ have frequently been studied for the polynomial family $f_c(x)=x^d+c$ in both char 0 and positive char $p$, but less is known about the dynamical behavior of other families. I am studying the smoothness and irreducibility of $Y_1(n)$ and $Y_0(n)$ [Read More...]

**Presenter:**Colette La Pointe, CUNY Graduate Center

**Authors:**Colette La Pointe

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**October 1, 2023; 2:00 pm

## On the proportion of everywhere locally soluble superelliptic curves

We investigate the proportion of superelliptic curves that have a $\mathbb{Q}_p$ point for every place $p$ of $\mathbb{Q}$. We show that this proportion is positive and given by the product of local densities, we provide lower bounds for this proportion in general, and for superelliptic curves of the form $y^3 = f(x,z)$ for an integral binary form $f$ of [Read More...]

**Presenter:**Lea Beneish, University of North Texas

**Authors:**Lea Beneish and Christopher Keyes

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**October 1, 2023; 2:25 pm

## Integers that are sums of rational sixth powers

We prove that $164634913$ is the smallest positive integer that is a sum of two rational sixth powers but not a sum of two integer sixth powers. If $C_{k}$ is the curve $x^{6} + y^{6} = k$, we use the existence of morphisms from $C_{k}$ to elliptic curves, together with the Mordell-Weil sieve, to rule out the existence of rational points on $C_{k}$ for [Read More...]

**Presenter:**Alexis Newton, Emory University

**Authors:**Alexis Newton, Jeremy Rouse

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**October 1, 2023; 2:50 pm

## A threefold violating a local-to-global principle for rationality

A variety satisfies the local-to-global principle for rational points if the existence of local points at all places of Q implies the existence of (global) rational points over Q. In this talk, we consider a local-to-global principle for rationality (i.e. the property of being birational to projective space). We construct an example of a smooth projective [Read More...]

**Presenter:**Lena Ji, University of Michigan

**Authors:**Sarah Frei, Lena Ji

**Symposium Year:**2023

**Session:**Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]

**Presentation Time:**October 1, 2023; 3:15 pm

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