The Siegel modular variety of degree two and level three

Abstract

Abstract. Let A2(n) denote the quotient of the Siegel upper half space of degree two by Γ2(n), the principal congruence subgroup of level n in Sp(4, Z). A2(n) is the moduli space of principally polarized abelian varieties of dimension two with a level n structure, and has a compactification A2(n) * first constructed by Igusa. When n ≥ 3 this is a smooth projective algebraic variety of dimension three.In this work we analyze the topology of A2 (3) * and the open subset A2(3). In this way we obtain the rational cohomology ring of Γ2(3). The key is that one has an explicit description of A2 (3) * : it is the resolution of the 45 nodes on a projective quartic threefold whose equation was first written down about 100 years ago by H. Burkhardt. We are able to compute the zeta function of this variety reduced modulo certain primes.

Descent via (3,3)-isogeny on Jacobians of genus 2 curves

“…. ,ξ 3 for φ * (O J (2Θ J )) that is the pullback of a basis of the type described by (8). We can then read off the curve C, at least up to quadratic twist, from the resulting equation forK.…”

The defect of Fano $3$-folds

“…The cohomology of Y is determined in [HW01] and b 2 ( Y ) = 61, so that the defect of the Burkhardt quartic is 15.…”

Theory of the Siegel Modular Variety

“…We note that the intersection cohomology IH * Ā g , C) contains a copy of H * (Y g , C). For the theory of the cohomology of the Siegel modular variety (in particular of degree two) we refer to [20], [58,59], [87], [88] and [124,125].…”