Non-vanishing of LL-functions associated to cusp forms of half-integral weight

In this article, we prove non-vanishing results for $L$-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen to forms of half-integral weight.Comment: 8 pages, Accepted for publication in Oman conference proceedings (Springer

On Shimura's decomposition

Let $k$ be an odd integer $\ge 3$ and $N$ a positive integer such that $4 \mid N$. Let $\chi$ be an even Dirichlet character modulo $N$. Shimura decomposes the space of half-integral weight cusp forms $S_{k/2}(N,\chi)$ as a direct sum of $S_0(N,\chi)$ (the subspace spanned by 1-variable theta- series) and $S_{k/2}(N,\chi,\phi)$ where $\phi$ runs through a certain family of integral weight newforms. The explicit computation of this decomposition is important for practical applications of a theorem of Waldspurger relating critical values of $L$-functions of quadratic twists of newforms of even weight to coefficients of modular forms of half-integral weight.Comment: 12 pages, to appear in the International Journal of Number Theor

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 correspondence of modular forms and applications to values of L-series

An interpretation of the Rogers–Zudilin approach to the Boyd conjectures is established. This is based on a correspondence of modular forms which is of independent interest. We use the reinterpretation for two applications to values of L-series and values of their derivatives

Approximations to seismic AVA responses: Validity and potential in glaciological applications

Amplitude-variation-with-angle (AVA) methods establish the seismic properties of material either side of a reflective interface, and their use is growing in glaciology. The AVA response of an interface is defined by the complex Knott-Zoeppritz (K-Z) equations, numerous approximations to which we typically assume weak interface contrasts and isotropic propagation, inconsistent with the strong contrasts at glacier beds and the vertically transverse isotropic (VTI) fabrics were associated with englacial reflectivity. We considered the validity of a suite of approximate K-Z equations for the exact P-wave reflectivity RP of ice overlying bedrock, sediment and water, and englacial interfaces between isotropic and VTI ice.We found that the approximations of Aki-Richards, Shuey, and Fatti match exact glacier bed reflectivity to within RP ± 0.05, smaller than the uncertainty in typical glaciological AVA analyses, but only for maximum incident angle θi limited to 30°. A stricter limit of θi ≤ 20° offered comparable accuracy to a hydrocarbon benchmark case of shale overlying gas-charged sand. The VTI-compliant Rüger approximation accurately described englacial reflectivity, to within RP ± 0.01, and it can be modified to give a quadratic expression in sin2 (θi)suitable for curve-matching operations. Having shown the circumstances under which AVA approximations were valid for glaciological applications, we have suggested that their interpretative advantages can be exploited in the future AVA interpretations

Изучение гидролитической устойчивости и растворимости стампирина

Представлены результаты исследования гидролитической устойчивости стампирина I (1-фенил-2,3-диметил-4-стеароиламино-5-пиразолона)-нового противовоспалительного средства в различных средах и условиях и его растворимости в некоторых органических растворителях. Показано, что наиболее подходящими условиями полного гидролиза I является кипячение его на воздушной бане в 25% растворе соляной кислоты в течение 45 минут. В водной и щелочной средах I является гидролитически устойчивым. Определена растворимость I в граммах на 100 мл раствора при 20° С весовым методом. Она равна 1,31 в этиловом спирте, 1,01 в изопропиловом спирте, 0,07 в диэтиловом эфире, 3,77 в бензоле, 0,79 в четыреххлористом углероде. В воде I практически не растворим

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

Modular differential equations for characters of RCFT

We discuss methods, based on the theory of vector-valued modular forms, to determine all modular differential equations satisfied by the conformal characters of RCFT; these modular equations are related to the null vector relations of the operator algebra. Besides describing effective algorithmic procedures, we illustrate our methods on an explicit example.Comment: 13 page

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.Comment: v5: Some typos correcte