Let $S_k^\sigma(N)$ denote the space of cusp forms of level $N$, weight $k$, and Atkin-Lehner sign pattern $\sigma$, and $S_k^{\mathrm{new},\sigma}$ denote its new subspace. In this paper, we study the asymptotic behavior of the coefficients of the $m$-th Hecke polynomial over $S_k^\sigma(N)$ and $S_k^{\mathrm{new},\sigma}(N)$. In particular, we show that in certain settings, all but finitely many of these coefficients take a particular sign. We also study settings in which the coefficients do not tend to any particular sign.
Let $H_k$ be an $L^2$-normalized Hecke basis for the space of all holomorphic cusp forms of weight $k$. We show that $\max_{f\in H_k}\Vert F\Vert_6\gg (\log\log k)^{\frac{1}{2}}$ where $F(z)=(\Im z)^{\frac{k}{2}}f(z).$ This confirms that the $L^6$-norm of Hecke eigenforms does not converge uniformly as the weight goes...
For a fixed prime $p$, let $\mu_p$ denote the $p$-adic Plancherel measure. Then the first main goal of this paper is to prove effective (and moreover explicit) $\mu_p$-equidistribution of the $p$-th Hecke eigenvalues over the Atkin--Lehner subspaces $S_k^\sigma(N) \subseteq S_k(N)$ and $S_k^{\operatorname{new}, \sigma}...
Aarya J. Kumar, Sargam Mondal, Erick Ross et al.· 2 citations
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the class...
We study the distribution of signs of the (real-valued) Hecke eigenvalues $A(m,1,...,1)$ of self-dual Hecke--Maass cusp forms for the group $\mathrm{SL}_n(\mathbb Z)$, where $n\geq 2$ is an integer. Our main result establishes, under Generalised Ramanujan--Petersson Conjecture, that for almost all $x\sim X$ the short i...
We prove that for every fixed $\lambda>0$ and all sufficiently large $n$, any $z_1,\dots,z_n\in\C$ with $|z_j|\geq1$ satisfy $\max_{2\leq k\leq n+1}|\sum_j z_j^k|>e^{-\lambda n}$. Consequently, the $n$th root of the optimal maximum tends to $1$, so no constant $C>1$ in Erd\H{o}s 973 can exist. The proof combines a trun...
We prove the Feldman--Ilmanen--Knopf conjecture for finite time Type I singularities of the K\"ahler--Ricci flow. More precisely, for any compact K\"ahler manifold $Y$ and its blow-up $\pi:\operatorname{Bl}_pY\longrightarrow Y$, if $[\omega_0]-Tc_1(M)=\pi^*[\omega_Y]$, then any Type I parabolic blow-up limit of the K\"...
Tong-Xin Xu, Zhen-Lei Zhang· 1 citation
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