COMPUTING POINTS ON BIELLIPTIC MODULAR CURVES OVER FIXED QUADRATIC FIELDS

An Algorithm for Modular Elliptic Curves over Real Quadratic Fields

Let F be a real quadratic field with narrow class number one, and f a Hilbert newform of weight 2 and level n with rational Fourier coefficients, where n is an integral ideal of F . By the Eichler-Shimura construction, which is still a conjecture in many cases when [F : Q] > 1, there exists an elliptic curve Ef over F attached to f . In this paper, we develop an algorithm that computes the (can...

Modular Forms and Elliptic Curves over Imaginary Quadratic Fields

On Hyperelliptic Modular Curves over Function Fields

We prove that there are only finitely many modular curves of Delliptic sheaves over Fq(T ) which are hyperelliptic. In odd characteristic we give a complete classification of such curves.

Rational Points on Curves over Finite Fields

Preface These notes treat the problem of counting the number of rational points on a curve defined over a finite field. The notes are an extended version of an earlier set of notes Aritmetisk Algebraisk Geometri – Kurver by Johan P. Hansen [Han] on the same subject. In Chapter 1 we summarize the basic notions of algebraic geometry, especially rational points and the Riemann-Roch theorem. For th...

Counting Points on Curves over Finite Fields

Stanford University) Abstract: A curve is a one dimensional space cut out by polynomial equations, such as y2=x3+x. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables (x and y in the example) in the finite field that satisfy the given polyn...