The Katz-Sarnak Density Conjecture states that zeros of families of $L$-functions are well-modeled by eigenvalues of random matrix ensembles. For suitably restricted test functions, this correspondence yields upper bounds for the families' order of vanishing at the central point. We generalize previous results on the $n$\textsuperscript{th} centered moment of the distribution of zeros to allow arbitrary test functions. On the computational side, we use our improved formulas to obtain significantly better bounds on the order of vanishing for cuspidal newforms, setting world records for the quality of the bounds. We also discover better test functions that further optimize our bounds. We see improvement as early as the $5$\textsuperscript{th} order, and our bounds improve rapidly as the rank grows (more than one order of magnitude better for rank 10 and more than four orders of magnitude for rank 50).
Building on the work of Iwaniec, Luo and Sarnak, we use the $n$-level density to bound the probability of vanishing to order at least $r$ at the central point for families of cuspidal newforms of prime level $N \to \infty$, split by sign. There are three methods to improve bounds on the order of vanishing: optimizing the test functions, increasing the support, and increasing the $n$-level density studied. Previous work determined the optimal test functions for the $1$ and $2$-level densities in certain support ranges, leading to marginal improvements in bounds and making it not a productive av…
Pseudonullity for fine Selmer groups over multiple $\mathbb{Z}_p^{d}$-extensions
Morningside Center of Mathematics, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Recently, the problem of bounding the sup norms of $L^2$-normalized cuspidal automorphic newforms $ϕ$ on $\text{GL}_2$ in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character $χ$ of $ϕ$ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general $χ$. If the level $N$ is a square, our result reduces to $$\|ϕ\|_\infty \ll N^{\frac14+ε},$$ at least under the Ramanujan Conjecture. In particular, when $χ$ has conductor $N$, this improves upon the previous best known bound…
We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.
Let $\mathitΠ$ be a cohomological irreducible cuspidal automorphic representation of ${\rm GL}_2(\mathbb{A}_{\mathbb F})$ with central character $ω_{\mathitΠ}$ over a totally real number field ${\mathbb F}$. In this paper, we prove the algebraicity of the near central non-critical value of the symmetric fourth $L$-function of $\mathitΠ$ twisted by $ω_{\mathitΠ}^{-2}$. The algebraicity is expressed in terms of the Petersson norm of the normalized newform of $\mathitΠ$ and the top degree Whittaker period of the Gelbart-Jacquet lift ${\rm Sym}^2\mathitΠ$ of $\mathitΠ$.
On the moments of one-level densities in families of holomorphic cusp forms in the level aspect
Massachusetts Institute of Technology（麻省理工学院）；University of California, San Diego（加州大学圣迭戈分校）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Michigan, Ann Arbor（密歇根大学安娜堡分校）；University College London（伦敦大学学院）；Williams College（威廉姆斯学院）；University of California, Berkeley（加州大学伯克利分校）；University of Georgia（佐治亚大学）；Stanford University（斯坦福大学）
We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We g…
Let $D$ be an indefinite quaternion division algebra over $\mathbb{Q}$. We approach the problem of bounding the sup-norms of automorphic forms $ϕ$ on $D^\times(\mathbb{A})$ that belong to irreducible automorphic representations and transform via characters of unit groups of orders of $D$. We obtain a non-trivial upper bound for $\|ϕ\|_\infty$ in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for $\|ϕ\|_\infty$ in the setting of newforms for Eichler orders. In the special case when the index of the order in a maximal order is …
