In this paper, we investigate the apparent singularities and the dual parameters of rank $2$ parabolic connections on ${\mathbb{P}}^1$ and rank $2$ (parabolic) Higgs bundles on ${\mathbb{P}}^1$. Then we obtain explicit descriptions of Zariski-open sets of the moduli space of the parabolic connections and the moduli space of the Higgs bundles. For $n=5$, we can give global descriptions of the moduli spaces in detail.

On the basis of fractional calculus, we introduce an integral of controlled paths against $\beta$-Hölder rough paths with $\beta \in (1/3,1/2]$. The integral is defined by the Lebesgue integrals for fractional derivative operators, without using any argument based on discrete approximation. We show in this article that the integral is consistent with that obtained by the usual integration in rough path analysis, given by the limit of the compensated Riemann–Stieltjes sums.

We study the category $Rep(Q,\mathcal{M})$ of representations of a quiver $Q$ with values in an abelian category $\mathcal{M}$. Under certain assumptions, we show that every cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces two (explicitly described) cotorsion pairs $(\Phi (\mathcal{A}),Rep(Q,\mathcal{B}\left)\right)$ and $(Rep(Q,\mathcal{A}),\Psi (\mathcal{B}\left)\right)$ in $Rep(Q,\mathcal{M})$. This is akin to a result by Gillespie, which asserts that a cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{M}$ induces cotorsion pairs $(\tilde{\mathcal{A}},dg\tilde{\mathcal{B}})$ and $(dg\tilde{\mathcal{A}},\tilde{\mathcal{B}})$ in the category $Ch\left(\mathcal{M}\right)$ of chain complexes in $\mathcal{M}$. Special cases of our results recover descriptions of the projective and injective objects in $Rep(Q,\mathcal{M})$ proved by Enochs, Estrada, and García Rozas.

Let $G$ be a Lie group, and let $\Gamma$ be a finite group. We show in this article that the space $Hom(\Gamma ,G)/G$ is discrete and—in addition—finite if $G$ has finitely many connected components. This means that in the case in which $\Gamma$ is a discontinuous group for the homogeneous space $G/H$, where $H$ is a closed subgroup of $G$, all the elements of Kobayashi’s parameter space are locally rigid. Equivalently, any Clifford–Klein form of finite fundamental group does not admit nontrivial continuous deformations. As an application, we provide a criterion of local rigidity in the context of compact extensions of ${\mathbb{R}}^n$.

Ichino proved a formula expressing global trilinear period integrals as products of central values of triple product $L$-functions and certain local trilinear integrals. In general, these local trilinear integrals are difficult to evaluate, and the main result in this article is to prove an identity relating local trilinear integrals and products of local Rankin–Selberg integrals in the twisted case. This result has applications to an explicit version of Ichino’s formula and the construction of $p$-adic $L$-functions for twisted triple products in our forthcoming works.

It follows from work of S. Mochizuki, F. Liu, and B. Osserman that there is a relationship between Ehrhart’s theory concerning rational polytopes and the geometry of the moduli stack classifying dormant indigenous bundles on a proper hyperbolic curve in positive characteristic. This relationship was established by considering the (finite) cardinality of the set consisting of certain colorings on a $3$-regular graph called spin networks. In the present article, we recall the correspondences between spin networks, lattice points of rational polytopes, and dormant indigenous bundles and present some identities and explicit computations of invariants associated with the objects involved.

In this article, we will show that if $S$ is an Inoue surface with $b_1=1$ and $b_2=0$, then the number of deformation equivalence classes of complex surfaces diffeomorphic to $S$ is at most $16$.

Let $A$ be a geometrically integral algebra over a field $k$. We prove that, for any affine $k$-domain $R$, if there exists an extension field $K$ of $k$ such that $R\subseteq K{\otimes }_{k}A$ and $R⊈K$, then there exists an extension field $L$ of $k$ such that $R\subseteq L{\otimes }_{k}A$ and ${trdeg}_k\left(L\right)<{trdeg}_k\left(R\right)$. This generalizes a result of Freudenburg, namely, the fact that this is true for $A=k^{\left[1\right]}$.

Iwasawa theory of Heegner points on abelian varieties of ${GL}_2$ type has been studied by, among others, Mazur, Perrin-Riou, Bertolini, and Howard. The purpose of this article is to describe extensions of some of their results in which abelian varieties are replaced by the Galois cohomology of Deligne’s $p$-adic representation attached to a modular form of even weight greater than $2$. In this setting, the role of Heegner points is played by higher-dimensional Heegner-type cycles that have been recently defined by Bertolini, Darmon, and Prasanna. Our results should be compared with those obtained, via deformation-theoretic techniques, by Fouquet in the context of Hida families of modular forms.