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A class of Artin-Schreier curves with many automorphisms Scheidler, Renate


Algebraic curves with many points are useful in coding theory, but are also of number theoretic and geometric interest in their own right. Their symmetries are described by their automorphism group. Other information, such as the number of rational points on the curve and on the associated Jacobian variety over any field, is encoded in their zeta function. Unfortunately, all these objects are generally notoriously difficult to compute. In this talk, we describe a class of Artin-Schreier curves whose unusually big automorphism group can be explicitly described. The automorphism group contains a large extraspecial subgroup, precise knowledge of which makes it possible to compute the zeta functions of these curves after extending the base field to contain the appropriate field of definition. We find that over fields of square cardinality, these curves are either maximal or minimal, and we classify which curves fall into which category. This is joint work with Irene Bouw, Wei Ho, Beth Malmskog, Padmavathi Srinivasan and Christelle Vincent.

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