Effective $p$-descent

Abstract

Given $E$, an elliptic curve defined over $K$, a field of positive characteristic, provided that $j$, the Weierstrass $j$-invariant, is not an element of $K^p$, we construct explicitly, that is, we give by a closed form formula, a non-trivial homomorphism, $\mu:E(K) \rightarrow K^+$, from the group of $K$-rational points of $E$ to $K^+$, the additive group of $K$. In the course of our analysis we discover a canonical differential, $\omega_q \in \Omega; K|{\bb F}_p$, associated to $E$ and we relate it to the differential $dq/q$ associated to the Tate curve. If the transcendence degree of $K$ over ${\bb F}_p$ is equal to one, as for example is the case for function fields in one variable, then $\mu$ is a $p$-descent map, that is, its kernel is equal to $pE(K)$ and the explicit formula for $\mu$ can be used to provide effective proofs of analogues of classical theorems on elliptic curves. For example, in the author's thesis at The University of Texas at Austin the analogue of Siegel's Theorem on the finiteness of integral points of $E(K)$ is proved effectively.