Spectral aspect subconvex bounds for ${\rm U}_{n+1} \times {\rm U}_{n}$

Abstract

Let (π, σ) traverse a sequence of pairs of cuspidal automorphic representations of an anistropic unitary Gan-Gross-Prasad pair (U n+1 , Un) over a number field. We assume that at some distinguished archimedean place, the pair stays away from the conductor dropping locus, while at every other place, the pair has bounded ramification and satisfies certain local conditions (in particular, temperedness). We prove that the subconvex boundholds for any fixed δ < 1 16n 5 + 56n 4 + 84n 3 + 72n 2 + 28n .Among other ingredients, the proof employs a refinement of the microlocal calculus for Lie group representations developed with A. Venkatesh and an observation of S. Marshall concerning the geometric side of the relative trace formula.

Hecke-Maass cusp forms on PGL_n of large levels with non-vanishing L-values

“…An advantage of our method is that one can deal with arbitrary GL n uniformly by a relatively soft method except that the exact evaluation of local zeta-integrals is required to compute the main term. We should mention recent researches [7], [29] and [54] done in a similar spirit to our work dealing with analytic problems related to automorphic forms on GL(n) by a soft method.…”

The density conjecture for principal congruence subgroups

“…It should be possible to treat these using more general constructions via induction from open compact subgroups. The methods in [NV] provide robust tools in the archimedean (microlocal) case, and it should be possible to translate them to the p-adic setting, see also [Ne,Section 2.5]. We hope to return to this in the future.…”

The second moment of $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions

“…dilated) or high center often exclude these narrow classes and thus, does not usually become fruitful to yield subconvex bound of an L-function which has conductor drop, see e.g. [2,26,25].…”

“…Previously, in [26] Nelson-Venkatesh asymptotically evaluated the first moment keeping Π fixed and letting π vary over a dilated Plancherel ball when (G, H) are Gan-Gross-Prasad pairs, more interestingly, allowing arbitrary weights in the spectral side. More recently in [25] Nelson proved a Lindelöf-consistent upper bound of the second moment for the groups (G, H) = (U(n + 1), U(n)) in the split case keeping π fixed and letting Π vary over a Plancherel ball with high center. The method in [25] also yields an asymptotic formula with power savings of a specific weighted second moment over this family.…”

“…More recently in [25] Nelson proved a Lindelöf-consistent upper bound of the second moment for the groups (G, H) = (U(n + 1), U(n)) in the split case keeping π fixed and letting Π vary over a Plancherel ball with high center. The method in [25] also yields an asymptotic formula with power savings of a specific weighted second moment over this family. Blomer in [1] obtained a Lindelöf-consistent upper bound of the second moment for G = H = GL(n) keeping Π a fixed cuspidal representation and letting π vary in a Plancherel ball, however, his method does not yield an asymptotic formula.…”