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.

HECKE OPERATORS AND THE COHERENT COHOMOLOGY OF SHIMURA VARIETIES

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain p-adic estimates of Satake parameters of certain nonregular self-dual automorphic representations of $\mathrm {GL}_n$ .

Average Bound Toward the Generalized Ramanujan Conjecture and Its Applications on Sato–Tate Laws for GL(n)

We give the 1st non-trivial estimate for the number of $GL(n)$ ($n\ge 3$) Hecke–Maass forms whose Satake parameters at any given prime $p$ fail the Generalized Ramanujan Conjecture and study some applications on the (vertical) Sato–Tate laws.

On the equality case of the Ramanujan Conjecture for Hilbert modular forms

The generalized Ramanujan Conjecture for cuspidal unitary automorphic representations [Formula: see text] on [Formula: see text] posits that [Formula: see text]. We prove that this inequality is strict if [Formula: see text] is generated by a Hilbert modular form of weight two, with complex multiplication, and [Formula: see text] is a finite place of degree one. Equivalently, the Satake parameters of [Formula: see text] are necessarily distinct. We also give examples where the equality case does occur for places [Formula: see text] of degree two.

Plancherel Distribution of Satake Parameters of Maass Cusp Forms on GL3

We prove an equidistribution result for the Satake parameters of Maass cusp forms on $\operatorname{GL}_{3}$ with respect to the p-adic Plancherel measure by using an application of the Kuznetsov trace formula. The techniques developed in this paper deal with the removal of arithmetic weight $L(1,F,\operatorname{Ad})^{-1}$ in the Kuznetsov trace formula on $\operatorname{GL}_{3}$.

SUR LA TORSION DANS LA COHOMOLOGIE DES VARIÉTÉS DE SHIMURA DE KOTTWITZ-HARRIS-TAYLOR

(Torsion in the cohomology of Kottwitz–Harris–Taylor Shimura varieties) When the level at $l$ of a Shimura variety of Kottwitz–Harris–Taylor is not maximal, its cohomology with coefficients in a $\overline{\mathbb{Z}}_{l}$-local system isn’t in general torsion free. In order to prove torsion freeness results of the cohomology, we localize at a maximal ideal $\mathfrak{m}$ of the Hecke algebra. We then prove a result of torsion freeness resting either on $\mathfrak{m}$ itself or on the Galois representation $\overline{\unicode[STIX]{x1D70C}}_{\mathfrak{m}}$ associated to it. Concerning the torsion, in a rather restricted case than Caraiani and Scholze (« On the generic part of the cohomology of compact unitary Shimura varieties », Preprint, 2015), we prove that the torsion doesn’t give new Satake parameters systems by showing that each torsion cohomology class can be raised in the free part of the cohomology of a Igusa variety.

Local spectral equidistribution for degree two Siegel modular forms in level and weight aspects

We prove an equidistribution result for the Satake parameters of the local representations attached to Siegel cusp forms of degree 2 of increasing level and weight, counted with a certain arithmetic weight. We then apply this to compute the symmetry type of a similarly weighted distribution of the low-lying zeros of L-functions attached to these cusp forms.