**Title:** Curves of Genus Two with Complex Multiplication

**Speaker:** Eyal Goren

**Speaker Info:** McGill

**Brief Description:**

**Special Note**: **The speaker will give a preseminar talk for graduate students at 11 am in TECH M120.**

**Abstract:**

Curves of genus 2 with complex multiplication are a natural analogue of elliptic curves with complex multiplication, parameterized by CM points of modular curves. The theory concerning values of modular functions at CM points of modular curves is well developed. Some applications include the construction of elliptic units, and construction of elliptic curves suitable for cryptography. In comparison, little is known about analogous questions for modular functions of genus 2. The study of such questions can be motivated by constructing units in abelian extensions of quartic CM fields (related to Stark's conjecture) and applications to cryptography. K. Lauter (Microsoft Research) and myself have investigated such questions and obtained several results, allowing in particular the applications to cryptography, and construction of S-units, where S is effectively controlled in terms of the CM data. We were also able to prove an analogue of the Gross-Zagier theorem in this setting. In this talk I will attempt to give an overview of the subject and indicated the main ideas going into the proofs.Information for the preseminar at 11 am:

Title: Curves of genus 2 with complex multiplication - warmup. Abstract: I will spend some time explaining why cryptographers care about curves of genus two whose Jacobian has complex multiplication, and how one might construct and parameterize curves of genus two. I will also explain some basic notion concerning the reduction type (modulo a prime) of a genus 2 curve and its Jacobian and what happens in the presence of complex multiplication. Finally, if we are still up to it, I will explain what is the main questions complex multiplication is occupied with in this setting.

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