Genus 3 hyperelliptic curves with CM via Shimura Reciprocity
| Autor: | Sorina Ionica, Bogdan Adrian Dina |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Abelian variety
Pure mathematics Mathematics - Number Theory Mathematics::Number Theory Reciprocity law symbols.namesake Mathematics::Algebraic Geometry Reciprocity (electromagnetism) Quartic function Jacobian matrix and determinant symbols FOS: Mathematics Number Theory (math.NT) Abelian group CM-field 14K22 (Primary) 14D22 (Secondary) Hyperelliptic curve Mathematics |
| DOI: | 10.48550/arxiv.2003.06386 |
| Popis: | Up to isomorphism over C, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic CM field K, we show that if there exists a hyperelliptic Jacobian with CM by K, then all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura's reciprocity law, we give an algorithm for computing approximations of the invariants of the initial curve, as well as their Galois conjugates. This allows us ton define and compute class polynomials for genus 3 hyperelliptic curves with CM. |
| Databáze: | OpenAIRE |
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