MATH Seminar
| Title: Isogenies of Elliptic Curves and Arithmetical Structures on Graphs |
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| Defense: Dissertation |
| Speaker: Tomer Reiter of Emory University |
| Contact: Tomer Reiter, tomer.reiter@emory.edu |
| Date: 2021-03-19 at 11:00AM |
| Venue: https://emory.zoom.us/j/92804829998?pwd=OFpZcWdlS2lrUFRLbDZQNklxZ3IwQT09 |
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| Abstract: In this defense, we prove two results that come from studying curves. The first is a classification result for elliptic curves. Let $\mathbf{Q}(2^{\infty})$ be the compositum of all quadratic extensions of $\mathbf{Q}$. Torsion subgroups of rational elliptic curves base changed to $ \mathbf{Q}(2^{\infty}) $ were classified by Laska, Lorenz, and Fujita. Recently, Daniels, Lozano-Robledo, Najman, and Sutherland classified torsion subgroups of rational elliptic curves base changed to $ \mathbf{Q} (3^{\infty})$, the compositum of all cubic extensions of $ \mathbf{Q} $. We classify cyclic isogenies of rational elliptic curves base changed to $\mathbf{Q}(2^{\infty}) $, for all but finitely many elliptic curves over $ \mathbf{Q}(2^{\infty}) $.\\ \\ Next, we turn to arithmetical structures, which Lorenzini introduced to model degenerations of curves. Let $G$ be a connected undirected graph on $n$ vertices with no loops but possibly multiedges. Given an arithmetical structure $(\textbf{r}, \textbf{d})$ on $G$, we describe a construction which associates to it a graph $G'$ on $n-1$ vertices and an arithmetical structure $(\textbf{r}', \textbf{d}')$ on $G'$. By iterating this construction, we derive an upper bound for the number of arithmetical structures on $G$ depending only on the number of vertices and edges of $G$. In the specific case of complete graphs, possibly with multiedges, we refine and compare our upper bounds to those arising from counting unit fraction representations. |
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