Magma Implementations for Polarizations of abelian varieties over finite fields via canonical liftings, by Jonas Bergström, Valentijn Karemaker and Stefano Marseglia.
Please send comments and bug reports to stefano.marseglia89@gmail.com.
This package requires stmar89/AbVarFq.
The file AlmOrd.m contains the function overorders_maximal_at_ss that given as input an IsogenyClassFq of almost-ordinary squarefree abelian varieties in odd characteristic returns the over-orders of ZZ[F,V] which are maximal at the supersingular part of the endomorphism algebra. These and only these are endomorphism rings of abelian varieties in such an isogeny class.
See Oswal-Shankar (below) for more details.
The file ResRefCond.m contains code to check wheter a CM-type satisfies the Shimura-Taniyama formula and the Residual Reflex Condition (RRC). Recall that a CMType PHI satisfies RRC if:
*) the CM-type satisfies the Shimura-Taniyama formula, and
*) the associated reflex field has residue field that can be realized as a subfield of the field of definition of AVh.
See Chai-Conrad-Oort (below) for more details about RRC.
The main intrinsics are:
ShimuraTaniyama(AVh,PHI): returns wheter a CM-type PHI satisfies the Shimura-Taniyama formula for the Forbenius of the Isogeny class AVh.pAdicReflexField(AVh,PHI): given an isogeny class AVh and a CM-type PHI returns the reflex field associated to PHI as a p-adic field.IsResidueReflexFieldEmbeddable(AVh,PHI): given an isogeny class AVh over FFq returns the if the residue field of reflex field associated to the CM-type can be embedded in FFq.ResidualReflexCondition(AVh,PHI): given an isogeny class AVh and a CM-type PHI returns whether PHI satisfies the residual reflex conditon (RRC).ResidualReflexCondition(AVh): given an isogeny class AVh returns the sequence of CM-types that satisfy RRC.
The package provides several auxiliary intrinsics, including RationalSplittingField, pAdicSplittingField, EmbeddingOfSplittingFields and ComplexRoots, that allow to perform intermediate computations. Severeal computations can be performed using different methods.
For complete complete descriptions and more details we refer to the List of functions.
In the file examples.txt there is the code for the examples in our paper. Such code is also intended as a guide on how to use the various functions and intrisics to compute isomorphism classes and principal polarizations.
The file additional_examples.pdf contains cumulative tables of computations for squarefree isogeny classes over small finite fields of small dimension.
Eventually this packages will become a submodule of stmar89/AbVarFq and the functionalities will be included in the correspoding intrinsic.
Ching-Li Chai, Brian Conrad, and Frans Oort, Complex multiplication and lifting problems, Mathematical Surveys and Monographs, vol. 195, American Mathematical Society, Providence, RI, 2014.
Abhishek Oswal and Ananth N. Shankar, Almost ordinary abelian varieties over finite fields, Journal of the London Mathematical Society 101 (2020), no. 3, 923–937.