Research

My research interests are in arithmetic geometry and algebraic number theory. I am interested in studying elliptic curves and Galois representations. My advisor is Álvaro Lozano-Robledo.

1. Let \(E/\mathbb{Q}\) be an elliptic curve with CM and let \(N\geq 2\) be an integer. In previous work, it has been shown that \(Gal(\mathbb{Q}(E[N])/\mathbb{Q})\) is only abelian for \(N=2,3\), and \(4\). In recent work, for a prime \(p\) and integer \(n \geq 1\), I classified the maximal abelian extension contained in \(\mathbb{Q}(E[p^n])/\mathbb{Q}\) for an elliptic curve \(E/\mathbb{Q}\) with CM. To do so, I bound the size of the commutator subgroups of \(Gal(\mathbb{Q}(E[p^n])/\mathbb{Q})\).

2. With my advisor, Álvaro Lozano-Robledo, I studied when the \(\mathcal{l}\)-adic image of a Galois representation attached to an elliptic curve \(E\) with CM can be abelian. Let \(K\) be an imaginary quadratic field, and let \(\mathcal{O}_{K,f}\) be an order in \(K\) of conductor \(f \geq 1\). Let \(E\) be an elliptic curve with CM by \(\mathcal{O}_{K,f}\) such that \(E\) is defined by a model over \(\mathbb{Q}(j_{K,f})\), where \(j_{K,f}=j(E)\). In our project, we classify the values of \(N\geq 2\) and the elliptic curves \(E/\mathbb{Q}(j_{K,f})\) such that (i) the division field \(\mathbb{Q}(j_{K,f},E[N])\) is an abelian extension of \(\mathbb{Q}(j_{K,f})\), and (ii) the \(N\)-division field coincides with the \(N\)-th cyclotomic extension of the base field.

3. In the spring of 2023, I participated in the Women in Numbers 6 workshop. Isogenous elliptic curves have the same conductor, but not necessarily the same minimal discriminant ideal. In our research project, we explicitly classify all \(p^2\)-isogenous elliptic curves defined over a number field with the same minimal discriminant ideal for odd prime \(p\) where \(X_0(p^2)\) has genus \(0\), i.e., \(p=3\) or \(5\). As a consequence, we give a list of all \(p^2\)-isogenous discriminant (ideal) twins over \(\mathbb{Q}\) for such \(p\). This is joint work with Alyson Deines, Andreea Iorga, Changningphaabi Namoijam, Manami Roy, and Lori D. Watson.

4. With Steven J. Miller and Eduardo Dueñez, I proved that the sum of the \(d^{th}\)-powers up to \(n\) is a polynomial of degree \(d+1\) in \(n\) using L’Hopital’s rule. For a positive integer \(d\), let \(p_d(n) := 0^d + 1^d + 2^d + \cdots + n^d\); i.e., \(p_d(n)\) is the sum of the first \(d^{th}\)-powers up to \(n\). It is well known that \(p_d(n)\) is a polynomial of degree \(d+1\) in \(n\). While this is usually proved by induction, once \(d\) is not small it is a challenge as one needs to know the polynomial for the inductive step. We show how this difficulty can be bypassed by giving a simple proof that \(p_d(n)\) is a polynomial of degree \(d+1\) in \(n\) using L’Hopital’s rule, and show how we can then determine the coefficients by Cramer’s rule.

5. In the summer of 2022, I participated in the Rethinking Number Theory 3 Workshop. Let \(E\) and \(E'\) be \(2\)-isogenous elliptic curves defined over \(\mathbb{Q}\). For our project, we were interested in studying the proportion of primes \(p\) for which \(E(\mathbb{F}_p)\cong E'(\mathbb{F}_p)\) and \(E(\mathbb{F}_{p^2}) \not\cong E'(\mathbb{F}_{p^2})\). When this happens, we call \(p\) anomalous. There has been previous work done on this by John Cullinan and Nathan Kaplan (see [1]). In our project, we complete the classification begun in [1]. We give an explicit formula for the proportion of anomalous primes, depending on the images \(G\) and \(G'\) of the \(2\)-adic representations of \(E\) and \(E'\), respectively. We consider both the non-CM and CM case. This is joint work with John Cullinan, Nathan Kaplan, Gabrielle Scullard, and others.

6. In fall 2019 to spring 2020, I worked on my undergraduate honors thesis under Andrew Obus. In my honors thesis, I calculated the probability that the gcd of a pair of quadratic integers \(n,m\) chosen randomly, uniformly, and independently from the set of quadratic integers of norm \(x\) or less, is \(k\). I also calculated the expected norm of the gcd( \(n,m\) ) as \(x\) tends to infinity, with explicit error terms. I determined the probability and expected norm of the gcd for quadratic integer rings that are UFDs. I also outlined a method to determine the probability and expected norm of the gcd of elements in quadratic integer rings that are not UFDs.

7. In the summer of 2019, I participated in the NSF REU at Texas A&M University. In our research project, we proved that the crank partition function is asymptotically equidistributed modulo \(Q\), for any odd number \(Q\). To prove this, we obtained effective bounds on the error term from Zapata Rolon’s asymptotic estimate for the crank function. We then used those bounds to prove the surjectivity and strict log-subadditivity of the crank function. This was joint work with Wei-Lun Tsai and Aaron Kreigman.

8. In the summer of 2018, I participated in the NSF REU at Oregon State University. In our research project, we showed that eta-quotients which are modular for any congruence subgroup of level \(N\) coprime to 6 can be viewed as modular for \(\Gamma_0(N)\). We then categorized when even weight eta-quotients can exist in \(M_k(\Gamma_1(p))\) and \(M_k(\Gamma_1(pq))\), for distinct primes \(p,q\). We also provided some new examples of elliptic curves whose corresponding modular forms can be written as a linear combination of eta-quotients, and we described an algorithmic method for finding additional examples. This was joint work with Holly Swisher, Michael Allen, Nicholas Anderson, and Benjamin Oltsik.