Computational aspects of modular parametrizations of elliptic curves

Abstract

\abstract{ We investigate computational problems related to modular parametrizations of elliptic curves defined over $\mathbb{Q}$. We develop algorithms to compute the Mazur Swinnerton-Dyer critical subgroup of elliptic curves, and verify that for all elliptic curves of rank two and conductor less than a thousand, the critical subgroup is torsion. We also develop algorithms to compute Fourier expansions of $\Gamma_0(N)$-newforms at cusps other than the cusp at infinity. In addition, we study properties of Chow-Heegner points associated to a pair of elliptic curves. We proved that the index of Chow-Heegner points are always divisible by two when the conductor $N$ has many prime divisors, .We also develop an algebraic algorithms to compute the Chow-Heegner points.