Curves with many symmetries

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Noam Elkies, Harvard University
Fine Hall 214

By a celebrated theorem of Hurwitz, a curve $X/{\bf C}$ of genus $g>1$ has at most $84(g-1)$ points. Curves that attain or come close to this bound, such as the modular curves ${\rm X}(N)$, often have a rich structure with diverse connections to and near number theory. We review the relevant basic theorems and give some explicit examples, including recent observations on continuous families of such curves, for which the number of automorphisms is bounded by $12(g-1)$.