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2021 | OriginalPaper | Chapter

# Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB

Authors : Taylor Dupuy, Kiran Kedlaya, David Roe, Christelle Vincent

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## Abstract

This document is intended to summarize the theory and methods behind fq_isog collection inside the ab_var database in the LMFDB as well as some observations gleaned from these databases. This collection consists of tables of Weil q-polynomials, which by the Honda-Tate theorem are in bijection with isogeny classes of abelian varieties over finite fields.
Appendix
Available only for authorised users
Footnotes
1
A Hodge class is an étale cohomology class that is invariant under the twisted action of Frobenius. A Hodge class is exceptional if it is not in the ring (under cup product) generated by Hodge classes of weight two. The Tate conjecture states that the space of Hodge classes is spanned by the images of algebraic cycles under the cycle class map to l-adic cohomology. This remains an open problem.

3
The particular function we use was written by Cremona: http://​doc.​sagemath.​org/​html/​en/​reference/​number_​fields/​sage/​rings/​number_​field/​unit_​group.​html. We also remark that there is an additional quite large bottleneck in this algorithm due to the need to compute the splitting field of the characteristic polynomial.

4
We also included (5,  3), where q g(g+1)∕2 ≈ 1.4 ⋅ 107, and we only included q up to 500 for g = 1, rather than 107.

5
Namely, we added powers of 2,  3,  5,  7 up to 1024 for g ∈{1,  2}, and raised the bound for g = 3 from 13 to 25.

6
We remark that the bounds used in op. cit. are also related to the original iterator described in [Ked08] which some reader might find enlightening.

7
It may be feasible to prove that the limiting distribution is Gaussian by bounding the difference in moments directly.

8
The radix point is the analogue of the decimal point for a general base expansion.

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