Pour un morphisme $T\to S$, plat et de présentation finie, l’enveloppe étale — un avatar du ${\pi }_0(T∕S)$ — peut ne pas exister ; par contre l’enveloppe étale séparée, i.e., celle qui est universelle pour les schémas étales et séparés sur $S$, existe dès que $S$ est localement connexe. On la note $T\to {\pi }^s(T∕S)$ ; c’est le quotient de $T$ par la relation d’équivalence minimale à graphe ouvert et fermé dans $T{\times }_{S}T$ ; cette construction permet de démontrer qu’un morphisme fpqc de changement de base $S^{\prime }\to S$ induit un isomorphisme ${\pi }^s(S^{\prime }{\times }_{S}T∕S^{\prime })\to S^{\prime }{\times }_S{\pi }^s(T∕S)$ si ses fibres géométriques sont connexes. Par ailleurs, lorsque $S$ est normal, disons intègre, et que le morphisme $T\to S$ est normal, on dispose d’un isomorphisme ${\pi }^s(T_{\xi }∕\xi )\simeq {\pi }^s{(T∕S)}_{\xi }$ ; il permet d’étendre à des morphismes normaux des résultats connus sur un corps de base. Toujours sous des hypothèses de normalité, le morphisme canonique ${\pi }_0(T∕S)\to {\pi }^s(T∕S)$ fait apparaître le schéma ${\pi }^s$ comme l’enveloppe séparée de l’espace algébrique ${\pi }_0$.

For a flat morphism $T\to S$ of finite presentation, the étale envelope — an avatar of ${\pi }_0(T∕S)$ — may fail to exist; but, the separated étale envelope, i.e., the one which is universal for the separated étale $S$-schemes only, is shown to exist as soon as $S$ is locally connected; denoted ${\pi }^s(T∕S)$, it is the quotient of $T$ by the equivalence relation which is minimal among those with clopen graph in $T{\times }_{S}T$; from that we prove that, for a fpqc base change $S^{\prime }\to S$, the morphism ${\pi }^s(S^{\prime }{\times }_{S}T∕S^{\prime })\to S^{\prime }{\times }_S{\pi }^s(T∕S)$ is an isomorphism if the geometric fibres of $S^{\prime }\to S$ are connected. When $S$ is integral with generic point $\xi$, and if both $S$ and the morphism $T\to S$ are normal, then one gets the isomorphism ${\pi }^s(T_{\xi }∕\xi )\simeq {\pi }^s{(T∕S)}_{\xi }$, on the strength of which one can often extend to $S$ results previously proven when the base is a field.

We obtain the last of the standard Kuznetsov formulas for $\text{SL}(3,\mathbb{Z})$. In the previous paper, we were able to exploit the relationship between the positive-sign Bessel function and the Whittaker function to apply Wallach’s Whittaker expansion; now we demonstrate the expansion of functions into Bessel functions for all four signs, generalizing Wallach’s theorem for $\text{SL}(3)$. As applications, we again consider the Kloosterman zeta functions and smooth sums of Kloosterman sums. The new Kloosterman zeta functions pose the same difficulties as we saw with the positive-sign case, but for the negative-sign case, we obtain some analytic continuation of the unweighted zeta function and give a sort of reflection formula that exactly demonstrates the obstruction to further continuation arising from the Kloosterman sums whose moduli are far apart. The completion of the remaining sign cases means this work now both supersedes the author’s thesis and completes the work started in the original paper of Bump, Friedberg and Goldfeld.

Under relatively mild and natural conditions, we establish integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the congruence ideal controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch–Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modular form, the cotangent complex of the corresponding Hecke algebra and the archimedean regulator attached to some Beilinson–Flach element.

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo–Serganova functor. Given a simple $\mathfrak{𝔭}(n)$-module $L$ and a certain odd element $x\in \mathfrak{𝔭}(n)$ of rank $1$, we give an explicit description of the composition factors of the $\mathfrak{𝔭}(n-1)$-module ${\text{DS}}_x(L)$, which is defined as the homology of the complex

$$\mathrm{\Pi }M\underset{}{\overset{x}{\to }}M\underset{}{\overset{x}{\to }}\mathrm{\Pi }M,$$

where $\mathrm{\Pi }$ denotes the parity-change functor $(-)\otimes {ℂ}^{0|1}$.

In particular, we show that this $\mathfrak{𝔭}(n-1)$-module is multiplicity-free.

We then use this result to give a simple explicit combinatorial formula for the superdimension of a simple integrable finite-dimensional $\mathfrak{𝔭}(n)$-module, based on its highest weight. In particular, this reproves the Kac–Wakimoto conjecture for $\mathfrak{𝔭}(n)$, which was proved earlier by the authors.

We define the covering gonality and separable covering gonality of varieties over arbitrary fields, generalizing the definition given by Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery for complex varieties. We show that, over an algebraically closed field, a smooth multidegree $(d_1,\dots ,d_k)$ complete intersection in ${\mathbb{P}}^N$ has separable covering gonality at least $d-N+1$, where $d=d_1+\cdots +d_k$. We also show that the very general such hypersurface has covering gonality at least $\frac{1}{2}(d-N+2)$.

Using twisted sheaves, formal-local methods, and elementary transformations we show that separated algebraic spaces which are constructed as pushouts or contractions (of curves) have enough Azumaya algebras. This implies (1) Under mild hypothesis, every cohomological Brauer class is representable by an Azumaya algebra away from a closed subset of codimension $\ge 3$, generalizing an early result of Grothendieck and (2) $\text{Br}(X)={\text{Br}}^{\prime }(X)$ when $X$ is an algebraic space obtained from a quasiprojective scheme by contracting a curve. This result is valid for all dimensions but if we specialize to surfaces, it solves the question entirely: there are always enough Azumaya algebras on separated surfaces.