Hecke action and the degree of the modular parameterization
Heegner cycles, modular forms and jacobi forms
We give a geometric interpretation of an arithmetic rule to generate explicit formulas for the Fourier coefficients of elliptic modular forms and their associated Jacobi forms. We discuss applications of these formulas and derive as an example a criterion similar to Tunnel's criterion for a number to be a congruent number.
Heegner points and derivatives of L-series.
Heuristics on Tate-Shafarevitch groups of elliptic curves defined over .
High rank eliptic curves of the form y2 = x3 + Bx.
Seven elliptic curves of the form y2 = x3 + B x and having rank at least 8 are presented. To find them we use the double descent method of Tate. In particular we prove that the curve with B = 14752493461692 has rank exactly 8.
Higher order operations in Deligne cohomology.