Let $p$ be an odd prime, and let $E_1$ and $E_2$ be two elliptic curves defined over a number field $K$, with good ordinary reduction at $p$. We compare the $\Lambda$-ranks and (generalized) Iwasawa invariants of the Pontryagin duals of the Selmer groups of $E_1$ and $E_2$ over ${\mathbb{Z}}_p^d$-extensions $\mathbb{L}_\infty$ of $K$ for general $d \ge 1$ under the hypothesis that $E_1[p^i] \cong E_2[p^i]$ as Galois modules for a sufficiently large $i$. This generalizes and complements previous work over ${\mathbb{Z}}_p$-extensions. The comparison of generalized Iwasawa invariants is related via an up-down approach to the comparison of the variation of classical Iwasawa invariants over the ${\mathbb{Z}}_p$-extensions of $K$ which are contained in $\mathbb{L}_\infty$.

Twisted arrow $\infty$-categories of $(\infty ,1)$-categories were introduced by Lurie, and they have various applications in higher category theory. Abellán García and Stern gave a generalization to twisted arrow $\infty$-categories of $(\infty ,2)$-categories. In this paper, we introduce another simple model for twisted arrow $\infty$-categories of $(\infty ,2)$-categories.

Let $f \in \mathbb{Q}[x]$ be a square-free polynomial of degree at least $3$, $m_i$, $i=1,2,3$, odd positive integers, and $a_i$, $i=1,2,3$, non-zero rational numbers. We show the existence of a rational function $D\in \mathbb{Q}(v_1,v_2,v_3,v_4)$ such that the Jacobian of the quadratic twist of $y^2=f(x)$ and the Jacobian of the $m_i$-twist, respectively, $2m_i$-twist, of $y^2=x^{m_i}+a_i^2$, $i=1,2,3$, by $D$ are all of positive Mordell–Weil ranks. As an application, we present families of hyperelliptic curves with large Mordell–Weil rank.

Let $\Bbbk$ be a field, $H$ a Hopf algebra over $\Bbbk$, and $R = (_iM_j)_{1 \leq i,j \leq n}$ a generalized matrix algebra. In this work, we establish necessary and sufficient conditions for $H$ to act partially on $R$. To achieve this, we introduce the concept of an opposite covariant pair and demonstrate that it satisfies a universal property. In the special case where $H = \Bbbk G$ is the group algebra of a group $G$, we recover the conditions given in [7] for the existence of a unital partial action of $G$ on $R$.

We derive faithful inclusions of C*-algebras from a coend-type construction in unitary tensor categories. This gives rise to different potential notions of discreteness for an inclusion in the non-irreducible case and provides a unified framework that encloses the theory of compact quantum group actions. We also provide examples coming from semi-circular systems and from factorization homology. In the irreducible case, we establish conditions under which the C*-discrete and W*-discrete conditions are equivalent.

For each prime $p$, this paper constructs compact complex hyperbolic $2$-manifolds with an isometric action of $\mathbb{Z} / p \mathbb{Z}$ that is not free and has only isolated fixed points. The case $p = 2$ is special, and finding general examples for $p=2$ is related to whether or not complex hyperbolic lattices are conjugacy separable on torsion.

Fulton’s matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen–Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper, we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.

Given a finite abelian group $G$ and $t\in \mathbb{N}$, there are two natural types of subsets of the Cartesian power $G^t$; namely, Cartesian powers $S^t$ where $S$ is a subset of $G$ and (cosets of) subgroups $H$ of $G^t$. A basic question is whether two such sets intersect. In this paper, we show that this decision problem is NP-complete. Furthermore, for fixed $G$ and $S$, we give a complete classification: we determine conditions for when the problem is NP-complete and show that in all other cases the problem is solvable in polynomial time. These theorems play a key role in the classification of algebraic decision problems in finitely generated rings developed in later work of the author.

In this paper, we first describe the cohomology theory of Lie supertriple systems by using the cohomology theory of the associated Leibniz superalgebras. Then we focus on Lie supertriple systems with superderivations, called LSTSDer pairs. We introduce the notion of representations of LSTSDer pairs and investigate their corresponding cohomology theory. We also construct a differential graded Lie algebra whose Maurer–Cartan elements are LSTSDer pairs. Moreover, we consider the relationship between a LSTSDer pair and the associated LeibSDer pair. Furthermore, we develop the 1-parameter formal deformation theory of LSTSDer pairs and prove that it is governed by the cohomology groups. At last, we study abelian extensions of LSTSDer pairs and show that equivalent abelian extensions of LSTSDer pairs are classified by the third cohomology groups.

This paper focuses on quadratic Hom–Leibniz algebras, defined as (left or right) Hom–Leibniz algebras equipped with symmetric, non-degenerate, and invariant bilinear forms. In particular, we demonstrate that every quadratic regular Hom–Leibniz algebra is symmetric, meaning that it is simultaneously a left and a right Hom–Leibniz algebra. We provide characterizations of symmetric (resp. quadratic) Hom–Leibniz algebras. We also investigate the $\mathrm{T}^*$-extensions of Hom–Leibniz algebras, establishing their compatibility with solvability and nilpotency. We study the equivalence of such extensions and provide the necessary and sufficient conditions for a nilpotent quadratic Hom–Leibniz algebra to be isometric to a $\mathrm{T}^*$-extension. Furthermore, through the procedure of double extension, which is a central extension followed by a generalized semi-direct product, we get an inductive description of all quadratic regular Hom–Leibniz algebras, allowing us to reduce their study to that of quadratic regular Hom–Lie algebras. Finally, we construct several non-trivial examples of symmetric (resp. quadratic) Hom–Leibniz algebras.

Let $\Gamma$ be a Schottky subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and let $X=\Gamma \backslash {\mathbb{H}}^2$ be the associated hyperbolic surface. We consider the family of Hecke congruence coverings of $X$, which we denote as usual by $ X_0(q) = \Gamma _0(q)\backslash {\mathbb{H}}^2$. Conditional on the Lindelöf Hypothesis for quadratic L-functions, we establish a uniform and explicit spectral gap for the Laplacian on $ X_0(q)$ for “almost” all prime levels $q$. Assuming the generalized Riemann hypothesis for quadratic $L$-functions, we obtain an even larger spectral gap.

In this paper, we establish an asymptotic formula for the twisted second moments of Dirichlet $L$-functions with one twist when averaged over all primitive Dirichlet characters of modulus $R$, where $R$ is a monic polynomial in $\mathbb{F}_q[T]$.

We prove the formula

\begin{equation*} \text{cat}_G(X\vee Y)=\max \{\text{cat}_G(X),\text{cat}_G(Y)\} \end{equation*}

$X\vee Y$

$\bigvee _{i=1}^m X_i$

$G$

$X_i$

$G$

$i=1,\ldots ,m$

$X/A$

$G$

$X$

$A$

$A\hookrightarrow X$

$G$

$\overline {x_0}\,:\,A\to X$

$x_0\in X^G$

\begin{align*} \text{TC}_G(X\vee Y)&=\max \{\text{TC}_G(X),\text{TC}_G(Y),\text{cat}_G(X\times Y)\},\\ \text{TC}^G(X\vee Y)&=\max \{\text{TC}^G(X),\text{TC}^G(Y),_{X\vee Y}\text{cat}_{G\times G}(X\times Y)\}, \end{align*}

$G$

$G$

$X$

$Y$

In the second half of the 19th century, Darboux obtained determinant formulae that provide the general solution for a linear hyperbolic second-order PDE with the finite Laplace series. These formulae played an important role in his study of the theory of surfaces and, in particular, in the theory of conjugate nets. During the last three decades, discrete analogues of conjugate nets (Q-nets) were actively studied. Laplace series can be defined also for hyperbolic difference operators. We prove discrete analogues of Darboux formulae for discrete and semi-discrete hyperbolic operators with finite Laplace series.

Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system $\Phi$ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory $\mathcal{F}(\Phi )$ of objects admitting a composition series-like filtration with factors in $\Phi$ has the Jordan-Hölder property on these filtrations. This article has two main aims. First, we introduce notions of subobjects, simple objects and composition series for an extriangulated category, in order to define a Jordan-Hölder extriangulated category. Moreover, we characterise Jordan-Hölder, length, weakly idempotent complete extriangulated categories in terms of the associated Grothendieck monoid and Grothendieck group. Second, we develop a theory of stratifying systems in extriangulated categories. We define projective stratifying systems and show that every stratifying system $\Phi$ in an extriangulated category is part of a minimal projective one $(\Phi ,Q)$. We prove that $\mathcal{F}(\Phi )$ is a length, Jordan-Hölder extriangulated category when $(\Phi ,Q)$ satisfies a left exactness condition. We give several examples and answer a recent question of Enomoto–Saito in the negative.

In this paper, we describe finite order entire solutions of nonlinear partial differential equation

\begin{equation*} \prod _{i=1}^n\left (a_{i 1} u_{z_1}+a_{i 2} u_{z_2}+\cdots +a_{i n} u_{z_n}\right )=1 \end{equation*}

We study the geometry induced on the local orbit spaces of Killing vector fields on (Riemannian) $G$-manifolds, with an emphasis on the cases $G=\textrm {Spin}(7)$ and $G=G_2$. Along the way, we classify the harmonic morphisms with one-dimensional fibres from $G_2$-manifolds to Einstein manifolds.

In 1954, B. H. Neumann discovered that if $G$ is a group in which all conjugacy classes have finite cardinality at most $m$, then the derived group $G'$ is finite of $m$-bounded order. In 2018, G. Dierings and P. Shumyatsky showed that if $|x^G| \le m$ for any commutator $x\in G$, then the second derived group $G''$ is finite and has $m$-bounded order. This paper deals with finite groups in which $|x^G|\le m$ whenever $x\in G$ is a commutator of prime power order. The main result is that $G''$ has $m$-bounded order.

Given a Hamiltonian torus action on a symplectic manifold, Teleman and Fukaya have proposed that the Fukaya category of each symplectic quotient should be equivalent to an equivariant Fukaya category of the original manifold. We lay out new conjectures that extend this story – in certain situations – to singular values of the moment map. These include a proposal for how, in some cases, we can recover the non-equivariant Fukaya category of the original manifold starting from data on the quotient.

To justify our conjectures, we pass through the mirror and work out numerous examples, using well-established heuristics in toric mirror symmetry. We also discuss the algebraic and categorical structures that underlie our story.

In this article, we study the Johnson homomorphisms of basis-conjugating automorphism groups of free groups. We construct obstructions for the surjectivity of the Johnson homomorphisms. By using it, we determine their cokernels of degree up to four and give further observations for degree greater than four. As applications, we give the affirmative answer to the Andreadakis problem for the basis-conjugating automorphism groups of free groups at degree four. Moreover, we calculate twisted first cohomology groups of the braid-permutation automorphism groups of free groups.