In joint work with Raf Cluckers, we propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new predicted uniform behavior. The exponential sums have summation sets consisting of integers modulo p^m lying p-adically close to y, and the proposed bounds are uniform in p, y, and m. We give evidence for the conjecture, by showing uniform bounds in p, y, and in some values for m. On the way, we prove new bounds for log-canonical thresholds which are closely related to the bounds predicted by the conjecture.

When a holomorphic modular form is a newform, its L-function has nice analytic properties and associates a cuspidal automorphic representation, which is a restricted product of local representations. To recover the newform from the representation, Casselman considered the fixed line of the congruence subgroups of GL(2) at the conductor level on the local representations. A vector on this line shall encode the conductor, the L-function and the \epsilon-factor of the representation. This is called the theory of newforms for GL(2). Similar theory has been established for some groups of small ranks as well as GL(n). In this talk I will introduce one for SO(2n+1).

This talk will explain some ways Iwasawa theory can be used to show that
elliptic curves have rank one when the ranks of the p-adic Selmer groups also predict this.

Shimura varieties are defined over complex numbers and generally have number fields as the field of definition. Motivated by an example constructed by Mumford, we find conditions which guarantee a curve in char. p lifts to a Shimura curve of Hodge type. The conditions are intrinsic in positive characteristics and thus they shed light on a definition of Shimura curves in positive characteristics. 

In this talk, I will start with modular curves, and discuss the moduli interpretation of Shimura curves. Then I will present such a condition in terms of isocrystals. Time permitting, I would show a deformation result on Barsotti-Tate groups, which serves as a key step in the proof. 
 

Contrary to their classical namesakes over the ring of integers, Pell equations over function rings in characteristic zero need not have infinitely many solutions. How often this occurs has been the theme of recent work of D. Masser and U. Zannier. The case of smooth curves is governed by the relative Manin-Mumford conjecture on abelian schemes. We pursue this study by considering singular curves and the associated generalized jacobians.