  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.
