Local root numbers of elliptic curves over dyadic fields

We consider an elliptic curve over a dyadic field with additive, potentially good reduction. We study the finite Galois extension of the dyadic field generated by the three-torsion points of the elliptic curve. As an application, we give a formula to calculate the local root number of the elliptic curve over the dyadic field.Comment: 11 page

Ramification and moduli spaces of finite flat models

We determine the type of the zeta functions and the range of the dimensions of the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This gives a generalization of Raynaud's theorem on the uniqueness of finite flat models in low ramifications.Comment: 26 page

Modeling uncertainty of flexible structures with unknown high-order modal parameters - a geometric characterization of frequency responses

A control-oriented uncertainty modeling on frequency domain is presented for a class of spectral systems with unknown high-order modal parameters. At any user-specified frequency, the set of all the frequency responses of the feasible systems is characterized on a complex plane in terms of the convex bull of several circle segments, where the system is said to be feasible if partial modal parameters are given and some other conditions are satisfied by the unknown parameters. We emphasize that such a characterization enables us to quantify the least upper bounds of errors for any nominal models, and to develop further efficient results using some additional information. It is shown that, the DC gain information of the system reduces the size of the feasible set to the half or smaller for all frequencies. The efficiency of the presented scheme is demonstrated by a simple example of ideal flexible beam. </p