AUTOMORPHIC LEFSCHETZ PROPERTIES FOR NONCOMPACT ARITHMETIC MANIFOLDS

We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math.192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math.125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.

ON SOME CONSEQUENCES OF A THEOREM OF J. LUDWIG

We prove some qualitative results about the p-adic Jacquet–Langlands correspondence defined by Scholze, in the $\operatorname {\mathrm {GL}}_2(\mathbb{Q}_p )$ residually reducible case, using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration, the global p-adic Jacquet–Langlands correspondence can also deal with automorphic forms with principal series representations at p in a nontrivial way, unlike its classical counterpart.

Applications of analytic newvectors for $$\mathrm {GL}(n)$$

We provide a few natural applications of the analytic newvectors, initiated in Jana and Nelson (Analytic newvectors for $$\text {GL}_n(\mathbb {R})$$ GL n ( R ) , arXiv:1911.01880 [math.NT], 2019), to some analytic questions in automorphic forms for $$\mathrm {PGL}_n(\mathbb {Z})$$ PGL n ( Z ) with $$n\ge 2$$ n ≥ 2 , in the archimedean analytic conductor aspect. We prove an orthogonality result of the Fourier coefficients, a density estimate of the non-tempered forms, an equidistribution result of the Satake parameters with respect to the Sato–Tate measure, and a second moment estimate of the central L-values as strong as Lindelöf on average. We also prove the random matrix prediction about the distribution of the low-lying zeros of automorphic L-function in the analytic conductor aspect. The new ideas of the proofs include the use of analytic newvectors to construct an approximate projector on the automorphic spectrum with bounded conductors and a soft local (both at finite and infinite places) analysis of the geometric side of the Kuznetsov trace formula.