We first briefly review Dwork's trace formula and Wan's decomposition theorems. As an application, we consider a family of Laurent polynomials which is a generalization of the Laurent polynomials appeared in Iwaniec's work, and determine $p$-adic valuations for all the roots of the $L$-functions associated to an Zariski open dense subset of the space of Laurent polynomials. For lower dimension cases, we represent the Zariski open subset explicitly by computing an explicit Hasse polynomial.

In the first talk we discussed some models that can be attacked via the trace map as well as some model-independent result. In this talk we shall apply our model-independent results to some specific models (Jacobi operators, CMV matrices, quantum and classical Ising models) and derive answers to questions that until quite recently were open. We will also present a connection between CMV matrices and Ising models. We shall state also some open problems and propose some routes for further development.

We give a Chabauty-like method for finding p-adic approximations to
integral points on hyperelliptic curves when the Mordell-Weil rank of
the Jacobian equals the genus. The method uses an interpretation of
the component at p of the p-adic height pairing in terms of iterated
Coleman integrals.  This is joint work with Amnon Besser and Steffen
Mueller.

We study the energy dissipation features of systems comprised of two components one of which is highly lossy and the other lossless. One of the principal results is that all the eigenmodes of any such system split into two distinct classes, high-loss and low-loss,according to their dissipative properties. Interestingly, this splitting is more pronounced the higher the loss of the lossy component. In addition to that, the real frequencies of the high-loss eigenmodes can become very small and even can vanish entirely, which is the case of overdamping. An exhaustive analytical study of the energy, dissipated power, and quality factor for such composite systems is given.