GG-odometers and their almost 1-1 extensions

In this paper we recall the concepts of $G$-odometer and $G$-subodometer for $G$-actions, where $G$ is a discrete finitely generated group, which generalize the notion of odometer in the case G=\ZZ. We characterize the $G$-regularly recurrent systems as the minimal almost 1-1 extensions of subodometers, from which we deduce that the family of the $G$-Toeplitz subshifts coincides with the family of the minimal symbolic almost 1-1 extensions of subodometers.Comment: 18 page

Horizontal non-vanishing of Heegner points and toric periods

Let $F/\mathbb{Q}$ be a totally real field and $A$ a modular \GL_2-type abelian variety over $F$. Let $K/F$ be a CM quadratic extension. Let $\chi$ be a class group character over $K$ such that the Rankin-Selberg convolution $L(s,A,\chi)$ is self-dual with root number $-1$. We show that the number of class group characters $\chi$ with bounded ramification such that $L'(1, A, \chi) \neq 0$ increases with the absolute value of the discriminant of $K$. We also consider a rather general rank zero situation. Let $\pi$ be a cuspidal cohomological automorphic representation over \GL_{2}(\BA_{F}). Let $\chi$ be a Hecke character over $K$ such that the Rankin-Selberg convolution $L(s,\pi,\chi)$ is self-dual with root number $1$. We show that the number of Hecke characters $\chi$ with fixed $\infty$-type and bounded ramification such that $L(1/2, \pi, \chi) \neq 0$ increases with the absolute value of the discriminant of $K$. The Gross-Zagier formula and the Waldspurger formula relate the question to horizontal non-vanishing of Heegner points and toric periods, respectively. For both situations, the strategy is geometric relying on the Zariski density of CM points on self-products of a quaternionic Shimura variety. The recent result \cite{Ts, YZ, AGHP} on the Andr\'e-Oort conjecture is accordingly fundamental to the approach.Comment: Adv. Math., to appear. arXiv admin note: text overlap with arXiv:1712.0214

Some intertemporal and informational aspects of economic theory

I present several theoretical models in which agents participate in environments where some relevant economic data is not known perfectly and decisions are taken within an intertemporal setting. In all the models I make extensive use of game theoretic concepts because agents are strategically interdependent. In other words in each model I describe, the action of one agent is known to affect at least one other agent and vice versa. I therefore commence with a description of the theoretical tools applied throughout. In particular, I focus on the refinements of the Nash equilibrium in games of incomplete or imperfect information. The first model I describe is one where a consumer is unable to discern the quality of a good at a particular store prior to consumption. I show that when a consumer locates a good match store the store will exercise price discrimination by increasing its price in the future. The model characterizes introductory offers. A number of extensions are considered, including the idea of interrelated prices. This is the theme of the next chapter where I show how price matching refunds can act as a price discriminatory mechanism if consumers differ over costs of acquiring information about prices. In the next chapter I present a model of multidimensional signalling, where refund promises or free trial periods when combined with the selling price can signal quality for certain. In the final chapter I analyse a bargain between a country and a foreign company and focus on the design of an incentive compatible tax schedule. The aim of the models is to provide new insights about economic relationships that feature intertemporal and informational aspects

Linear independence of Gamma values in positive characteristic

We investigate the arithmetic nature of special values of Thakur's function field Gamma function at rational points. Our main result is that all linear independence relations over the field of algebraic functions are consequences of the known relations of Anderson and Thakur arising from the functional equations of the Gamma function.Comment: 51 page