For $a∕q\in \mathbb{Q}$ the Estermann function is defined as $D\left(s,a∕q\right):={\sum }_{n\ge 1}d\left(n\right)n^{-s}e\left(n\frac{a}{q}\right)$ if $\Re \left(s\right)>1$ and by meromorphic continuation otherwise. For $q$ prime, we compute the moments of $D\left(s,a∕q\right)$ at the central point $s=1∕2$, when averaging over $1\le a<q$.

As a consequence we deduce the asymptotic for the iterated moment of Dirichlet $L$-functions ${\sum }_{{\chi }_1,\dots ,{\chi }_k\left(modq\right)}{\left|L\left(\frac{1}{2},{\chi }_1\right)\right|}^2\cdots {\left|L\left(\frac{1}{2},{\chi }_k\right)\right|}^2{\left|L\left(\frac{1}{2},{\chi }_1\cdots {\chi }_k\right)\right|}^2$, obtaining a power saving error term.

Also, we compute the moments of certain functions defined in terms of continued fractions. For example, writing $f_{\pm }\left(a∕q\right):={\sum }_{j=0}^r{\left(\pm 1\right)}^j{b}_j$ where $\left[0;b_0,\dots ,b_r\right]$ is the continued fraction expansion of $a∕q$ we prove that for $k\ge 2$ and $q$ primes one has ${\sum }_{a=1}^{q-1}{f}_{\pm }{\left(a∕q\right)}^k\sim 2\left(\zeta {\left(k\right)}^2∕\zeta \left(2k\right)\right)q^k$ as $q\to \infty$.

Soit $K$ un corps de nombres totalement réel. Pour tout nombre premier $p\ge 5$, notons $F_p$ la courbe de Fermat d’équation $x^p+y^p+z^p=0$. Sous l’hypothèse que $2$ est totalement ramifié dans $K$, on établit quelques résultats sur l’ensemble $F_p\left(K\right)$ des points de $F_p$ rationnels sur $K$. On obtient un critère pour que le théorème de Fermat asymptotique soit vrai sur $K$, critère relatif à l’ensemble des newforms modulaires paraboliques de Hilbert sur $K$, de poids parallèle $2$ et de niveau l’idéal premier au-dessus de $2$. Il peut souvent se tester simplement numériquement, notamment quand le nombre de classes restreint de $K$ vaut $1$. Par ailleurs, en utilisant la méthode modulaire, on démontre le théorème de Fermat de façon effective, sur certains corps de nombres dont les degrés sur $\mathbb{Q}$ sont $3,4,5,6$ et $8$.

Let $K$ be a totally real number field. For all prime number $p\ge 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$, we establish some results about the set $F_p\left(K\right)$ of points of $F_p$ rational over $K$. We obtain a criterion so that the asymptotic Fermat’s last theorem is true over $K$, criterion related to the set of Hilbert modular cuspidal newforms over $K$, of parallel weight $2$ and of level the prime ideal above $2$. It is often simply testable numerically, particularly if the narrow class number of $K$ is $1$. Furthermore, using the modular method, we prove Fermat’s last theorem effectively, over some number fields whose degrees over $\mathbb{Q}$ are $3,4,5,6$ and $8$.

We study $G$-valued Galois deformation rings with prescribed properties, where $G$ is an arbitrary (not necessarily connected) reductive group over an extension of ${\mathbb{Z}}_l$ for some prime $l$. In particular, for the Galois groups of $p$-adic local fields (with $p$ possibly equal to $l$) we prove that these rings are generically regular, compute their dimensions, and show that functorial operations on Galois representations give rise to well-defined maps between the sets of irreducible components of the corresponding deformation rings. We use these local results to prove lower bounds on the dimension of global deformation rings with prescribed local properties. Applying our results to unitary groups, we improve results in the literature on the existence of lifts of mod $l$ Galois representations, and on the weight part of Serre’s conjecture.

We prove functorial weak factorization of projective birational morphisms of regular quasiexcellent schemes in characteristic 0 broadly based on the existing line of proof for varieties. From this general functorial statement we deduce factorization results for algebraic stacks, formal schemes, complex analytic germs, Berkovich analytic and rigid analytic spaces, answering a present need in nonarchimedean geometry. Techniques developed for this purpose include a method for functorial factorization of toric maps, variation of GIT quotients relative to general noetherian qe schemes, and a GAGA theorem for Stein compacts.

We study pushforwards of log pluricanonical bundles on projective log canonical pairs $\left(Y,\mathrm{\Delta }\right)$ over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when $Y$ surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when $Y$ is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective nonvanishing statements. We also prove an effective weak positivity statement for log pluricanonical bundles in this setting, which may be of independent interest. In each context we indicate over which loci positivity holds. Finally, using the description of such loci, we show an effective vanishing theorem for pushforwards of certain log-sheaves under smooth morphisms.

Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph $G$:

• the Lovász-Saks-Schrijver ideal defining the $d$-dimensional orthogonal representations of the graph complementary to $G$, and

• the determinantal ideal of the $\left(d+1\right)$-minors of a generic symmetric matrix with $0$ in positions prescribed by the graph $G$.

In characteristic $0$ these two ideals turn out to be closely related and algebraic properties such as being radical, prime or a complete intersection transfer from the Lovász–Saks–Schrijver ideal to the determinantal ideal. For Lovász–Saks–Schrijver ideals we link these properties to combinatorial properties of $G$ and show that they always hold for $d$ large enough. For specific classes of graphs, such a forests, we can give a complete picture and classify the radical, prime and complete intersection Lovász–Saks–Schrijver ideals.

We study the connection between the singularities of a finite type $\mathbb{Z}$-scheme $X$ and the asymptotic point count of $X$ over various finite rings. In particular, if the generic fiber $X_{\mathbb{Q}}=X{\times }_{Spec\mathbb{Z}}Spec\mathbb{Q}$ is a local complete intersection, we show that the boundedness of $\left|X\left(\mathbb{Z}∕p^n\mathbb{Z}\right)\right|∕p^{ndim{X}_{\mathbb{Q}}}$ in $p$ and $n$ is in fact equivalent to the condition that $X_{\mathbb{Q}}$ is reduced and has rational singularities. This paper completes a recent result of Aizenbud and Avni.

We give a structural proof of the fact that the realization of the degree-zero part of the polylogarithm on abelian schemes in analytic Deligne cohomology can be described in terms of the Bismut–Köhler higher analytic torsion form of the Poincaré bundle. Furthermore, we provide a new axiomatic characterization of the arithmetic Chern character of the Poincaré bundle using only invariance properties under isogenies. For this we obtain a decomposition result for the arithmetic Chow group of independent interest.

Let $k$ be a base field, $K$ be a field containing $k$, and $L∕K$ be a field extension of degree $n$. The essential dimension $ed\left(L∕K\right)$ over $k$ is a numerical invariant measuring “the complexity” of $L∕K$. Of particular interest is

$$\tau \left(n\right)=max\left\{ed\left(L∕K\right)\mid L∕K\text{ is a separable extension of degree }n\right\},$$

also known as the essential dimension of the symmetric group $S_n$. The exact value of $\tau \left(n\right)$ is known only for $n\le 7$. In this paper we assume that $k$ is a field of characteristic $p>0$ and study the essential dimension of inseparable extensions $L∕K$. Here the degree $n=\left[L:K\right]$ is replaced by a pair $\left(n,e\right)$ which accounts for the size of the separable and the purely inseparable parts of $L∕K$, respectively, and $\tau \left(n\right)$ is replaced by

$$\tau \left(n,e\right)=max\left\{ed\left(L∕K\right)\mid L∕K\text{ is a field extension of type }\left(n,e\right)\right\}.$$

The symmetric group $S_n$ is replaced by a certain group scheme $G_{n,e}$ over $k$. This group scheme is neither finite nor smooth; nevertheless, computing its essential dimension turns out to be easier than computing the essential dimension of $S_n$. Our main result is a simple formula for $\tau \left(n,e\right)$.