We prove several multiplicity one theorems for generalized modular functions (GMF), in terms of their $q$-exponents, and make a precise statement about the nature of values that the prime $q$-exponents of a GMF can take. We shall also study the integrality of general $q$-exponents of a GMF and give an upper bound on the first sign change of these $q$-exponents.

We show that for every Banach space $X, \dim X>1$, every maximal commutative subalgebra of $L(X)$ has uncountably many copies between maximal commutative subalgebras of $L(X)$. Answering to a question of Aleksander Pe{\l}czy\'nski, we show also that for an arbitrary infinite dimensional Banach space $X$ there are at least countably many multiplications making of $X$ a commutative unital Banach algebra.

Let $\ell>2$ be a positive integer, $\zeta_\ell$ a primitive $\ell$-th root of unity, and $K$ a number field containing $\zeta_\ell+\zeta_\ell\inv$ but not $\zeta_\ell$. In a recent paper, Chonoles et. al. study iterated towers of number fields over $K$ generated by the generalized Rikuna polynomial, $r_n(x,t;\ell) \in K(t)[x]$. They note that when $K = \mathbb{Q}$, $t \in \{0,1\}$, and $\ell=3$, the only ramified prime in the resulting tower is 3, and they ask under what conditions is the number of ramified primes small. In this paper, we apply a theorem of Guàrdia, Montes, and Nart to derive a formula for the discriminant of $\mathbb{Q}(\theta)$ where $\theta$ is a root of $r_n(x,t;3)$, answering the question of Chonoles et. al. in the case $K = \mathbb{Q}$, $\ell=3$, and $t \in \mathbb{Z}$. In the latter half of the paper, we identify some cases where the dynamics of $r_n(x,t;\ell)$ over finite fields yields an explicit description of the decomposition of primes in these iterated extensions.

Let $G$ be a semisimple almost simple group defined over a global function field $K$, not anisotropic of type $A_n$. We express the (relative) Tamagawa number of $G$ in terms of local data, including the number $t_\infty(G)$ of types in one orbit of a special vertex in the Bruhat--Tits building of $G_\infty(\hat{K}_\infty)$ for some place $\infty$ and the class number $h_\infty(G)$ of $G$ at $\infty$.

We estimate the sum of products or quotients of $L$-functions, where the sum is taken over all quadratic extensions of given genus over a fixed global function field. Our estimate for the sum of the quotient of two $L$-functions is analogous to a result of Schmidt where he estimates the sum of the quotient of two $L$-series, where the sum is over quadratic extensions of $\mathbb{Q}$ with absolute value of the discriminant less than a given bound.

Expressions for a family of integrals involving the Hurwitz zeta function are established using standard properties of the Fourier transform.

Let $X$ be a Noetherian space, let $\Phi:X\longrightarrow X$ be a continuous function, let $Y\subseteq X$ be a closed set, and let $x\in X$. We show that the set $S:=\{n\in\mathbb{N}\colon \Phi^n(x)\in Y\}$ is a union of at most finitely many arithmetic progressions along with a set of Banach density zero. In particular, we obtain that given any quasi-projective variety $X$, any rational map $\Phi:X\longrightarrow X$, any subvariety $Y\subseteq X$, and any point $x\in X$ whose orbit under $\Phi$ is in the domain of definition for $\Phi$, the set $S$ is a finite union of arithmetic progressions together with a set of Banach density zero. This answers a question posed by Laurent Denis [7]. We prove a similar result for the backward orbit of a point and provide some quantitative bounds.