Computing isogenies between abelian varieties

Abstract

We describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let A be an abelian variety of dimension g defined over a field of odd characteristic. Our algorithm comprises two principal steps. First, given a theta null point for A and a subgroup K isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian variety A/K. Then, from the knowledge of a theta null point of A/K, we present an algorithm to obtain a rational expression for an isogeny from A to A/K. The algorithm that results from combining these two steps can be viewed as a higher-dimensional analog of the well-known algorithm of Vélu for computing isogenies between elliptic curves. In the case where K is isomorphic to (Z/ Z) g for ∈ N * , the overall time complexity of this algorithm is equivalent to O(log ) additions in A and a constant number of th root extractions in the base field of A. In order to improve the efficiency of our algorithms, we introduce a compressed representation that allows us to encode a point of level 4 of a g-dimensional abelian variety using only g(g + 1)/2 · 4 g coordinates. We also give formulas for computing the Weil and commutator pairings given input points in theta coordinates.

Isogeny graphs of ordinary abelian varieties

“…The ( , )-isogenies are important for the following reason: algorithms for computing isogenies of elliptic curves from a given kernel (such as Vélu's formulae [8]) are difficult to generalize in higher dimension, as cyclic isogenies do not preserve the property of being principally polarizable in genus 2. The known methods such as [9][10][11][12][13][14][15][16][17] apply only to ( , )-isogenies. Only the very recent method of [18] allows to compute isogenies of certain cyclic kernels.…”

Computing separable isogenies in quasi-optimal time

“…Our algorithm is very similar to that of [11,Theorem 1.1] which is based, on the one hand, on the algorithm described in [24] to compute an isogeny f : A → B between A together with a line bundle of level n and B with a line bundle of level n , and, on the other hand, on the Koizumi general addition formula [19] from which a change of level formula is deduced (see [11,Proposition 4.1]). Our main improvement consists in working with 'formal points' rather than with geometric points of the kernel K.…”

Arithmetic on abelian and Kummer varieties

“…We can verify (13) by expressing the left hand side and right hand side of the equation in term of the U L 2 χ,i basis using (11). Then (14) and (15) are immediate consequences of (13) (using that θ i (−z 2 ) = θ −i (z 2 )) For more details, see [LR12] or [Mum66b]. A slightly different form is also given in [Mum66b, p. 334-335]; see also [Mum83;Koi76] for an analytic proof.…”

“…Now we fix an affine lift λ i π(z + e i ) where λ i is an unknown projective factor. By computing differential additions, and since π(z + me i ) = π(z) we recover λ m i as in [LR12;CR13]. We choose λ i satisfying these equations; by Theorem 4.4 we can then recover one of the element z (or −z) in the preimage.…”