Identities for Anderson generating functions for Drinfeld modules

Anderson generating functions are generating series for division values of points on Drinfeld modules, and they serve as important tools for capturing periods, quasi-periods, and logarithms. They have been fundamental in recent work on special values of positive characteristic L-series and in transcendence and algebraic independence problems. In the present paper we investigate techniques for expressing Anderson generating functions in terms of the defining polynomial of the Drinfeld module and determine new formulas for periods and quasi-periods.Comment: 18 page

On the Atkin Polynomials

We identify the Atkin polynomials in terms of associated Jacobi polynomials. Our identificationthen takes advantage of the theory of orthogonal polynomials and their asymptotics to establish many new properties of the Atkin polynomials. This shows that co-recursive polynomials may lead to interesting sets of orthogonal polynomials.Comment: 18 pages. Accepted for publication at the Pacific Journal of Mathematic

On pseudo-real finite subgroups of PGL⁡3(C)\operatorname{PGL}_3(\mathbb{C})

Let $G$ be a finite subgroup of $\operatorname{PGL}_3(\mathbb{C})$, and let $\sigma$ be the generator of $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$. We say that $G$ has a \emph{real field of moduli} if $^{\sigma}G$ and $G$ are $\operatorname{PGL}_3(\mathbb{C})$-conjugates, that is, if $\exists\,\phi\in\operatorname{PGL}_3(\mathbb{C})$ such that \phi^{-1}\,G\,\phi=\,^{\sigma}G. Furthermore, we say that $\mathbb{R}$ is \emph{a field of definition for $G$} or that \emph{$G$ is definable over $\mathbb{R}$} if $G$ is $\operatorname{PGL}_3(\mathbb{C})$-conjugate to some $G'\subset\operatorname{PGL}_3(\mathbb{R})$. In this situation, we call $G'$ \emph{a model for $G$ over $\mathbb{R}$}. If $G$ has $\mathbb{R}$ as a field of definition but is not definable over $\mathbb{R}$, then we call $G$ \emph{pseudo-real}. In this paper, we first show that any finite cyclic subgroup $G=\mathbb{Z}/n\mathbb{Z}$ in $\operatorname{PGL}_3(\mathbb{C})$ has {a real field of moduli} and we provide a necessary and sufficient condition for $G=\mathbb{Z}/n\mathbb{Z}$ to be definable over $\mathbb{R}$; see Theorems 2.1, 2.2, and 2.3. We also prove that any dihedral group $\operatorname{D}_{2n}$ with $n\geq3$ in $\operatorname{PGL}_3(\mathbb{C})$ is definable over $\mathbb{R}$; see Theorem 2.4. Furthermore, we study all six classes of finite primitive subgroups of $\operatorname{PGL}_3(\mathbb{C})$, and show that all of them except the icosahedral group $\operatorname{A}_5$ are pseudo-real; see Theorem 2.5, whereas $\operatorname{A}_5$ is definable over $\mathbb{R}$. Finally, we explore the connection of these notions in group theory with their analogues in arithmetic geometry; see Theorem 2.6 and Example 2.7.Comment: 11 page