We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.

Let $f(x_{1},\ldots ,x_{n})$ be a regular indefinite integral quadratic form with $n\geqslant 9$, and let $t$ be an integer. Denote by $\mathbb{U}_{p}$ the set of $p$-adic units in $\mathbb{Z}_{p}$. It is established that $f(x_{1},\ldots ,x_{n})=t$ has solutions in primes if (i) there are positive real solutions, and (ii) there are local solutions in $\mathbb{U}_{p}$ for all prime $p$.

Let $S$ be a smooth minimal complex surface of general type with $p_{g}=0$ and $K^{2}=7$. We prove that any involution on $S$ is in the center of the automorphism group of $S$. As an application, we show that the automorphism group of an Inoue surface with $K^{2}=7$ is isomorphic to $\mathbb{Z}_{2}^{2}$ or $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$. We construct a $2$-dimensional family of Inoue surfaces with automorphism groups isomorphic to $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$.

In this paper, we investigate the nontrivial zeros of quadratic $L$-functions near the real axis. Assuming the generalized Riemann hypothesis, we give an asymptotic formula for the weighted pair correlation function of quadratic $L$-functions associated to the Kronecker symbols. From this formula, we obtain several results on the rate of simple zeros of quadratic $L$-functions and on the average distance of such nontrivial zeros.

We study certain integer valued length functions on triangulated categories, and establish a correspondence between such functions and cohomological functors taking values in the category of finite length modules over some ring. The irreducible cohomological functions form a topological space. We discuss its basic properties, and include explicit calculations for the category of perfect complexes over some specific rings.

A semialgebraic map $f:X\rightarrow Y$ between two real algebraic sets is called blow-Nash if it can be made Nash (i.e., semialgebraic and real analytic) by composing with finitely many blowings-up with nonsingular centers.

We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition.

The proof relies on motivic integration arguments and on the virtual Poincaré polynomial of McCrory–Parusiński and Fichou. In particular, we need to generalize Denef–Loeser change of variables key lemma to maps that are generically one-to-one and not merely birational.