Many interesting questions can be asked about the decomposition of Jacobians of curves. For instance, we may want to know which curves have completely decomposable Jacobians (Jacobians which are the product of g elliptic curves) [4]. We may ask about number theoretic properties of the elliptic curves that show up in the decomposition of Jacobians of curves [2]. We would also like to know how ma...

In 2000, Galbraith and McKee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points. We show how their heuristics can be generalized to Jacobians of curves of higher genus. We then elaborate this in genus g = 2 and study various related issues, such as the probability of cyclicity...

A notion of additive dilogarithm for a field k is introduced, based on the K-theory and higher Chow groups of the affine line relative to 2(0). Analogues of the K2-regulator, the polylogarithm Lie algebra, and the `-adic realization of the dilogarithm motive are discussed. The higher Chow groups of 0-cycles in this theory are identified with the Kähler differential forms Ωk. It is hoped that th...

We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of dimension 2.

We describe an algorithm to compute the cardinality of Jacobians of ordinary hyperelliptic curves of small genus over finite fields F2n with cost O(n ). This algorithm is derived from ideas due to Mestre. More precisely, we state the mathematical background behind Mestre’s algorithm and develop from it a variant with quasiquadratic time complexity. Among others, we present an algorithm to find ...

In this paper we study genus 2 curves whose Jacobians are (4, 4)-isogenous to a product of elliptic curves. Such Jacobians are called (4, 4)-split. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We give a generic model such that any genus 2 curve with optimally (4, 4)-split Jacobian can be obtained as a specialization. We als...

In this paper we study the dynamics and kinematics of manipulators that have fewer actuators than degrees of freedom. These under{actuated manipulators arise in a number of important applications such as free{ ying space robots, hyper{redundant manipulators, manipulators with structural exibility, etc. In our analysis we decompose such under{actuated manipulators into component active and passi...

In this work are studied the Jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.