1. CMB 2007 (vol 50 pp. 409)
|Discriminants of Complex Multiplication Fields of Elliptic Curves over Finite Fields |
We show that, for most of the elliptic curves $\E$ over a prime finite field $\F_p$ of $p$ elements, the discriminant $D(\E)$ of the quadratic number field containing the endomorphism ring of $\E$ over $\F_p$ is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over $\F_p$.
Categories:11G20, 11N32, 11R11
2. CMB 2003 (vol 46 pp. 149)
|The Ramification Polygon for Curves over a Finite Field |
A Newton polygon is introduced for a ramified point of a Galois covering of curves over a finite field. It is shown to be determined by the sequence of higher ramification groups of the point. It gives a blowing up of the wildly ramified part which separates the branches of the curve. There is also a connection with local reciprocity.
3. CMB 1999 (vol 42 pp. 78)
|Fermat Jacobians of Prime Degree over Finite Fields |
We study the splitting of Fermat Jacobians of prime degree $\ell$ over an algebraic closure of a finite field of characteristic $p$ not equal to $\ell$. We prove that their decomposition is determined by the residue degree of $p$ in the cyclotomic field of the $\ell$-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.