Using Automorphism Groups of Curves to Control the Slopes of their Jacobians
I’ve felt for a long time that automorphisms of curves should control or at least exert serious force on the slopes on their Jacobians. Symmetry forces height, as I’ve written about previously in Models of Formal Groups Laws of Every Height, and Endomorphisms Directly Control Slope.
In this post, I conjecture that the Frobenius eigenvalues of Artin-Schreier-Witt Curves (\( \mathbb{Z}/p^k \)-covers) are Gauss sums, generalizing an old and classic theorem of Davenport and Hasse (fuck nazis though). I explain and expand (i.e., make the paper type check and fill in details) a gorgeous reproof of this theorem using local systems by Robert Coleman.
This playful note is toward exploring this force, and outlining a conjectural approach for further exploiting it. Enjoy and click. Read about my Gauss Sums :P