We consider t-structures that naturally arise on elliptic fibrations. By filtering the category of coherent sheaves on an elliptic fibration using the torsion pairs corresponding to these t-structures, we prove results describing equivalences of t-structures under Fourier–Mukai transforms.

Consider $A$ an abelian variety of dimension $r$ defined over $\mathbb{Q}$. Assume that ${rank}_{\mathbb{Q}}A\ge g$, where $g\ge 0$ is an integer, and let $a_1,\dots ,a_g\in A\left(\mathbb{Q}\right)$ be linearly independent points. (So, in particular, $a_1,\dots ,a_g$ have infinite order, and if $g=0$, then the set $\{a_1,\dots ,a_g\}$ is empty.) For $p$ a rational prime of good reduction for $A$, let $\overline{A}$ be the reduction of $A$ at $p$, let ${\overline{a}}_i$ for $i=1,\dots ,g$ be the reduction of $a_i$ (modulo $p$), and let $\langle {\overline{a}}_1,\dots ,{\overline{a}}_g\rangle$ be the subgroup of $\overline{A}\left({\mathbb{F}}_p\right)$ generated by ${\overline{a}}_1,\dots ,{\overline{a}}_g$. Assume that $\mathbb{Q}\left(A\right[2\left]\right)=\mathbb{Q}$ and $\mathbb{Q}\left(A\right[2],2^{-1}{a}_1,\dots ,2^{-1}{a}_g)\ne \mathbb{Q}$. (Note that this particular assumption $\mathbb{Q}\left(A\right[2\left]\right)=\mathbb{Q}$ forces the inequality $g\ge 1$, but we can take care of the case $g=0$, under the right assumptions, also.) Then in this article, in particular, we show that the number of primes $p$ for which $\frac{\overline{A}\left({\mathbb{F}}_p\right)}{\langle {\overline{a}}_1,\dots ,{\overline{a}}_g\rangle }$ has at most $(2r-1)$ cyclic components is infinite. This result is the right generalization of the classical Artin’s primitive root conjecture in the context of general abelian varieties; that is, this result is an unconditional proof of Artin’s conjecture for abelian varieties. Artin’s primitive root conjecture (1927) states that, for any integer $a\ne \pm 1$ or a perfect square, there are infinitely many primes $p$ for which $a$ is a primitive root $(modp)$. (This conjecture is not known for any specific $a$.)

For a Jacobian elliptic surface $S_0$ over a finite field $k$ and a prime $\ell$ different from the characteristic of $k$, the points of period ${\ell }^r$ on the smooth fibers of $S_0$ yield, for each $r\in {\mathbb{Z}}_{\ge 0}$, a smooth projective curve $C_r$ over $k$ by taking Zariski closure in $S_0$ and normalization. We consider the restriction map in $\ell$-adic étale cohomology $H^2(S_0,{\mathbb{Z}}_{\ell }(1\left)\right)\to H^2({\bigsqcup }_{r\ge 0}{C}_r,{\mathbb{Z}}_{\ell }(1\left)\right)={\prod }_{r\ge 0}{H}^2(C_r,{\mathbb{Z}}_{\ell }(1\left)\right)$. By using an earlier result of ours we prove that, except for at most a finite number of such primes $\ell$, this map is faithful on the submodule $F^1{H}^2(S_0,{\mathbb{Z}}_{\ell }(1){)}^0$ of those classes vanishing on the geometric fibers and on the zero section of $S_0$, and that it gives an isomorphism between this submodule and the subgroup of $Pic\left({\bigsqcup }_{r\ge 0}{C}_r\right)={\prod }_{r\ge 0}Pic\left(C_r\right)$ of primitive elements in the sense of Serre.

The relation between the Auslander (resp., Bass) class and the class of modules with finite Gorenstein projective (resp., injective) dimension is well known when these mentioned classes are built with a dualizing module over Noetherian $n$-perfect rings. Basically, the results are necessary conditions to ensure that both classes coincide. In this article we try to extend and sometimes improve some of these results by weakening the condition of being dualizing. Among other results, we prove that a Wakamatsu tilting module with some extra conditions is precisely a module ${}_{R}C$ such that the Bass class ${\mathcal{B}}_C\left(R\right)$ coincides with the class of modules of finite Gorenstein injective dimension.

Let $\ell$ and $p$ be odd primes. For a positive integer $\mu$, let $k_{\mu }$ be the ray class field of $k=\mathbb{Q}\left(e^{2\pi i/\ell }\right)$ modulo $2p^{\mu }$. We present certain class fields $K_{\mu }$ of $k$ such that $k_{\mu }\subset K_{\mu }\subset k_{\mu +1}$, and we provide a necessary and sufficient condition for $K_{\mu }=k_{\mu +1}$. We also construct, in the sense of Hilbert, primitive generators of the field $K_{\mu }$ over $k_{\mu }$ by using Shimura’s reciprocity law and special values of theta constants.

In this article, the fractional Hardy-type operator of variable order $\beta \left(x\right)$ is shown to be bounded from the variable-exponent Herz–Morrey spaces $M{\dot{K}}_{p_1,q_1(\cdot )}^{\alpha (\cdot ),\lambda }\left({\mathbb{R}}^n\right)$ into the weighted space $M{\dot{K}}_{p_2,q_2(\cdot )}^{\alpha (\cdot ),\lambda }({\mathbb{R}}^n,\omega )$, where $\alpha \left(x\right)\in L^{\infty }\left({\mathbb{R}}^n\right)$ is log-Hölder continuous both at the origin and at infinity, $\omega =(1+|x|{)}^{-\gamma \left(x\right)}$ with some $\gamma \left(x\right)>0$, and $1/q_1\left(x\right)-1/q_2\left(x\right)=\beta \left(x\right)/n$ when $q_1\left(x\right)$ is not necessarily constant at infinity.

Let $p$ be an odd prime, and let $\lambda \in \mathbb{Z}$. Consider the loop space $Y_t=S_{\left(p\right)}^{2t-1}$ for $t\ge 1$ with $t\left|\right(p-1)$. Then we first determine the condition for the power map ${\Phi }_{\lambda }$ on $Y_t$ to be an $A_p$-map. We next assume that $X$ is a simply connected ${\mathbb{F}}_p$-finite $A_p$-space and that $\lambda$ is a primitive $(p-1)$st root of unity mod $p$. Our results show that if the reduced power operations $\{{\mathcal{P}}^i{\}}_{i\ge 1}$ act trivially on the indecomposable module $Q{H}^{\ast }(X;{\mathbb{F}}_p)$ and the power map ${\Phi }_{\lambda }$ on $X$ is an $A_n$-map with $n>(p-1)/2$, then $X$ is ${\mathbb{F}}_p$-acyclic.

When a singular projective variety $X_{\mathrm{sing}}$ admits a projective crepant resolution $X_{\mathrm{res}}$ and a smoothing $X_{\mathrm{sm}}$, we say that $X_{\mathrm{res}}$ and $X_{\mathrm{sm}}$ are related by extremal transition. In this article, we study a relationship between the quantum cohomology of $X_{\mathrm{res}}$ and $X_{\mathrm{sm}}$ in some examples. For $3$-dimensional conifold transition, a result of Li and Ruan implies that the quantum cohomology of a smoothing $X_{\mathrm{sm}}$ is isomorphic to a certain subquotient of the quantum cohomology of a resolution $X_{\mathrm{res}}$ with the quantum variables of exceptional curves specialized to one. We observe that similar phenomena happen for toric degenerations of $Fl(1,2,3)$, $Gr(2,4)$, and $Gr(2,5)$ by explicit computations.