The Shimura–Taniyama formula

Abstract: We state and prove the Shimura–Taniyama formula, under some technical restrictions. Let $A$ be an abelian variety with complex multiplication by a CM algebra $E$. Roughly speaking, the Shimura–Taniyama formula asserts that the Frobenius action on the reduction of $A$ arises from an element of $E$. This can be viewed as a weak way to access information about the Galois representation associated to $A$. Time-permitting, we will state (but not prove) the Main theorem of complex multiplication, which gives complete access to this Galois representation.

algebraic geometrynumber theory

Audience: advanced learners

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STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Fall 2026 topic: Explicit class field theory.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

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