For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For instance, we prove that if $f\in S_k\left({\Gamma }_1\left(N\right)\right)$ is a classical holomorphic modular form whose $L$-function does not vanish for $\Re \left(s\right)>\left(k+1\right)∕2$, then $f$ is a Hecke eigenform. Our proof adapts and extends work of Saias and Weingartner, who proved a similar result for degree-$1$ $L$-functions.

We develop a framework to apply tropical and nonarchimedean analytic methods to multiplication maps for linear series on algebraic curves, studying degenerations of these multiplications maps when the special fiber is not of compact type. As an application, we give a new proof of the Gieseker–Petri theorem, including an explicit tropical criterion for a curve over a valued field to be Gieseker–Petri general.

We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese remainder theorem conditions, obtaining an exponent of distribution $\frac{1}{2}+\frac{7}{300}$.

We discuss the relation between crystalline Dieudonné theory and Dieudonné displays of $p$-divisible groups. The theory of Dieudonné displays is extended to the prime 2 without restriction, which implies that the classification of finite locally free group schemes by Breuil–Kisin modules holds for the prime 2 as well.

We prove that the universal unramified deformation ring $R^{unr}$ of a continuous Galois representation $\bar{\rho }:G_{F^{+}}\to {GL}_n\left(k\right)$ (for a totally real field $F^{+}$ and finite field $k$) is finite over $\mathcal{O}=W\left(k\right)$ in many cases. We also prove (under similar hypotheses) that the universal deformation ring $R^{univ}$ is finite over the local deformation ring $R^{loc}$.

We show that techniques inspired by Kollár and Viehweg’s study of weak positivity, combined with vanishing for log-canonical pairs, lead to new generation and vanishing results for direct images of pluricanonical bundles. We formulate the strongest such results as Fujita conjecture-type statements, which are then shown to govern a range of fundamental properties of direct images of pluricanonical and pluriadjoint line bundles, like effective vanishing theorems, weak positivity, or generic vanishing.