Algebraic Geometry for Coding Theory and Cryptography by Everett W. Howe Kristin E. Lauter & Judy L. Walker

Author:Everett W. Howe, Kristin E. Lauter & Judy L. Walker
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

3.4.3 Invariants for Curves and Abelian Surfaces

We need to relate the RM invariants (J 1, J 2, J 3) to the invariants for plain old principally polarized abelian surfaces, and in particular Jacobians of genus-2 curves without any special RM structure. The moduli space of principally polarized abelian surfaces is coarsely represented by the Siegel modular space , where

and the symplectic group

acts on via

Every rational function on is a quotient of elements of the graded ring of holomorphic Siegel modular forms for . Igusa proved in [15] that this ring is generated by ψ 4, ψ 6, χ 10, and χ 12, where

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