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Modular curves and finite groups: building connections via computation

By David Roe

Appears in collection : COUNT - Computations and their Uses in Number Theory / Les calculs et leurs utilisations en théorie des nombres

The study of rational points on modular curves has a long history in number theory. Mazur's 1970s papers that describe the possible torsion subgroups and isogeny degrees for rational elliptic curves rest on a computation of the rational points on $X_{0}(N)$ and $X_{1}(N)$, and a large body of work since then continues this tradition.

Modular curves are parameterized by open subgroups $H$ of $\mathrm{GL}_{2}(\hat{\mathbb{Z}})$, and correspondingly parameterize elliptic curves $E$ whose adelic Galois representation $\displaystyle \lim_{ \leftarrow }E[n]$ is contained in $H$. For general $H$, the story of when $X_{H}$ has non-cuspidal rational or low degree points (and thus when there exist elliptic curves with the corresponding level structure) becomes quite complicated, and one of the best approaches we have for understanding it is large-scale computation.

I will describe a new database of modular curves, including rational points, explicit models, and maps between models, along with some of the mathematical challenges faced along the way. The close connection between modular curves and finite groups also arises in other areas of number theory and arithmetic geometry. Most well known are Galois groups associated to field extensions, but one attaches automorphism groups to algebraic varieties and Sato-Tate groups to motives. Building on existing tables of groups, we have added a new finite groups section to the L-functions and modular forms database, which we hope will prove useful both to number theorists and to others who are using and studying finite groups.

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Citation data

  • DOI 10.24350/CIRM.V.20007203
  • Cite this video Roe, David (02/03/2023). Modular curves and finite groups: building connections via computation. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20007203
  • URL https://dx.doi.org/10.24350/CIRM.V.20007203

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