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Decompositions of an Abelian surface and quadratic forms

Shouhei Ma (2011)

Annales de l’institut Fourier

When a complex Abelian surface can be decomposed into a product of two elliptic curves, how many decompositions does the Abelian surface admit? We provide arithmetic formulae for the number of such decompositions.

Denominators of Igusa class polynomials

Kristin Lauter, Bianca Viray (2014)

Publications mathématiques de Besançon

In [22], the authors proved an explicit formula for the arithmetic intersection number CM ( K ) . G 1 on the Siegel moduli space of abelian surfaces, under some assumptions on the quartic CM field K . These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. One of the main tools in the proof was a previous result of the authors [21] generalizing the singular moduli formula of Gross...

Derived invariance of the number of holomorphic 1 -forms and vector fields

Mihnea Popa, Christian Schnell (2011)

Annales scientifiques de l'École Normale Supérieure

We prove that smooth projective varieties with equivalent derived categories have isogenous Picard varieties. In particular their irregularity and number of independent vector fields are the same. This implies that all Hodge numbers are the same for arbitrary derived equivalent threefolds, as well as other consequences of derived equivalence based on numerical criteria.

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