Publication Date

5-2014

Advisor(s)

Christopher J. Rasmussen

Department

Mathematics

Language

English

Abstract

Let A be an abelian variety defined over a number field F. For a prime number 𝓁, we consider the field extension of F generated by the 𝓁-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we require this field to be both a pro-𝓁 extension of F(ΞΌπ“βˆž) and unramified away from 𝓁, examples are quite rare. Indeed, it is expected that for a fixed dimension and field of definition, there exists such an abelian variety for only a finite number of primes.

We prove a uniform version of the conjecture in the case where the abelian varieties are elliptic curves with complex multiplication. In addition, we provide explicit bounds in cases where the number field has degree less than or equal to 100.

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