Band 2024, Heft 813

In this paper, we study 𝑝-adic endoscopy on eigenvarieties for SL 2 \mathrm{SL}_{2} over totally real fields, taking a geometric perspective. We show that non-automorphic members of endoscopic 𝐿-packets of regular weight contribute eigenvectors to overconvergent cohomology at critically refined endoscopic points on the eigenvariety, and we precisely quantify this contribution. This gives a new perspective on and generalizes previous work of the second author. Our methods are geometric, and are based on showing that the SL 2 \mathrm{SL}_{2} -eigenvariety is locally a quotient of an eigenvariety for GL 2 \mathrm{GL}_{2} , which allows us to explicitly describe the local geometry of the SL 2 \mathrm{SL}_{2} -eigenvariety. In particular, we show that it often fails to be Gorenstein.

In this paper, we compute the first and second variation of the normalized Einstein–Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres. While the situation is quite similar to the Riemannian case in dimension greater than or equal to five, in three dimensions, we observe a crucial difference, which mainly depends on the embeddable character of the perturbed CR structure.

We derive precise formulas for the archimedean Euler factors occurring in certain standard Langlands 𝐿-functions for unitary groups. In the 1980s, Paul Garrett, as well as Ilya Piatetski-Shapiro and Stephen Rallis (independently of Garrett), discovered integral representations of automorphic 𝐿-functions that are Eulerian but, in contrast to the Rankin–Selberg and Langlands–Shahidi methods, do not require that the automorphic representations to which the 𝐿-functions are associated are globally generic. Their approach, the doubling method , opened the door to a variety of applications that could not be handled by prior methods. For over three decades, though, the integrals occurring in the Euler factors at archimedean places for unitary groups eluded precise computation, except under particular simplifications (such as requiring certain representations to be one-dimensional, as Garrett did in the first major progress on this computation and only prior progress for general signatures). We compute these integrals for holomorphic discrete series of general vector weights for unitary groups of any signature. This has consequences not only for special values of 𝐿-functions in the archimedean setting, but also for 𝑝-adic 𝐿-functions, where the corresponding term had remained open.

In this paper, we use a regularized theta lifting to construct harmonic Maass forms corresponding to binary theta functions of weight k ≥ 2 k\geq 2 under the 𝜉-operator. As a result, we show that their holomorphic parts have algebraic Fourier coefficients, with compatible Galois action. As an application, we prove rationality properties of coefficients of harmonic Maass forms corresponding to CM newforms, answering a question of Bruinier, Ono and Rhoades.

Let M be a smooth, compact manifold and let 𝒩 μ {{\mathcal{N}_{\mu}}} denote the set of Riemannian metrics on M with smooth volume density μ. For g 0 ∈ 𝒩 μ {g_{0}\in{\mathcal{N}_{\mu}}} , we show that if dim ⁡ ( M ) ≥ 5 {\dim(M)\geq 5} , then there exists an open and dense subset 𝒴 g 0 ⊂ T g 0 ⁢ 𝒩 μ {{\mathcal{Y}_{g_{0}}}\subset T_{g_{0}}{\mathcal{N}_{\mu}}} (in the C ∞ {C^{\infty}} topology) so that, for each h ∈ 𝒴 g 0 {h\in{\mathcal{Y}_{g_{0}}}} , the ( 𝒩 μ , L 2 ) {({\mathcal{N}_{\mu}},L^{2})} Ebin geodesic γ h ⁢ ( t ) {\gamma_{h}(t)} with γ h ⁢ ( 0 ) = g 0 {\gamma_{h}(0)=g_{0}} and γ h ′ ⁢ ( 0 ) = h {\gamma_{h}^{\prime}(0)=h} satisfies lim t → ∞ ⁡ R ⁢ ( γ h ⁢ ( t ) ) = - ∞ {\lim_{t\to\infty}R(\gamma_{h}(t))=-\infty} , uniformly, where R : 𝒩 μ → C ∞ ⁢ ( M ) {R\colon\mathcal{N}_{\mu}\to C^{\infty}(M)} is the scalar curvature.

For a compact orientable surface Σ g , 1 \Sigma_{g,1} of genus 𝑔 with one boundary component and for an odd prime number 𝑝, we study the homology of the unordered configuration spaces C ∙ ( Σ g , 1 ) : = ∐ n ≥ 0 C n ( Σ g , 1 ) C_{\bullet}(\Sigma_{g,1}):=\coprod_{n\geq 0}C_{n}(\Sigma_{g,1}) with coefficients in F p \mathbb{F}_{p} . We describe H ∗ ⁢ ( C ∙ ⁢ ( Σ g , 1 ) ; F p ) H_{*}(C_{\bullet}(\Sigma_{g,1});\mathbb{F}_{p}) as a bigraded module over the Pontryagin ring H ∗ ⁢ ( C ∙ ⁢ ( D ) ; F p ) H_{*}(C_{\bullet}(D);\mathbb{F}_{p}) , where 𝐷 is a disc, and compute in particular the bigraded dimension over F p \mathbb{F}_{p} . We also consider the action of the mapping class group Γ g , 1 \Gamma_{g,1} and prove that the mod-𝑝 Johnson kernel K g , 1 ⁢ ( p ) ⊆ Γ g , 1 \mathcal{K}_{g,1}(p)\subseteq\Gamma_{g,1} is the kernel of the action on H ∗ ⁢ ( C ∙ ⁢ ( Σ g , 1 ; F p ) ) H_{*}(C_{\bullet}(\Sigma_{g,1};\mathbb{F}_{p})) .

In Riemannian geometry, the Cheng’s maximal diameter rigidity theorem says that if a complete n -manifold M of Ricci curvature, Ric M ≥ ( n - 1 ) {\operatorname{Ric}_{M}\geq(n-1)} , has the maximal diameter π, then M is isometric to the unit sphere S 1 n {S^{n}_{1}} . The main result in this paper is a quantitative maximal diameter rigidity: if M satisfies that Ric M ≥ n - 1 {\operatorname{Ric}_{M}\geq n-1} , diam ⁡ ( M ) ≈ π {\operatorname{diam}(M)\approx\pi} , and the Riemannian universal cover of every metric ball in M of a definite radius satisfies a Reifenberg condition, then M is diffeomorphic and bi-Hölder close to S 1 n {S^{n}_{1}} .

We produce a fully faithful functor from finite type nilpotent spaces to cosimplicial binomial rings, thus giving an algebraic model of integral homotopy types. As an application, we construct an integral version of the Grothendieck–Teichmüller group.