We prove a semiample generalization of Poonen’s Bertini theorem over a finite field that implies the existence of smooth sections for wide new classes of divisors. The probability of smoothness is computed as a product of local probabilities taken over the fibers of the morphism determined by the relevant divisor. We give several applications including a negative answer to a question of Baker and Poonen by constructing a variety (in fact one of each dimension) which provides a counterexample to Bertini over finite fields in arbitrarily large projective spaces. As another application, we determine the probability of smoothness for curves in Hirzebruch surfaces and the distribution of points on those smooth curves.

The object of this paper is to initiate a study of the Fourier coefficients of a weight $1$ harmonic Maass form and relate them to the complex Galois representation associated to a weight $1$ newform, which is the form’s image under a certain differential operator. In this paper, our focus will be on weight $1$ dihedral newforms of prime level $p\equiv 3(mod4)$. In this case we give properties of the Fourier coefficients that are similar to (and sometimes reduce to) cases of Stark’s conjectures on derivatives of $L$-functions. We also give a new modular interpretation of certain products of differences of singular moduli studied by Gross and Zagier. Finally, we provide some numerical evidence that the Fourier coefficients of a mock-modular form whose shadow is exotic are similarly related to the associated complex Galois representation.

Consider homogeneous $G/H$ and $G/F$, for an $S$-algebraic group $G$. A lattice $\Gamma$ acts on the left strictly conservatively. The following rigidity results are obtained: morphisms, factors, and joinings defined a priori only in the measurable category are in fact algebraically constrained. Arguing in an elementary fashion, we manage to classify all the measurable $\Phi$ commuting with the $\Gamma$-action: assuming ergodicity, we find that they are algebraically defined.

Recently, it has become well known that the conjectural integral identity is of crucial importance in the motivic Donaldson–Thomas invariants theory for noncommutative Calabi–Yau threefolds. The purpose of this article is to consider different versions of the identity, for regular functions and formal functions, and to give them the positive answer for the algebraically closed ground fields. Technically, the result on the motivic Milnor fiber by Hrushovski–Loeser using Hrushovski–Kazhdan’s motivic integration and Nicaise’s computations on motivic integrals on special formal schemes are main tools.