We construct maps from moduli spaces of vector bundles on a Riemann surface X to opers on X, using nonabelian theta functions. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle...

We study systems of equations, F (x) = 0, given by piecewise differentiable functions F : R ! R, k " n. The focus is on the representability of the solution set locally as an (n # k)-dimensional Lipschitz manifold. For that, nonsmooth versions of inverse function theorems are applied. It turns out that their applicability depends on the choice of a particular basis. To overcome this obstacle we...

The technique of Weil restriction of scalars has signiicant implications for elliptic curve cryptography. In this paper we apply these ideas to the case of the discrete logarithm problem in the Jacobian of a curve of genus greater than one over a nite eld F q n where n > 1.

In his previous papers [25, 26, 28] the author proved that in characteristic 6= 2 the jacobian J(C) of a hyperelliptic curve C : y = f(x) has only trivial endomorphisms over an algebraic closure Ka of the ground field K if the Galois group Gal(f) of the irreducible polynomial f(x) ∈ K[x] is either the symmetric group Sn or the alternating group An. Here n ≥ 9 is the degree of f . The goal of th...

The Jacobian group (also known as the critical or sandpile group) is an important invariant of a finite, connected graph $X$; it finite abelian whose cardinality equal to number spanning trees $X$ (Kirchhoff's Matrix Tree Theorem). A specific type covering graph, called derived that constructed from voltage with $G$ object interest in this paper. Towers graphs are studied by using aspects class...