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

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

The remaining cases of the Kramer-Tunnell conjecture

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some when $K$ is of characteristic $2$, and we complete its proof by reducing the positive characteristic case to characteristic $0$. For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_p$--we find an elliptic curve $E'$ defined over a $p$-adic field such that all the terms in the Kramer-Tunnell formula for $E'$ are equal to those for $E$.Comment: 13 pages; final version, to appear in Compositio Mathematic

Cohomology of rigid curves with semi-stable coverings

We construct a semi-stable formal model of a wide open rigid curve with a semi-stable covering, and study the l-adic cohomology of the rigid curve. We describe the l-adic cohomology of the rigid curve using the l-adic cohomology of the irreducible components of a semi-stable reduction, and homology and cohomology of some graphs. We also prove the functoriality of the description for a finite flat morphism that is compatible with semi-stable coverings of wide open rigid curves.Comment: 13 page