We describe a method for determining the minimal length of elements in the generalized Thompson's groups F( p). We compute the length of an element by constructing a tree pair diagram for the element, classifying the nodes of the tree and summing associated weights from the pairs of node classifications. We use this method to effectively find minimal length representatives of an element.

Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over Fp in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove that every ordinary hyperelliptic curve C of genus g ≥ 2 has level 2. We provide a good number of examples and raise a conjecture.

We consider the question of when the L-polynomial of one curve divides the L-polynomial of another curve. A theorem of Tate gives an answer in terms of jacobians. We consider the question in terms of the curves. The last author gave an invited talk at the 12th International Conference on Finite Fields and Their Applications on this topic, and stated two conjectures. In this article we prove one of those conjectures.

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