Rankin-Cohen Type Differential Operators for Siegel Modular Forms

Let H_n be the Siegel upper half space and let F and G be automorphic forms on H_n of weights k and l, respectively. We give explicit examples of differential operators D acting on functions on H_n x H_n such that the restriction of D(F(Z_1) G(Z_2)) to Z = Z_1 = Z_2 is again an automorphic form of weight k+l+v on H_n. Since the elliptic case, i.e. n=1, has already been studied some time ago by R. Rankin and H. Cohen we call such differential operators Rankin-Cohen type operators. We also discuss a generalisation of Rankin-Cohen type operators to vector valued differential operators.Comment: 19 pages LaTeX2e using amssym.de

Generic differential operators on Siegel modular forms and special polynomials

Holomorphic vector valued differential operators acting on Siegel modular forms and preserving automorphy under the restriction to diagonal blocks are important in many respects, including application to critical values of L functions. Such differential operators are associated with vectors of new special polynomials of several variables defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular forms. We will give formulas for all such polynomials in two different ways. One is to describe them using polynomials characterized by monomials in off-diagonal block variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We construct an explicit generating series of polynomials mutually related under certain mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential operators of this sort are obtained by certain projections. This process exhausts all the differential operators in question. This is also generic in the sense that for any number of variables and block partitions, it is given by a recursive unified expression. As an application, we prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are essentially vector valued Siegel modular forms of lower degrees, which are obtained as images of the differential operators given above. We also show that the original forms are recovered by the images of our operators. This is an ultimate generalization of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results and practical construction are also given

Evolution of Collisionally Merged Massive Stars

We investigate the evolution of collisionally merged stars with mass of ~100 Msun which might be formed in dense star clusters. We assumed that massive stars with several tens Msun collide typically after ~1Myr of the formation of the cluster and performed hydrodynamical simulations of several collision events. Our simulations show that after the collisions, merged stars have extended envelopes and their radii are larger than those in the thermal equilibrium states and that their interiors are He-rich because of the stellar evolution of the progenitor stars. We also found that if the mass-ratio of merging stars is far from unity, the interior of the merger product is not well mixed and the elemental abundance is not homogeneous. We then followed the evolution of these collision products by a one dimensional stellar evolution code. After an initial contraction on the Kelvin-Helmholtz (thermal adjustment) timescale (~10^{3-4} yr), the evolution of the merged stars traces that of single homogeneous stars with corresponding masses and abundances, while the initial contraction phase shows variations which depend on the mass ratio of the merged stars. We infer that, once runaway collisions have set in, subsequent collisions of the merged stars take place before mass loss by stellar winds becomes significant. Hence, stellar mass loss does not inhibit the formation of massive stars with mass of ~1000Msun

Eisenstein Congruences for SO(4, 3), SO(4, 4), Spinor, and Triple Product L-values

We work out instances of a general conjecture on congruences between Hecke eigenvalues of induced and cuspidal automorphic representations of a reductive group, modulo divisors of certain critical L-values, in the case that the group is a split orthogonal group. We provide some numerical evidence in the case that the group is SO(4, 3) and the L-function is the spinor L-function of a genus 2, vector-valued, Siegel cusp form. We also consider the case that the group is SO(4, 4) and the L-function is a triple product L-function

On the special values of certain L-series related to half-integral weight modular forms

Let h be a cuspidal Hecke eigenform of half-integral weight, and En/2+1/2 be Cohen’s Eisenstein series of weight n/2+1/2. For a Dirichlet character χ we define a certain linear combination R(χ)(s, h,En/+1/2) of the Rankin-Selberg convolution products of h and En/2+1/2 twisted by Dirichlet characters related with χ. We then prove a certain algebraicity result for R(χ)(l, h,En/2+1/2) with l integers

Relations between some invariants of algebraic varieties in positive characteristic

We discuss relations between certain invariants of varieties in positive characteristic, like the a-number and the height of the Artin-Mazur formal group. We calculate the a-number for Fermat surfacesComment: 13 page

The Saito-Kurokawa lifting and Darmon points

Let E_{/_\Q} be an elliptic curve of conductor $Np$ with $p\nmid N$ and let $f$ be its associated newform of weight 2. Denote by $f_\infty$ the $p$-adic Hida family passing though $f$, and by $F_\infty$ its $\Lambda$-adic Saito-Kurokawa lift. The $p$-adic family $F_\infty$ of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients $\{\widetilde A_T(k)\}_T$ indexed by positive definite symmetric half-integral matrices $T$ of size $2\times 2$. We relate explicitly certain global points on $E$ (coming from the theory of Stark-Heegner points) with the values of these Fourier coefficients and of their $p$-adic derivatives, evaluated at weight $k=2$.Comment: 14 pages. Title change

A new geometric description for Igusa's modular form (azy)5(azy)_5

The modular form $(azy)_5$ notably appears in one of Igusa's classic structure theorems as a generator of the ring of full modular forms in genus 2, being exhibited by means of a complicated algebraic expression. In this work a different description for this modular form is provided by resorting to a peculiar geometrical approach.Comment: 10 page

Sodium abundances in nearby disk stars

We present sodium abundances for a sample of nearby stars. All results have been derived from NLTE statistical equilibrium calculations. The influence of collisional interactions with electrons and hydrogen atoms is evaluated by comparison of the solar spectrum with very precise fits to the Na I line cores. The NLTE effects are more pronounced in metal-poor stars since the statistical equilibrium is dominated by collisions of which at least the electronic component is substantially reduced. The resulting influence on the determination of sodium abundances is in a direction opposite to that found previously for Mg and Al. The NLTE corrections are about -0.1 in thick-disk stars with [Fe/H] about -0.6. Our [Na/Fe] abundance ratios are about solar for thick- and thin-disk stars. The increase in [Na/Fe] as a function of [Fe/H] for metal-rich stars found by Edvardsson et al. (1993) is confirmed. Our results suggest that sodium yields increase with the metallicity, and quite large amounts of sodium may be produced by AGB stars. We find that [Na/Fe]ratios, together with either [Mg/Fe] ratio, kinematic data or stellar evolutionary ages, make possible the individual discrimination between thin- and thick-disk membership.Comment: 11pages, 11 figures. A&A accepte

How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?

Single centered supersymmetric black holes in four dimensions have spherically symmetric horizon and hence carry zero angular momentum. This leads to a specific sign of the helicity trace index associated with these black holes. Since the latter are given by the Fourier expansion coefficients of appropriate meromorphic modular forms of Sp(2,Z) or its subgroup, we are led to a specific prediction for the signs of a subset of these Fourier coefficients which represent contributions from single centered black holes only. We explicitly test these predictions for the modular forms which compute the index of quarter BPS black holes in heterotic string theory on T^6, as well as in Z_N CHL models for N=2,3,5,7.Comment: LaTeX file, 17 pages, 1 figur