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.

We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland's integrable $N$-body Schrödinger operators and their generalizations. The first is an explicit computation of the Etingof-Kirillov traces of intertwining operators, the second an integral representation of hypergeometric type, and the third is a formula of Bethe ansatz type. The last two formulas are degenerations of elliptic formulas obtained previously in connection with the Knizhnik-Zamolodchikov-Bernard equation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctions are parametrized by a ‘Hermite-Bethe’ variety, a generalization of the spectral variety of the Lamé operator. We also give the $q$-deformed version of our first formula. In the scalar ${\rm sl}_N$ case, this gives common eigenfunctions of the commuting Macdonald-Rujsenaars difference operators.

We prove that an indecomposable principally polarized complex abelian variety $X$ is the Jacobian of a smooth curve if and only if there exist points $a, b, c$of $X$ whose images under the Kummer map $X \rightarrow |2\Theta|^{\ast}$ are distinct and collinear, and such that the subgroup of X generated by $a - b$ and $b - c$ is dense in $X$.

In this work, we study the diophantine equation\renewcommand{\theequation}{1.1}\begin{equation}f(X) = Y^m,\end{equation} where $m \geqslant 2$ is an integer and $f(X)$ is a polynomial with coefficients in a number field ${\bf K}$. The first important result on this topic is due to Siegel [19], who showed that if $m = 2$ and $f$ has at least three simple roots or if $m \geqslant 3$ and $f$ has at least two simple roots, then (1.1) has only finitely many integral solutions. Three years later, he proved [20] that if the algebraic curve defined by (1.1) is of positive genus, then (1.1) has only finitely many integral solutions. The $p$-adic analogue of this theorem was established independently by Lang [9] and LeVeque [12], who showed that, under the same conditions, (1.1) has only finitely many $S$-integral solutions. After that, LeVeque [13] gave a necessary and sufficient condition for the algebraic curve defined by (1.1) to have positive genus. However, all these results are based on Thue's method, and hence are ineffective.

A topological sphere theorem is obtained from the point of view of submanifold geometry. An important scalar is defined by the mean curvature and the squared norm of the second fundamental form of an oriented complete submanifold $M^n$ in a space form of nonnegative sectional curvature. If the infimum of this scalar is negative, we then prove that the Ricci curvature of $M^n$ has a positive lower bound. Making use of the Lawson–Simons formula for the nonexistence of stable $k$-currents, we eliminate $H_k (M^n, {\bb Z})$ for all $1 < k < n - 1$. We then observe that the fundamental group of $M^n$ is trivial. It should be emphasized that our result is optimal.