In this paper we list all the weight 2 newforms f(τ) that are products and quotients of the Dedekind eta-function η(τ) := q ∞ Y n=1 (1− q), where q := e2πiτ . There are twelve such f(τ), and we give a model for the strong Weil curve E whose Hasse-Weil L−function is the Mellin transform for each of them. Five of the f(τ) have complex multiplication, and we give elementary formulae for their Four...

Let E denote an elliptic curve over Q and Y E (7) the modular curve classifying the elliptic curves E over Q such that the representations of Gal(Q/Q) in the 7-torsion points of E and of E are symplectically isomorphic. In case E is given by a Weierstraß equation such that the c 4 invariant is a square, we exhibit here nontrivial points of Y E (7)(Q). From this we deduce an infinite family of c...

For each proper divisor d of (q − √q + 1), q being a square power of a prime, maximal curves Fq-covered by the Hermitian curve of genus 1 2 ( q− √ q+1 d − 1) are constructed.

Let q be a prime power, and let E be an elliptic curve over the field F q of q elements. As usual we associate to E a finite set called the set of rational points of E over F q. We denote this set by E(F q). We will explain these terms in Chapter 2. Once we know that E(F q) actually is a finite Abelian group, we may define the discrete logarithm problem in E(F q) as usual. However, since the us...