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Wednesday 11/20/02 1:00-2:00 PM in Hylan 1106B:
Ken McMurdy- Regions of Reduction Type on X_0(N) ABSTRACT: One way to study the geometry of the modular curve X_0(N) (as a rigid-analytic space over C_p) is to categorize moduli-theoretic points (E,C) based on the reduction type (mod p) of the elliptic curve E. For example, the points where E has bad reduction simply comprise a disjoint union of discs corresponding to families of Tate curves. In this talk I will construct a specific family of elliptic curves over C_5 and show explicitly how this family moves through the regions of ordinary, supersingular, and too-supersingular reduction. The focus will be on tracking the 5-torsion to see how it relates to the kernel of reduction. An alternative way to define supersingular elliptic curves is by their endomorphism rings, which are always isomorphic to a maximal order in some quaternion algebra. So if there is time, I will give explicit generators for the endomorphism ring (over F_5) of the supersingular elliptic curve at the heart of the constructed family. |