Valentin Blomer, Péter Maga
Let F be a Hecke-Maass cusp form for the group SL(4, Z) with Laplace eigenvalue lambda. Assume that F satisfies the Ramanujan conjecture at infinity (this is satisfied by almost all cusp forms). We show a power-saving sup-norm bound in terms of lambda (relative to the generic bound) for F restricted to a compact set.
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We prove the subconvexity conjecture for sup-norms of automorphic forms for congruence subgroups of SL(n, Z) that satisfy the Ramanujan conjecture at infinity.
Let $ϕ$ be an $L^2$-normalized Hecke--Maaß cusp form for $\mathrm{PGL}_n(\mathbb{Z}[i])$ on the locally symmetric space $X:=\mathrm{PGL}_n(\mathbb{Z}[i])\backslash \mathrm{PGL}_n(\mathbb{C}) / \mathrm{PU}_n$. If $Ω$ is a compact subset of $X$, then we prove the bound $\|ϕ|_Ω\|_{\infty}\ll_Ω λ_ϕ^{n(n-1)/4-δ}$ for some $δ>0$ depending only on $n$, where $λ_ϕ$ is the Laplace eigenvalue of $ϕ$.
In this short paper we give the sub-local upper bound for the sup norm of an automorphic form on $\text{PGL}_n$, whose associated automorphic representation has finite conductor $C(π)=p^c$ with $c\rightarrow \infty$, and its local component at the place of ramification is a minimal vector belonging to an irreducible representation with generic induction datum.
Blomer and Maga recently proved that, if $F$ is an $L^2$-normalized Hecke Maass cusp form for $\mathrm{SL}_n(\mathbb Z)$, and $Ω$ is a compact subset of $\mathrm{PGL}_n(\mathbb R)/\mathrm{PO}_n(\mathbb R)$, then we have $\|F|_Ω\|_\infty\ll_Ωλ_F^{n(n-1)/8-δ_n}$ for some $δ_n>0$, where $λ_F$ is the Laplacian eigenvalue of $F$. In the present paper, we prove an explicit version of their result.
The sup-norm problem for newforms of large level on $\operatorname{PGL}(n)$
本文研究了大水平新形式在PGL(n)上的上范数问题，通过新的降维理论和Atkin-Lehner算子推广，证明了次下界并基于GRH条件扩展结果。
The orbit method in number theory through the sup-norm problem for $\operatorname{GL}(2)$
本文通过轨道方法研究GL(2)的supnorm问题，证明了新形式的混合界并改进了非紧致设置下的局部界。
Vertex volumes, lattice-minima tails, and height zeta functions for the standard arithmetic quotient of $\operatorname{PGL}_d$
该研究针对 $\text{PGL}_d$ 的标准算术商，基于建筑理论计算顶点体积，引入同伦不变归一性格极小值高度，确定其可积阈值与尖点尾估计，推导高度泽塔函数的收敛域与解析延拓性质，明确计算了 $d=3,4,5$ 时的有理函数。
On the global sup-norm of GL(3) cusp forms
本文研究了非紧空间上GL(3)尖点形式的全局上确界范数问题，通过建立关于Laplace特征值的逐点上界，证明了tempered形式的全局上确界范数界，并推导出GL(3) Jacquet-Whittaker函数的新一致上界。
Cohn and Kumar showed the quadratic character of $q$ modulo $p$ gives the sign of the permutation of Hurwitz primes of norm $p$ induced by the Hurwitz primes of norm $q$ under metacommutation. We demonstrate that these permutations are equivalent to those induced by the right standard action of $\operatorname{PGL}_2 (\mathbb{F}_p)$ on $\mathbb{P}^1 (\mathbb{F}_p)$. This equivalence provides simpler proofs of the results of Cohn and Kumar and characterizes the cycle structure of the aforementioned permutations. Our methods are general enough to extend to all orders over the quaternions with a d…
Let G be a connected simple adjoint p-adic group not isomorphic to a projective linear group PGL(m,D) of a division algebra D, or an adjoint ramified unitary group of a split hermitian form in 3 variables. We prove that G admits an irreducible admissible supercuspidal (i.e. supersingular) representation over any field of characteristic p.
Let $π$ be a cuspidal automorphic representation of $PGL_2(\mathbb{A}_\mathbb{Q})$ of arithmetic conductor $C$ and archimedean parameter $T$, and let $ϕ$ be an $L^2$-normalized automorphic form in the space of $π$. The sup-norm problem asks for bounds on $\| ϕ\|_\infty$ in terms of $C$ and $T$. The quantum unique ergodicity (QUE) problem concerns the limiting behavior of the $L^2$-mass $|ϕ|^2 (g) \, d g$ of $ϕ$. All previous work on these problems in the conductor-aspect has focused on the case that $ϕ$ is a newform. In this work, we study these problems for a class of automorphic forms that a…
Classical Hecke operators on Maass forms are unitarely equivalent, up to a commuting phase, to completely positive maps on II$_1$ factors, associated to a pair of isomorphic subfactors, and an intertwining unitary. This representation is obtained through a quantized representation of the Hecke operators. in this representation, the Hecke operators act on the Berezin's quantization, deformation algebra of the fundamental domain of $\PSL(2,\Z)$ in the upper halfplane. The Hecke operators are inheriting from the ambient, non-commutative algebra on which they act, a rich structure of matrix inequa…
We develop an algorithm that determines, for a given squarefree binary form $F$ with real coefficients, a smallest representative of its orbit under $\operatorname{SL}(2,\mathbb Z)$, either with respect to the Euclidean norm or with respect to the maximum norm of the coefficient vector. This is based on earlier work of Cremona and Stoll. We then generalize our approach so that it also applies to the problem of finding an integral representative of smallest height in the $\operatorname{PGL}(2,\mathbb Q)$ conjugacy class of an endomorphism of the projective line. Having a small model of such an …
The cubic moment of $L$-functions for specified local component families
本文研究了 $L$-函数中心值的立方矩在特定局部分量家族上的 Lindelöf 平均上界，通过新发展的 Petersson/Bruggeman-Kuznetsov 公式，推导了 Weyl 强子凸界，尤其在平方满和深度方面。
Automorphic functions for square-zero extensions of curves over finite fields
该研究针对有限域上曲线的平方零扩张，推广了PGL₂的自守函数相关结果，证明了PGL₃下某猜想的新案例，还确定了对应球面尖与赫克有限函数的最优支撑界。
An approximate form of Artin's holomorphy conjecture and non-vanishing of Artin $L$-functions
该研究证明了对称群或素数次传递群对应的数域族中，除极少例外，阿廷$L$-函数存在符合猜想预测的全纯非零区域，并给出类群大小、周期轨道分布等四个重要数论应用。
Analytic properties of spherical cusp forms on GL(n)
本文研究了 $\mathrm{PGL}_n$ 上球型尖点自守形式，通过结合分析与算术工具，建立了与 Laplace 特征值一致相关的显式估计，包括 Whittaker 函数、全局上确界范数及本质支撑的一致界。
Branch points of split degenerate superelliptic curves II: on a conjecture of Gerritzen and van der Put
本文针对 Gerritzen 和 van der Put 关于分裂退化超椭圆曲线分支点簇数据的猜想，指出其需稍作修改，并证明了修改后猜想的更强版本，适用于任意素数 p 和任意剩余特征。
Bounding sup-norms of modular forms in terms of the level has been the focus of much recent study. In this work the sup norm of a half integral weight cusp form is bounded in terms of the level.
This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set.
We prove a sup norm bound on the real-analytic Eisenstein series, of the form $E(z, 1/2 + iT) \ll T^{3/8 + \varepsilon}$, uniformly for $z$ in a fixed compact subset of $\mathbb{H}$.
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