We show that the $L^1$ norm of an exponential sum of length X, and with coefficients equal to the Liouville or Möbius function, is at least $\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$. For the Liouville function, this improves on the lower bound $\gg X^{1/\log\log X}$ due to Balog and Perelli. For the Möbius function, this improves the lower bound $\gg X^{1/6}$ due to Balog and Ruzsa. The large discrepancy between these lower bounds is due to the method employed by Balog and Ruzsa, as it crucially relies on the vanishing of $\mu(n)$. Instead, our proof puts the two cases on an equal footing by exploiting the connection of these coefficients with zeros of Dirichlet L-functions. In the second paper in this series, we will obtain a lower bound $\gg X^{\delta}$ for some small $\delta$ but for general (non-pretentious) multiplicative functions.

A Laurent polynomial in two variables is tempered if its edge polynomials are cyclotomic. Variation of coefficients leads to a family of smooth complete genus-g curves carrying a canonical algebraic $K_2$-class over a g-dimensional base S, hence to an extension of admissible variations of mixed Hodge structure (or normal function) on S. We prove that the $\mathbb{R}$-split locus of this extension is finite. Consequently, the torsion locus of the normal function and the A-factor locus for the family of curves are also finite.

We describe the $\mathbb{F}_p$-cohomology of the congruence subgroups $\mathrm{SL}_n(\mathbb{Z}, p^m)$ in degrees $* \lt p-1$, for all large enough n, establishing a formula proposed by F. Calegari. Along the way, we also establish a formula for the stable cohomology of $\mathrm{SL}_n(\mathbb{Z}/p)$ with certain twisted coefficients.

We establish dimension formulae for the Witt-vector affine Springer fibers associated to a reductive group over a mixed characteristic local field, under the assumption that the group is essentially tamely ramified and the residue characteristic is not bad. In addition to the discriminant valuations that show up in classical works on the usual affine Springer fibers, our formulae also involve the Artin conductors and the Kottwitz invariants of the relevant conjugacy classes.

In this follow-up paper we show that smooth Hodge-proper stacks over $\mathcal O_K$ are $\mathbb Q_p$-locally acyclic: namely the natural map between étale $\mathbb Q_p$-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the $\mathbb Q_p$-case of general conjectures made in D. Kubrak and A. Prikhodko [p-adic Hodge theory for Artin stacks, Mem. Amer. Math. Soc. 304 (2024), 1174]. As a corollary, we get that if a smooth Artin stack over K has a smooth Hodge-proper model over $\mathcal O_K$, its $\mathbb Q_p$-étale cohomology is a crystalline Galois representation. We then also establish a truncated version of the above results in more general setting of smooth d-de Rham-proper stacks over $\mathcal O_K$: here we only require first d-de Rham cohomology groups be finitely generated over $\mathcal O_K$. As an application, we deduce a certain purity-type statement for étale $\mathbb Q_p$-cohomology of Raynaud generic fiber, as well as crystallinity of a first several étale cohomology groups in the presence of a Cohen–Macauley model over $\mathcal O_K$ in the schematic setting.