### The main theorem(s) of complex multiplication and beyond

Linfoot Number Theory Seminar

23rd May 2018, 11:00 am – 12:00 pm

Howard House, 4th Floor Seminar Room

The theory of complex multiplication on elliptic curves has many

fascinating number-theoretic applications. For example, it can be used

to construct all abelian extensions of an imaginary quadratic number field.

At the heart of the theory lies the so-called main theorem of complex

multiplication. This talk will start by stating the main theorem and

then sketch the above-mentioned application.

Furthermore, it will explain how the main theorem can be used to

understand the Galois action on the moduli space of elliptic curves.

Time permitting, it will also discuss the higher-dimensional case of CM

abelian varieties, state various other versions of the main theorem and

mention a curious discovery by J. Nekovar.

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