We first provide here a very short proof of a refinement of a theorem of Kodiyalam and Cutkosky, Herzog and Trung on the regularity of powers of ideals. This result implies a conjecture of Hà and generalizes a result of Eisenbud and Harris concerning the case of ideals primary for the graded maximal ideal in a standard graded algebra over a field. It also implies a new result on the regularities of powers of ideal sheaves. We then compare the cohomology of the stalks and the cohomology of the fibers of a projective morphism to the effect of comparing the maximums over fibers and over stalks of the Castelnuovo–Mumford regularities of a family of projective schemes.

Let $X$ be a curve over ${\mathbb{F}}_q$ with function field $F$. In this paper, we define a graph for each Hecke operator with fixed ramification. A priori, these graphs can be seen as a convenient language to organize formulas for the action of Hecke operators on automorphic forms. However, they will prove to be a powerful tool for explicit calculations and proofs of finite dimensionality results.

We develop a structure theory for certain graphs ${\mathcal{G}}_x$ of unramified Hecke operators, which is of a similar vein to Serre’s theory of quotients of Bruhat–Tits trees. To be precise, ${\mathcal{G}}_x$ is locally a quotient of a Bruhat–Tits tree and has finitely many components. An interpretation of ${\mathcal{G}}_x$ in terms of rank $2$ bundles on $X$ and methods from reduction theory show that ${\mathcal{G}}_x$ is the union of finitely many cusps, which are infinite subgraphs of a simple nature, and a nucleus, which is a finite subgraph that depends heavily on the arithmetic of $F$.

We describe how one recovers unramified automorphic forms as functions on the graphs ${\mathcal{G}}_x$. In the exemplary cases of the cuspidal and the toroidal condition, we show how a linear condition on functions on ${\mathcal{G}}_x$ leads to a finite dimensionality result. In particular, we reobtain the finite-dimensionality of the space of unramified cusp forms and the space of unramified toroidal automorphic forms.

In an appendix, we calculate a variety of examples of graphs over rational function fields.

Let $B$ be a Noetherian normal local ring and $G\subset Aut\left(B\right)$ be a cyclic group of local automorphisms of prime order. Let $A$ be the subring of $G$-invariants of $B$ and assume that $A$ is Noetherian. We prove that $B$ is a monogenous $A$-algebra if and only if the augmentation ideal of $B$ is principal. If in particular $B$ is regular, we prove that $A$ is regular if the augmentation ideal of $B$ is principal.

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb{Q}$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic $5$-manifolds constructed by quaternions using Eisenstein series. In the appendix, V. Emery computes these volumes using Prasad’s general formula. We use hyperbolic geometry in dimension $5$ to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

The article studies the compatibility of the refined Gross–Prasad (or Ichino–Ikeda) conjecture for unitary groups, due to Neal Harris, with Deligne’s conjecture on critical values of $L$-functions. When the automorphic representations are of motivic type, it is shown that the $L$-values that arise in the formula are critical in Deligne’s sense, and their Deligne periods can be written explicitly as products of Petersson norms of arithmetically normalized coherent cohomology classes. In some cases this can be used to verify Deligne’s conjecture for critical values of adjoint type (Asai) $L$-functions.

We study the groups $U_i$ in the unit filtration of a finite abelian extension $K$ of ${\mathbb{Q}}_p$ for an odd prime $p$. We determine explicit generators of the $U_i$ as modules over the ${\mathbb{Z}}_p$-group ring of $Gal\left(K∕{\mathbb{Q}}_p\right)$. We work in eigenspaces for powers of the Teichmüller character, first at the level of the field of norms for the extension of $K$ by $p$-power roots of unity and then at the level of $K$.

We show that a tilted algebra $A$ is tame if and only if for each generic root $d$ of $A$ and each indecomposable irreducible component $C$ of $mod\left(A,d\right)$, the field of rational invariants $k{\left(C\right)}^{GL\left(d\right)}$ is isomorphic to $k$ or $k\left(x\right)$. Next, we show that the tame tilted algebras are precisely those tilted algebras $A$ with the property that for each generic root $d$ of $A$ and each indecomposable irreducible component $C\subseteq mod\left(A,d\right)$, the moduli space $\mathcal{ℳ}{\left(C\right)}_{\theta }^{ss}$ is either a point or just ${\mathbb{P}}^1$ whenever $\theta$ is an integral weight for which $C_{\theta }^s\ne \varnothing$. We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space $\mathcal{ℳ}{\left(C\right)}_{\theta }^{ss}$ being smooth for each generic root $d$ of $A$, each indecomposable irreducible component $C\subseteq mod\left(A,d\right)$, and each integral weight $\theta$ for which $C_{\theta }^s\ne \varnothing$. As a consequence of this latter description, we show that the smoothness of the various moduli spaces of modules for a strongly simply connected algebra $A$ implies the tameness of $A$.

Along the way, we explain how moduli spaces of modules for finite-dimensional algebras behave with respect to tilting functors, and to theta-stable decompositions.

We investigate the period function of ${\sum }_{n=1}^{\infty }{\sigma }_a\left(n\right)e\left(nz\right)$, showing it can be analytically continued to $|argz|<\pi$ and studying its Taylor series. We use these results to give a simple proof of the Voronoi formula and to prove an exact formula for the second moments of the Riemann zeta function. Moreover, we introduce a family of cotangent sums, functions defined over the rationals, that generalize the Dedekind sum and share with it the property of satisfying a reciprocity formula.