Arithmetic on Abelian and Kummer Varieties
Lubicz , David
Robert , Damien
- Publisher: HAL CCSD
[ INFO.INFO-SC ] Computer Science [cs]/Symbolic Computation [cs.SC]
arxiv: Mathematics::Number Theory
International audience; A Kummer variety is the quotient of an abelian variety by the automorphism $(-1)$ acting on it. Kummer varieties can be seen as a higher dimensional generalisation of the $x$-coordinate representation of a point of an elliptic curve given by its Weierstrass model. Although there is no group law on the set of points of a Kummer variety, there remains enough arithmetic to enable the computation of exponentiations via a Montgomery ladder based on differential additions. In this paper, we explain that the arithmetic of a Kummer variety is much richer than usually thought. We describe a set of composition laws which exhaust this arithmetic and show that these laws may turn out to be useful in order to improve certain algorithms. We explain how to compute efficiently these laws in the model of Kummer varieties provided by level $2$ theta functions. We also explain how to recover the full group law of the abelian variety with a representation almost as compact and in many cases as efficient as the level $2$ theta functions model of Kummer varieties.