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 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](https://alozano.clas.uconn.edu/). 
 
-With my advisor, Álvaro Lozano-Robledo, I am studying 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 $$E$$ such that the division field $$\mathbb{Q}(j_{K,f},E[N])$$ is an abelian extension of $$\mathbb{Q}(j_{K,f})$$.
+With my advisor, Álvaro Lozano-Robledo, I am studying 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 the division field $$\mathbb{Q}(j_{K,f},E[N])$$ is an abelian extension of $$\mathbb{Q}(j_{K,f})$$.
 
 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](http://faculty.bard.edu/cullinan/about.html) and [Nathan Kaplan](https://www.math.uci.edu/~nckaplan/index.html) (see [[1]](https://arxiv.org/abs/2301.09176)). 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 and [Gabrielle Scullard](https://science.psu.edu/math/people/gns49).