Dispersionless Hirota equations and the genus 3 hyperelliptic divisor

Equations of dispersionless Hirota type have been thoroughly investigated in the mathematical physics and differential geometry literature. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional and the action of the natural equivalence group Sp(6, R) on the parameter space has an open orbit. However the structure of the `master-equation' corresponding to this orbit remained elusive. Here we prove that the master-equation is specified by the vanishing of any genus 3 theta constant with even characteristic. The rich geometry of integrable Hirota type equations sheds new light on local differential geometry of the genus 3 hyperelliptic divisor, in particular, the integrability conditions can be viewed as local differential-geometric constraints that characterise the hyperelliptic divisor uniquely modulo Sp(6, C)-equivalence.Comment: amended version, to appear in Comm. Math. Phys., 15 page

Constructing vector-valued Siegel modular forms from scalar-valued Siegel modular forms

This paper gives a simple method for constructing vector-valued Siegel modular forms from scalar-valued ones. The method is efficient in producing the siblings of Delta, the smallest weight cusp forms that appear in low degrees. It also shows the strong relations between these modular forms of different genera. We illustrate this by a number of examples.Comment: 21 pages; misprints corrected; to appear in PAM

Covariants of binary sextics and vector-valued Siegel modular forms of genus two

We extend Igusa’s description of the relation between invariants of binary sextics and Siegel modular forms of degree 2 to a relation between covariants and vector-valued Siegel modular forms of degree 2. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree 2

Siegel modular forms of genus 2 and level 2

We study vector-valued Siegel modular forms of genus 2 and level 2. We describe the structure of certain modules of vector-valued modular forms over rings of scalar-valued modular forms.Comment: 46 pages. To appear in International Journal of Mathematic

Covariants of binary sextics and vector-valued Siegel modular forms of genus two

We extend Igusa's description of the relation between invariants of binary sextics and Siegel modular forms of degree two to a relation between covariants and vector-valued Siegel modular forms of degree two. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree two.Comment: 19 page

Gravitational production of matter and radiation during reheating

I present the production of matter and radiation during reheating after inflation, considering only gravitational interactions between the inflaton background and the other sectors. Processes considered are the following: i) the exchange of a graviton, $h_{\mu \nu}$, involved in the scattering of the inflaton or particles in the newly created radiation bath; ii) scattering of the inflaton background and particles in the radiation bath including the effects of non-minimal couplings to curvature of the Higgs boson and the inflaton. Requiring the existence of heavy right-handed neutrinos (RHN), I show that a minimal scenario utilizing only these "gravitational portals" is able to generate simultaneously the observed relic density of Dark Matter (DM), the baryon asymmetry through leptogenesis, as well as a sufficiently hot thermal bath after inflation, for generic models of large field inflation.Comment: Contribution to the 34th Rencontres de Blois on Particle Physics and Cosmology (Blois 2023

On vector-valued Siegel modular forms of degree 2 and weight (j,2)

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes

Dimension formulas for spaces of vector-valued Siegel modular forms of degree two and level two

Using a description of the cohomology of local systems on the moduli space of abelian surfaces with a full level two structure, together with a computation of Euler characteristics we find the isotypical decomposition, under the symmetric group on 6 letters, of spaces of vector-valued Siegel modular forms of degree two and level two.Comment: 15 page