A Little Course On Correspondences

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Sections

In this little course we want to show how Deuring's correspondences between algebraic function fields over finite fields works using Magma V2.14-14. We don't want to give an introduction to the theory of abelian varieties, complex multiplication, algebras or algebraic geometry. We expect that the reader has some solid knowledge about this topics. We start with function fields of genus one and give several examples. In the elliptic case those, who are famililar with isogenies of elliptic curves, will recognize, that correspondences of isomorphic function fields are actually the same as isogenies, just written in another terminology. After we got some first experience with correspondences in the genus one case, we will step to genus two and afterwards to genus three, where non-hyperelliptic curves can occur. At the end of this course we should able to compute the endomorphismsring of arbitary curves over a finite field with irreducible Jacobian, also if the endomorphismalgebra is not commutative.

The aim of this course is to introduce in the representation of endomorphisms by correspondences and elementary arithmetic for correspondeces, especially the multiplication of them. We will see, that correspondence-classes correlate with the elements of the endomorphismring of the jacobian of our given curve. Then we want to show, how we can compute some non-trivial correspondences. To do that, we use some techniques the author used in his thesis. For the sake of simplicity we assume, that the Jacobian J_X of the non-singular projective, irreducible and reduced curve X/K, where K is a finite, is K-isogen to a product J_X = A^r, where A is a simple abelian variety over K and r is a positive rational integer.

At the end of the course we will see, that actually it is quite easy to compute correspondences other then those, which are known by the frobenius and his Rosati. We denote our function field with F/K, where K is a finite field. The correspondences are represented as ideals of the finite maximal order of a suitable transcendent constant field extension L of F. We define correspondence classes for divisors of L. Two divisors A and B of L are in the same correspondence class, if A- B = C + (F), where C is a constant divisor of L and (F) a suitable principal divisor. A prime divisor of L is constant, if his restriction to F is not trivial and a divisor is called constant, if its support contains only constant prime divisors. For our purpose we just have to understand, that in each correspondence class [A]_C we can choose an unique representant C(A) with useful properties, where A is a divisor of L. This representant C(A) will we recovered by interpolation of the coefficients of the norm of C(A) by suitable places of F/K of degree one.