**Arithmetic on Abelian and Kummer Varieties**

*David Lubicz and Damien Robert*

**Abstract: **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.

**Category / Keywords: **foundations / Arithmetic Kummer Varieties

**Date: **received 20 Jun 2014

**Contact author: **damien robert at inria fr

**Available format(s): **PDF | BibTeX Citation

**Version: **20140626:205356 (All versions of this report)

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