We study the modularity of the genus zero open Gromov-Witten potentials and its generating matrix factorizations for elliptic orbifolds. These objects constructed by Lagrangian Floer theory are a priori well-defined only around the large volume limit. It follows from modularity that they can be analytically continued over the global Kähler moduli space.

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.

No abstract uploaded!

Let$u$_{$n$}be the$n$th term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of$u$_{$n$}which is of the form$n$exp(log$n$/104 log log$n$). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdős from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.

No abstract uploaded!