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arXiv · 2305.15164

Quadratic $\ell$-adic sheaf and its Heisenberg group

Daichi Takeuchi
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Abstract

In this paper, we introduce a new class of $\ell$-adic sheaves, which we call quadratic $\ell$-adic sheaves, on connected unipotent commutative algebraic groups over finite fields. They are sheaf-theoretic enhancements of quadratic forms on finite abelian groups in the spirit of the function-sheaf dictionary. We show that a certain finite Heisenberg group acts on a quadratic sheaf and that the cohomology of the quadratic sheaf gives an irreducible representation of the group. We also compute the Frobenius eigenvalues of the cohomology groups. As a byproduct, we find a large number of examples of affine supersingular varieties.

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Daichi Takeuchi. 2023-05-24. Quadratic $\ell$-adic sheaf and its Heisenberg group. https://arxiv.org/abs/2305.15164

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