We prove a conjecture of Colliot-Thélène that implies the Ax–Kochen Theorem on $p$-adic forms. We obtain it as an easy consequence of a diophantine excision theorem whose proof forms the body of the present paper.

We prove that the best possible lower bound for the Artin conductor is exponential in the degree.

We introduce the Frobenius–Perron dimension of an endofunctor of a $\mathbb{k}$-linear category and provide some applications.

We prove that given a flat generically smooth morphism between smooth projective varieties with $F$- pure closed fibers, if the source space is Fano, weak Fano or a variety with nef anticanonical divisor, respectively, then so is the target space. We also show that, in arbitrary characteristic, a generically smooth surjective morphism between smooth projective varieties cannot have nef and big relative anticanonical divisor, if the target space has positive dimension.

We study systems of parameters over finite fields from a probabilistic perspective and use this to give the first effective Noether normalization result over a finite field. Our central technique is an adaptation of Poonen’s closed point sieve, where we sieve over higher dimensional subvarieties, and we express the desired probabilities via a zeta function-like power series that enumerates higher dimensional varieties instead of closed points. This also yields a new proof of a recent result of Gabber, Liu and Lorenzini (2015) and Chinburg, Moret-Bailly, Pappas and Taylor (2017) on Noether normalizations of projective families over the integers.

We study the asymptotic behaviour of higher order correlations

$${\mathbb{E}}_{n\le X∕d}{g}_1\left(n+a{h}_1\right)\cdots g_k\left(n+a{h}_k\right)$$

as a function of the parameters $a$ and $d$, where $g_1,\dots ,g_k$ are bounded multiplicative functions, $h_1,\dots ,h_k$ are integer shifts, and $X$ is large. Our main structural result asserts, roughly speaking, that such correlations asymptotically vanish for almost all $X$ if $g_1\cdots g_k$ does not (weakly) pretend to be a twisted Dirichlet character $n\mapsto \chi \left(n\right)n^{it}$, and behave asymptotically like a multiple of $d^{-it}\chi \left(a\right)$ otherwise. This extends our earlier work on the structure of logarithmically averaged correlations, in which the $d$ parameter is averaged out and one can set $t=0$. Among other things, the result enables us to establish special cases of the Chowla and Elliott conjectures for (unweighted) averages at almost all scales; for instance, we establish the $k$-point Chowla conjecture ${\mathbb{E}}_{n\le X}\lambda \left(n+h_1\right)\cdots \lambda \left(n+h_k\right)=o\left(1\right)$ for $k$ odd or equal to $2$ for all scales $X$ outside of a set of zero logarithmic density.

Let VI be the category of finite dimensional ${\mathbb{F}}_q$-vector spaces whose morphisms are injective linear maps and let $k$ be a noetherian ring. We study the category of functors from VI to $k$-modules in the case when $q$ is invertible in $k$. Our results include a structure theorem, finiteness of regularity, and a description of the Hilbert series. These results are crucial in the classification of smooth irreducible ${GL}_{\infty }\left({\mathbb{F}}_q\right)$-representations in nondescribing characteristic which is contained in Part II of this paper (VI-modules in nondescribing characteristic, part II, arxiv:1810.04592).

The degree of irrationality of a projective variety $X$ is defined to be the smallest degree of a rational dominant map to a projective space of the same dimension. For abelian surfaces, Yoshihara computed this invariant in specific cases, while Stapleton gave a sublinear upper bound for very general polarized abelian surfaces $\left(A,L\right)$ of degree $d$. Somewhat surprisingly, we show that the degree of irrationality of a very general polarized abelian surface is uniformly bounded above by 4, independently of the degree of the polarization. This result disproves part of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery.

In this paper we show there exists an infinite family of number fields $L$, Galois over $\mathbb{Q}$, for which the smallest prime $p$ of $\mathbb{Q}$ which splits completely in $L$ has size at least ${\left(log\left(\left|D_L\right|\right)\right)}^{2+o\left(1\right)}$. This gives a converse to various upper bounds, which shows that they are best possible.

We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli stacks of polarized varieties of this sort are Brody hyperbolic, answering a special case of a question of Viehweg and Zuo. For two-dimensional bases, we show analogous results in the more general case of families of varieties admitting a good minimal model.