No abstract uploaded!

We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role of upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms replacing the classical $\Delta$. This leads to analytic means of deriving a Colmez-type formula for "stable Taguchi height" of CM Drinfeld modules having arbitrary rank. A Lerch-Type formula for "totally real" function fields is also obtained, with the Heegner cycle on the Bruhat-Tits buildings intervene. Also our limit formula is naturally applied to the special values of both the Rankin-Selberg L-functions and the Godement-Jacquet L-functions associated to automorphic cuspidal representations over global function fields.

This project was sponsored through the Schiff Fellowship program of Brandeis University. This project involved using the power series method to construct a third order nonlinear ordinary differential equation, a Schwarzian equation, for each of the "genus zero" modular functions, described in the Conway-Norton paper. We first use the Borcherd recursion formuli to generate, in each case, a modular function up to whatever degree we desire, and then use the fact that there is a Schwarzian equation, determined by a single rational function we call a Q-value. By similar power series methods, we compute the coefficients of our rational function, and hence have all the necessary data to create a Schwarzian differential equation for each modular function. This equation can, in turn, be used to recover the modular function itself.

Abstract We present a relative trace formula approach to the Gross–Zagier formula and its generalization to higher-dimensional unitary Shimura vari- eties. As a crucial ingredient, we formulate a conjectural arithmetic funda- mental lemma for unitary Rapoport–Zink spaces. We prove the conjecture when the Rapoport–Zink space is associated to a unitary group in two or three variables.

We provide a family of counterexamples to a first formulation of the dynamical Manin–Mumford conjecture. We propose a revision of this conjecture and prove it for arbitrary subvarieties of Abelian varieties under the action of group endomorphisms and for lines under the action of diagonal endomorphisms of $\mathbb{P}^1 \times \mathbb{ P}^1$.