Algebraic Geometry of Abelian Varieties over Finite Fields

Abelian varieties over finite fields form a rich nexus between algebraic geometry, number theory and arithmetic applications. These higher‐dimensional analogues of elliptic curves admit a group structure and are classified up to isogeny by their Frobenius endomorphisms via the Honda–Tate theorem. Their point counts obey the Hasse–Weil bounds and give rise to Weil q-numbers, whose combinatorial arrangements determine Newton polygons and slope decompositions. Integral structures such as Tate modules and Dieudonné modules encode p-adic and ℓ-adic cohomological invariants, allowing one to probe endomorphism rings, monodromy actions and Tate conjecture cases. Practical applications range from explicit point-count realisations for cryptographic protocols to the construction of Jacobians of curves with extreme rational point counts. Recent advances have deepened understanding of lattice structures in cohomology, effective bounds on multiplicative relations among Frobenius eigenvalues and the realisation of prescribed orders within the Hasse–Weil interval, thus solidifying the conceptual framework and computational toolkit for this field.

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Algebraic Geometry of Abelian Varieties over Finite Fields publication trend

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Technical terms

Abelian variety: A complete algebraic variety with a commutative group law defined over a field.

Isogeny: A surjective morphism of abelian varieties with finite kernel, preserving group structure.

Tate module: The inverse limit of ℓ-power torsion points of an abelian variety, carrying a Galois action.

Frobenius endomorphism: The q-power map on varieties over a finite field of order q, central to point-count formulas.

Newton polygon: A graphical device recording slopes of Frobenius action on cohomology, indicating isogeny types.

Dieudonné module: A module with Frobenius and Verschiebung operators encoding p-power torsion in characteristic p.

References

1. Lattices in Tate modules. Proceedings of the National Academy of Sciences of the United States of America (2021).
2. Angle ranks of abelian varieties. Mathematische Annalen (2023).
3. Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points. Journal of Symbolic Computation (2024).
4. Abelian varieties of prescribed order over finite fields. Mathematische Annalen (2025).