Subconvex bounds for Hecke–Maass forms on compact arithmetic quotients of semisimple Lie groups

Let H be a semisimple algebraic group, K a maximal compact subgroup of $$G:=H({{\mathbb {R}}})$$ G : = H ( R ) , and $$\Gamma \subset H({{\mathbb {Q}}})$$ Γ ⊂ H ( Q ) a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke–Maass forms on the locally symmetric space $$\Gamma \backslash G/K$$ Γ \ G / K to corresponding bounds on the arithmetic quotient $$\Gamma \backslash G$$ Γ \ G for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial K-types, yielding such bounds for new classes of automorphic representations. They constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.

Special geometry and the swampland

In the context of 4d effective gravity theories with 8 supersymmetries, we propose to unify, strenghten, and refine the several swampland conjectures into a single statement: the structural criterion, modelled on the structure theorem in Hodge theory. In its most abstract form the new swampland criterion applies to all 4d $$ \mathcal{N} $$ N = 2 effective theories (having a quantum-consistent UV completion) whether supersymmetry is local or rigid: indeed it may be regarded as the more general version of Seiberg-Witten geometry which holds both in the rigid and local cases.As a first application of the new swampland criterion we show that a quantum-consistent $$ \mathcal{N} $$ N = 2 supergravity with a cubic pre-potential is necessarily a truncation of a higher-$$ \mathcal{N} $$ N sugra. More precisely: its moduli space is a Shimura variety of ‘magic’ type. In all other cases a quantum-consistent special Kähler geometry is either an arithmetic quotient of the complex hyperbolic space SU(1, m)/U(m) or has no local Killing vector.Applied to Calabi-Yau 3-folds this result implies (assuming mirror symmetry) the validity of the Oguiso-Sakurai conjecture in Algebraic Geometry: all Calabi-Yau 3-folds X without rational curves have Picard number ρ = 2, 3; in facts they are finite quotients of Abelian varieties. More generally: the Kähler moduli of X do not receive quantum corrections if and only if X has infinite fundamental group. In all other cases the Kähler moduli have instanton corrections in (essentially) all possible degrees.