This paper is the second of a series devoted to the study of the rank of J0(q) (the Jacobian of the modular curve X0(q)), from the analytic point of view stemming from the Birch and Swinnerton-Dyer conjecture, which is tantamount to the study, on average, of the order of vanishing at the central critical point of the L-functions of primitive weight two forms f of level q (q prime). We prove tha...

Let E be an optimal elliptic curve of conductor N , such that the L-function LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split. The Gross-Zagier theorem gives a formula that expresses the Birch and Swinnerton-Dyer conjectural order the Shafarevich-Tate group of E over K as a rational number. We extract an integer factor from...

1 Algebraic Groups 7 1.1 Group Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Restriction of Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Algebraic Groups over a Nonalgebraically Closed Field K . . . . . . . . . . . 13 1.4 Structure of Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Topologizing G(R...

We define a new E∞ operad StLGT ree based on surfaces with foliations which contains Ek sub–operads. We construct CW models for these operads and provide applications of these models by giving actions on Hochschild complexes –thus making contact with string topology–, by giving explicit cell representatives for the Dyer-LashofCohen operations for the 2-cubes and by constructing new Ω spectra. T...

Abstract This study provides the first contribution to knowledge of non-calcareous grasslands arenaceous sector Monti Sibillini National Park (central Italy). We conducted 55 relevés using Braun-Blanquet phytosociological method and analysed ecology plant communities by Redundancy analysis topographic variables Ellenberg Indicator Values as predictors. identified nine communities, belonging fou...