We construct and study Sarkisov links obtained by blowing up smooth space curves lying on smooth cubic surfaces. We restrict our attention to the case where the blowup is not weak Fano. Together with the results of [5], which cover the weak Fano case, we provide a classification of all such curves. This is achieved by computing all curves which satisfy certain necessary criteria on their multisecant curves and then constructing the Sarkisov link step by step.

Given a three-dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,\theta )$, we study the existence of a contact structure conformal to $\theta$ for which the logarithmic Hardy–Littlewood–Sobolev (LHLS) inequality holds. Our approach closely follows [30] in the Riemannian setting, yet the differential operators that we are dealing with are of very different nature. For this reason, we introduce the notion of Robin mass as the constant term appearing in the expansion of the Green’s function of the $P^{\prime }$-operator. We show that the LHLS inequality appears when we study the variation of the total mass under conformal change. This can be tied to the value of the regularized Zeta function of the operator at $1$ and hence we prove a CR version of the results in [27]. We also exhibit an Aubin-type result guaranteeing the existence of a minimizer for the total mass which yields the classical LHLS inequality.

Two elements $a$, $b$ in a ring $R$ form a right coprime pair, written $\langle a,b\rangle$, if $aR+bR=R$. Right coprime pairs have shown to be quite useful in the study of left cotorsion or exchange rings. In this paper, we define the class of right strong exchange rings in terms of descending chains of them. We show that they are semiregular and that this class of rings contains left injective, left pure-injective, left cotorsion, local, and left continuous rings. This allows us to give a unified study of all these classes of rings in terms of the behaviour of descending chains of right coprime pairs.

In the article [25] a general procedure to study solutions of the equations $x^4-d{y}^2=z^p$ was presented for negative values of $d$. The purpose of the present article is to extend our previous results to positive values of $d$. On doing so, we give a description of the extension $\mathbb{Q}(\sqrt{d},\sqrt{ϵ})/\mathbb{Q}(\sqrt{d})$ (where $ϵ$ is a fundamental unit) needed to prove the existence of a Hecke character over $\mathbb{Q}(\sqrt{d})$ with prescribed local conditions. We also extend some “large image” results due to Ellenberg regarding images of Galois representations coming from $\mathbb{Q}$-curves from imaginary to real quadratic fields.

A result of G. Godefroy asserts that a Banach space $X$ contains an isomorphic copy of ${\ell }_1$ if and only if there is an equivalent norm $\left|\right||·|\left|\right|$ such that, for every finite-dimensional subspace $Y$ of $X$ and every , there exists $x\in S_X$ so that $\left|\right||y+rx|\left|\right|\ge (1-\epsilon )(|\left|\right|y\left|\right||+|r|)$ for every $y\in Y$ and every $r\in ℝ$. In this paper we generalise this result to larger cardinals, showing that if $\kappa$ is an uncountable cardinal, then a Banach space $X$ contains a copy of ${\ell }_1(\kappa )$ if and only if there is an equivalent norm $\left|\right||·|\left|\right|$ on $X$ such that for every subspace $Y$ of $X$ with there exists a norm-one vector $x$ so that $\left|\right||y+rx|\left|\right|=\left|\right|\left|y\right|\left|\right|+\left|r\right|$ whenever $y\in Y$ and $r\in ℝ$. This result answers a question posed by S. Ciaci, J. Langemets, and A. Lissitsin, where the authors wonder whether the above statement holds for infinite successor cardinals. We also show that, in the countable case, the result of Godefroy cannot be improved to take $\epsilon =0$.

We prove an explicit formula for the second moment of symmetric square $L$-functions associated to Maass forms for the full modular group. In particular, we show how to express the considered second moment in terms of dual second moments of symmetric square $L$-functions associated to Maass cusp forms of levels $4$, $16$, and $64$.

We characterize dyadic little BMO via the boundedness of the tensor commutator with a single well-chosen dyadic shift. It is shown that several proof strategies work for this problem, both in the unweighted case and with Bloom weights. Moreover, we address the flexibility of one of our methods.

The main goal of the paper is to provide new insight into compactness in $L^p$-spaces on locally compact groups. The article begins with a brief historical overview and the current state of literature regarding the topic. Subsequently, we “take a step back” and investigate the Arzelà–Ascoli theorem on a non-compact domain together with one-point compactification. The main idea comes in Section 3, where we introduce the “$L^p$-properties” ($L^p$-boundedness, $L^p$-equicontinuity, and $L^p$-equivanishing) and study their “behaviour under convolution”. The paper proceeds with an analysis of Young’s convolution inequality, which plays a vital role in the final section. During the “grand finale”, all the pieces of the puzzle are brought together as we lay down a new approach to compactness in $L^p$-spaces on locally compact groups.

We provide several extensions of the modular method which were motivated by the problem of completing previous work to prove that, for any integer $n\ge 2$, the equation

$$x^{13}+y^{13}=3z^n$$

has no non-trivial primitive solutions. In particular, we present four elimination techniques which are based on: (1) establishing reducibility of certain residual Galois representations over a totally real field; (2) generalizing image of inertia arguments to the setting of abelian surfaces; (3) establishing congruences of Hilbert modular forms without the use of often impractical Sturm bounds; and (4) a unit sieve argument which combines information from classical descent and the modular method.

The extensions are of broader applicability and provide further evidence that it is possible to obtain a complete resolution of a family of generalized Fermat equations by remaining within the framework of the modular method. As a further illustration of this, we complete a theorem of Anni–Siksek to show that, for $\ell ,m\ge 5$, the only primitive solutions to the equation $x^{2\ell }+y^{2m}=z^{13}$ are trivial.

For the spin group $G={\text{Spin}}_{2n+1}$ with arbitrary $n$, a generic $G$-torsor $E$ over a field, and a parabolic subgroup $P\subset G$, we consider the generic flag variety $E/P$ and describe its Chow ring modulo torsion. This description determines the index of $E/P$, completing results of [3], where the index has been determined for most $P$.

This paper contributes to the proof of the conjecture posed in [5], stating that a Nichols algebra of diagonal type with finite Gelfand–Kirillov dimension has a finite (generalized) root system. We prove the conjecture assuming that the rank is $3$ or that the braiding is of Cartan type.

We describe some new ways to construct saturated fusion subsystems, including, as a special case, the normalizer of a set of components of the ambient fusion system. This was motivated in part by Aschbacher’s construction of the normalizer of one component, and in part by joint work with three other authors where we had to construct the normalizer of all of the components.

We give a characterization of $L^p(\sigma )$ for uniformly rectifiable measures $\sigma$ using Tolsa’s $\alpha$-numbers, by showing, for and $f\in L^p(\sigma )$, that

$$\Vert f{\Vert }_{L^p(\sigma )}~{\Vert ({\displaystyle {\int }_0^{\infty }({\alpha }_{f\sigma }(}x,r)+|f{|}_{x,r}{\alpha }_{\sigma }(x,r){)}^2\frac{dr}{r}{)}^{\frac{1}{2}}\Vert }_{L^p(\sigma )}.$$

We describe the real forms of Gizatullin surfaces of the form $xy=p(z)$ and of Koras–Russell threefolds of the first kind. The former admit zero, two, three, four, or six isomorphism classes of real forms, depending on the degree and the symmetries of the polynomial $p$. The latter, which are threefolds given by an equation of the form $x^{d}y+z^k+x+t^{\ell }=0$, all admit exactly one real form up to isomorphism.

We classify homogeneous CR submanifolds in complex hyperbolic spaces arising as orbits of a subgroup of the solvable part of the Iwasawa decomposition of the isometry group of the ambient space.