**I am interested in algebraic number theory and arithmetic geometry.**

This is a paper that uses ramification in division fields to preclude certain supersingular elliptic curves from corresponding to sporadic points on modular curves.

**Ramification in the Division Fields of Elliptic Curves and an Application to Sporadic Points on Modular Curves**[ show abstract | arXiv: 1810.04809 ] Consider an elliptic curve E over a number field K and let

**be a prime of O**

*p*_{K}lying above a prime p of

**Z**. Suppose E has supersingular reduction at

**. Fix a positive integer n and define L to be a minimal extension of K such that E(L) has a point of exact order p**

*p*^{n}. If

**is unramified over p, we show that L/K is an extension of degree p**

*p*^{2n}−p

^{2n−2}that is totally ramified over

**. If**

*p***is ramified over p, we are still able to show that φ(p**

*p*^{n}) properly divides [L:Q]. We apply our stronger bound to show that sporadic points on the modular curve X

_{1}(p

^{n}) cannot correspond to elliptic curves that are supersingular at

**with**

*p***unramified over p. Our methods are then generalized to X**

*p*_{1}(N) with N composite. We also describe ramification at and away from p in the full p

^{n}-th division field K(E[p

^{n}]). In the course of our investigation, we correct a theorem of Cassels.

Here are my slides from a presentation about this work.

This is a paper classifying two infinite families of monogenic S

_{4}Quartic fields.

**Two families of Monogenic S**[ show abstract | arXiv: 1802.09599 ] Consider the integral polynomials f

_{4}Quartic Number Fields_{a,b}(x)=x

^{4}+ax+b and g

_{c,d}(x)=x

^{4}+cx

^{3}+d. Suppose f

_{a,b}(x) and g

_{c,d}(x) are irreducible, b|a, and the integers b, d, 256d-27c

^{4}, and (256b

^{3}-27a

^{4})/gcd(256b

^{3},27a

^{4}) are all square-free. Using the Montes algorithm, we show that a root of f

_{a,b}(x) or g

_{c,d}(x) defines a monogenic extension of

**Q**and serves as a generator for a power integral basis of the ring of integers. In fact, we show monogeneity for slightly more general families. Further, we obtain lower bounds on the density of polynomials generating monogenic S

_{4}fields within the families f

_{b,b}(x) and g

_{1,d}(x).

It has been accepted and will appear in Acta Arithmetica.

Below is a paper I wrote with Alden Gassert and Katherine Stange.

**A Family of Monogenic Quartic Fields Arising from Elliptic Curves**[ show abstract | arXiv: 1708.03953 ] We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial 3-torsion fields for a certain one-parameter family of non-CM elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic S

_{4}fields given by T

^{4}− 6T

^{2}− αT − 3 for α ϵ Z such that α ± 8 are squarefree, are monogenic.

It has been accepted and will appear in The Journal of Number Theory.

Here are my slides from a presentation about this work.

Here is a paper I helped with while at an REU at Grand Valley State University. William Dickinson was our project advisor.

**Optimal Packings of Two to Four Equal Circles on any Flat Torus**[ show abstract | arXiv: 1708.05395 ] We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.

Some math related website I like:

Keith Conrad's Expository Papers

Andrew Snowden's Course on Mazur's Theorem

David Mumford's Algebraic Geometry Work

J.S. Milne's Course Notes

Katherine Stange's Website

Terry Tao's Blog

**L**-functions and

**M**odular

**F**orms

**D**ata

**B**ase

Kiran Kedlaya's Conferences Page