Taylor coefficients of Anderson generating functions and Drinfeld torsion extensions

We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating functions and their hyperderivatives at roots of unity. We also give a direct proof that the image of the Galois representation attached to the [Formula: see text]-adic Tate module lies in the [Formula: see text]-adic points of the motivic Galois group. This is a generalization of the corresponding result of Chang and Papanikolas for the [Formula: see text]-adic case.

FRACTAL SETS IN THE FIELD OF p-ADIC ANALOGUE OF THE COMPLEX NUMBERS

The [Formula: see text]-adic number field [Formula: see text] and the [Formula: see text]-adic analogue of the complex number field [Formula: see text] have a rich algebraic and geometric structure that in some ways rivals that of the corresponding objects for the real or complex fields. In this paper, we attempt to find and understand geometric structures of general sets in a [Formula: see text]-adic setting. Several kinds of fractal measures and dimensions of sets in [Formula: see text] are studied. Some typical fractal sets are constructed. It is worthwhile to note that there exist some essential differences between [Formula: see text]-adic case and classical case.

Towards the Finite Slope Part for GLn

Let $L$ be a finite extension of ${\mathbb{Q}}_p$ and $n\geq 2$. We associate to a crystabelline $n$-dimensional representation of ${\operatorname{Gal}}(\overline L/L)$ satisfying mild genericity assumptions a finite length locally ${\mathbb{Q}}_p$-analytic representation of ${\operatorname{GL}}_n(L)$. In the crystalline case and in a global context, using the recent results on the locally analytic socle from [6], we prove that this representation indeed occurs in spaces of $p$-adic automorphic forms. We then use this latter result in the ordinary case to show that certain “ordinary” $p$-adic Banach space representations constructed in our previous work appear in spaces of $p$-adic automorphic forms. This gives strong new evidence to our previous conjecture in the $p$-adic case.

Analytic Continuation of Equivariant Distributions

We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun.

On the smooth transfer for Guo–Jacquet relative trace formulae

We establish the existence of smooth transfer for Guo–Jacquet relative trace formulae in the $p$-adic case. This kind of smooth transfer is a key step towards a generalization of Waldspurger’s result on central values of L-functions of $\text{GL}_{2}$.