We develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque, Enriquez and Etingof (2009) and Levin and Racinet (2007). We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.

We determine the order of magnitude of $\mathbb{E}{\left|{\sum }_{n\le x}f\left(n\right)\right|}^{2q}$ up to factors of size $e^{O\left(q^2\right)}$, where $f\left(n\right)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1\le q\le clogx∕loglogx$.

In the Steinhaus case, we show that $\mathbb{E}{\left|{\sum }_{n\le x}f\left(n\right)\right|}^{2q}=e^{O\left(q^2\right)}{x}^q{\left(logx∕\left(qlog\left(2q\right)\right)\right)}^{{\left(q-1\right)}^2}$ on this whole range. In the Rademacher case, we find a transition in the behavior of the moments when $q\approx \left(1+\sqrt{5}\right)∕2$, where the size starts to be dominated by “orthogonal” rather than “unitary” behavior. We also deduce some consequences for the large deviations of ${\sum }_{n\le x}f\left(n\right)$.

The proofs use various tools, including hypercontractive inequalities, to connect $\mathbb{E}{\left|{\sum }_{n\le x}f\left(n\right)\right|}^{2q}$ with the $q$-th moment of an Euler product integral. When $q$ is large, it is then fairly easy to analyze this integral. When $q$ is close to 1 the analysis seems to require subtler arguments, including Doob’s $L^p$ maximal inequality for martingales.

We provide a complete proof of a duality theorem for the fppf cohomology of either a curve over a finite field or a ring of integers of a number field, which extends the classical Artin–Verdier Theorem in étale cohomology. We also prove some finiteness and vanishing statements.

We show that asymptotically the first Betti number $b_1$ of a Shimura curve satisfies the Gauss–Bonnet equality $2\pi \left(b_1-2\right)=vol$ where $vol$ is hyperbolic volume; equivalently $2g-2=\left(1+o\left(1\right)\right)vol$ where $g$ is the arithmetic genus. We also show that the first Betti number of a congruence hyperbolic 3-orbifold asymptotically vanishes relatively to hyperbolic volume, that is $b_1∕vol\to 0$. This generalizes previous results obtained by Frączyk, on which we rely, and uses the same main tool, namely Benjamini–Schramm convergence.

Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum ${\sum }_{\left|h\right|<n\le x}f\left(n\right)\tau \left(n-h\right)$, where $\tau$ denotes the divisor function and $h\in \mathbb{Z}\setminus \left\{0\right\}$. We consider in particular the special cases where $f$ is the generalized divisor function ${\tau }_z$ with $z\in ℂ$, and the characteristic function of sums of two squares (or more generally, ideal norms of abelian extensions). As another application, we deduce a full asymptotic expansion in the generalized Titchmarsh divisor problem ${\sum }_{\left|h\right|<n\le x,\omega \left(n\right)=k}\tau \left(n-h\right)$, where $\omega \left(n\right)$ counts the number of distinct prime divisors of $n$, thus extending a result of Fouvry and Bombieri, Friedlander and Iwaniec.

We present two different proofs: The first relies on an effective combinatorial formula of Heath-Brown’s type for the divisor function ${\tau }_{\alpha }$ with $\alpha \in \mathbb{Q}$, and an interpolation argument in the $z$-variable for weighted mean values of ${\tau }_z$. The second is based on an identity of Linnik type for ${\tau }_z$ and the well-factorability of friable numbers.

For any integer $m\ge 2$ and any dimension $n\ge 1$, we show that any $n$-dimensional Hodge diamond with values in $\mathbb{Z}∕m\mathbb{Z}$ is attained by the Hodge numbers of an $n$-dimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of $n$-dimensional smooth complex projective varieties besides the ones induced by the Hodge symmetries, which answers a question raised by Kollár in 2012.