We investigate the Gross–Prasad conjecture and its refinement for the Bessel periods in the case of $(\mathrm {SO}(5), \mathrm {SO}(2))$. In particular, by combining several theta correspondences, we prove the Ichino–Ikeda-type formula for any tempered irreducible cuspidal automorphic representation. As a corollary of our formula, we prove an explicit formula relating certain weighted averages of Fourier coefficients of holomorphic Siegel cusp forms of degree two, which are Hecke eigenforms, to central special values of $L$-functions. The formula is regarded as a natural generalization of the Böcherer conjecture to the non-trivial toroidal character case.

We show that a virtually residually finite rationally solvable (RFRS) group $G$ of type $\mathtt {FP}_n(\mathbb {Q})$ virtually algebraically fibres with kernel of type $\mathtt {FP}_n(\mathbb {Q})$ if and only if the first $n$ $\ell ^2$-Betti numbers of $G$ vanish, that is, $b_p^{(2)}(G) = 0$ for $0 \leqslant p \leqslant n$. This confirms a conjecture of Kielak. We also offer a variant of this result over other fields, in particular in positive characteristic. As an application of the main result, we show that amenable virtually RFRS groups of type $\mathtt {FP}(\mathbb {Q})$ are virtually Abelian. It then follows that if $G$ is a virtually RFRS group of type $\mathtt {FP}(\mathbb {Q})$ such that $\mathbb {Z} G$ is Noetherian, then $G$ is virtually Abelian. This confirms a conjecture of Baer for the class of virtually RFRS groups of type $\mathtt {FP}(\mathbb {Q})$, which includes (for instance) the class of virtually compact special groups.

Coinciding with the 75th anniversary of John Cade's seminal publication that first reported lithium's antimanic efficacy, we briefly recount the salient findings of the historic paper and draw attention to the important psychiatric research in Britain that reinforced its findings and the critical British opinions that likely impeded its clinical use.

We describe the $J$-invariant of a semisimple algebraic group $G$ over a generic splitting field of a Tits algebra of $G$ in terms of the $J$-invariant over the base field. As a consequence we prove a 10-year-old conjecture of Quéguiner-Mathieu, Semenov, and Zainoulline on the $J$-invariant of groups of type $\mathrm {D}_n$. In the case of type $\mathrm {D}_n$ we also provide explicit formulas for the first component and in some cases for the second component of the $J$-invariant.

Peasant transformation has often been framed in dichotomous, linear, and predominantly a-historical models. This article adopts a dynamic perspective and shows how the developments that have often been regarded as constituent to the long-term process of the decline of agriculture and peasant worlds were in fact part of the spread of more diversified labour and income strategies of the peasantries. In the movement towards a worldwide enclosure of the rural worlds after 1870, a global peasantry emerged through a wide range of regional trajectories and narratives. By the first half of the twentieth century, peasant households all over the world had become heavily involved in the capitalist market economy. In this article, we argue that the (re)creation of peasantries as a social group is part of a diverse complex of reciprocal exchanges, regional and extra-regional market transactions, actions of public forces, and social conflict. Also we stress that social class formation should be understood in its specific world-historical co-ordinates using a particular set of transhistorical concepts like (re)peasantization, peasant frontiers, and peasant regimes.

JR87, the first respondent, now nine, attended a controlled primary school in Belfast and took part in non-denominational Christian religious education (RE) and collective worship (CW). Her parents described themselves as ‘broadly humanist’ and had not raised their daughter in any religious tradition. Once she began attending school, however, they noticed that she would say a prayer before eating and ask them questions about God and religion. They voiced their concerns to the school about its RE teaching and were told that its provision of RE and CW was ‘bible-based’, followed the core syllabus for education and complied with the relevant legislation. They challenged this, arguing that the relevant legislation contravened their Convention rights under Article 9 of the European Convention on Human Rights and Article 2 of Protocol 1 (‘A2P1’). They were successful in the lower court and the Department of Education appealed.