We study the Rankin L-series of a cuspidal automorphic representation of GL(2) of even weight over the rational numbers, twisted by a character of a real quadratic field. When the sign of the functional equation is +1, we give an explicit formula for the central value of the L-series, analogous to the formulae obtained by Gross, Zhang, and Xue in the imaginary case. The proof uses a version of the Rankin-Selberg method in which the theta correspondence plays an important role. We give two applications, to computing the order of the Tate-Shafarevich group of the base change to a real quadratic field of an elliptic curve defined over the rationals, and to proving the equidistribution of individual closed geodesics on modular curves.

Abstract. We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for Γ1(N ): an extension of the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficients Theorem of Manin.

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as well as arithmetic information about the associated reciprocal classes of binary quadratic forms. One striking feature is the absence of a gap beyond zero in the limiting distribution, contrasting with the analog Euclidean situation.

In order to obtain highly dispersed heteropolyacids (HPAs) species, H3PMo12O40 and H4PVMo11O40 were supported on various supports: silica (Aerosil - Degussa and Romsil types) and TiO2. The structure and thermal decomposition of supported and unsupported HPAs were followed by different techniques (TGA-DTA, FTIR, XRD, low temperature nitrogen adsorption, scanning electron microscopy). All the supported HPAs were prepared by impregnation using the incipient wetness technique with a 1:1 mixture of water-ethanol. Samples were prepared with different concentrations to examine the effect of loading on the thermal behavior of the supported acid catalysts. The thermal stability was evaluated with reference to the bulk solid acids and mechanical mixtures. After deposition on silica types supports, an important decrease in thermal stability was observed on the Romsil types and a small decrease on the Aerosil type. The stability of the heteropolyacids supported on titania increased due to an anion-support interaction, as the thermal decomposition proceeded in two steps. The structure of the HPAs was not totally destroyed at 450 ?C as some IR bands were still preserved. A relatively uniform distribution of HPAs on the support surface was observed for all compositions of the active phase. No separate crystallites of solid phase HPAs were found in the SEM images.

We prove explicit formulas decomposing cusp forms of even weight for the modular group, in terms of generators having rational periods, and in terms of generators having rational Fourier coefficients. Using the Shimura correspondence, we also give a decomposition of Hecke cusp forms of half integral weight k + 1/2 with k even in terms of forms with rational Fourier coefficients, given by RankinCohen brackets of theta series with Eisenstein series.