arXiv subjects
Publications and source records attributed to Shenghao Hua.
Let $p_j$ be the prime with index $j$. We prove that the ratios $m/n$ for which $p_m+p_n$ is a square are dense in $\mathbb R_{>0}$. The same is true when $|p_m-p_n|$ is a square. In every nonempty open interval, almost every index can be used as a numerator and as a denominator. We also give a quantitative lower bound for the number of possible partners.
Prime coefficients of elliptic curves exhibit murmurations, statistical patterns that support the prediction of arithmetic labels. We conjecture that root number weighted averages of unitary normalized coefficients at primes and prime powers sample the same leading profile when placed at the effective position $p^k/X$, where $X$ is the conductor scale. For weight $2$ newforms of squarefree level in the level aspect, we prove this principle for every fixed $k$ under suitable short-window and growth conditions, extending Zubrilina's prime case and the square-case analysis of Kundu and Müller. Experiments with elliptic curve isogeny class representatives show that the resulting prime power features improve root number prediction and give a smaller gain in distinguishing ranks $0$ and $1$.
We introduce a special class of multiple Dirichlet series whose terms are supported on a variety and which admit an Euler product structure.
Let $f$ be a primitive holomorphic newform of weight at least $2$ and arbitrary level. We consider the representations obtained from its symmetric powers by taking tensor products, symmetric powers, and isobaric sums. Classical $\mathrm{GL}(2)$ representation theory decomposes each such expression into symmetric powers with determinant twists. This gives an automorphic realization and a factorization of the associated $L$-function. At a prime dividing the level, the same decomposition must be applied to the full Weil-Deligne parameter before inertia invariants and the kernel of monodromy are taken. For elliptic curves over $\mathbb{Q}$, we record the resulting local factors, conductors, and root numbers using the calculations of Dummigan-Martin-Watkins and Martin-Watkins.
In this paper, we study quadratic forms in spaces of holomorphic cusp forms. We show, conditionally, that when two quadratic forms in Hecke eigenforms share no common diagonal terms, their inner product is expected to converge to the sum of the products of their common off-diagonal coefficients. This phenomenon could be interpreted as a mixed $L^4$-norm problem. We also define the $\ell^p$-norm of a holomorphic cusp form via its expansion with respect to an orthonormal Hecke basis. We then establish a conditional upper bound for the $\ell^p$-norm, and deduce that the coefficients of quadratic forms of holomorphic cusp forms in the Hecke basis are not uniformly small, being dominated by small-amplitude components. This behavior is consistent with the expected distribution of orthogonal families of $L$-functions.
There is a one-sided central limit theorem for the logarithms of $L$-derivatives of a fixed rational non-CM elliptic curve $E$ over imaginary quadratic fields of rank one, analogous to a result of Radziwiłł and Soundararajan. There are also many $L$-derivatives that are not small.
The Keating--Snaith conjecture for orthogonal families may be viewed as analogous to a Gaussian distribution with a negative mean, and the possibility that mixed moments resemble a composition of independent moments, these two insights were combined and applied in Lester and Radziwi{łł}'s proof of quantum unique ergodicity for half-integral weight automorphic forms, via Soundararajan's method under the Generalized Riemann Hypothesis (GRH). This observation also yields a crucial and nontrivial saving in the resolution of certain arithmetic problems. Inspired by this, we select a series of typical mixed orthogonal families of $L$-functions: $\mathrm{GL}_2$ quadratic twisted families, Gao and Zhao established a sharp upper bound by building upon Harper's method, and one can replace square-free numbers with primes in this argument. Under the assumptions of the GRH and the Generalized Ramanujan Conjecture, we present the following three arithmetic applications: i) The decorrelation of Fourier coefficients of half-integral weight modular forms, specifically, a variant of Selberg orthogonality for distinct half-integral weight modular forms. ii) The decorrelation of automorphic periods averaged over prime imaginary quadratic fields. iii) The decorrelation of the analytic orders of isotropy subgroups of Tate--Shafarevich groups of elliptic curves under prime quadratic twists.
We demonstrate that for symmetric cubes of algebraic regular Hecke eigenforms on $\mathrm{GL}(2)$, a decorrelation phenomenon occurs for global Bessel periods of $\mathrm{SO}(5)\times \mathrm{SO}(2)$ averaged over imaginary quadratic fields, conditional on the Generalized Riemann Hypothesis.
In this paper we prove a lower bound result for extremely large values of $L(\frac{1}{2},χ_p)$ with prime numbers $p\equiv 1\pmod 8$.
In this paper, we formulate a conjecture on joint distribution of Hecke--Maass cusp forms. To support our conjecture, we prove two conditional results on joint moments of two Hecke--Maass cusp forms, which confirms statistical independence of orthogonal cusp forms.
In this paper, we prove the extreme values of $L$-functions at the central point for almost prime quadratic twists of an elliptic curve. As an application, we get the extreme values for the Tate--Shafarevich groups in the quadratic twist family of an elliptic curve under the Birth--Swinnerton-Dyer conjecture.
In this paper, we prove the conjectured order lower bound for the $k$-th moment of central values of quadratic twisted self-dual $\textrm{GL}(3)$ $L$-functions for all $k\geq 1$, based on our recent work on the twisted first moment of central values in this family of $L$-functions.
Let $ϕ$ and $ϕ'$ be two $\textrm{GL}(3)$ Hecke--Maass cusp forms. In this paper, we prove that $ϕ=ϕ'\textrm{ or }\widetilde{ϕ'}$ if there exists a nonzero constant $κ$ such that $$L(\frac{1}{2},ϕ\otimes χ_{8d})=κL(\frac{1}{2},ϕ'\otimes χ_{8d})$$ for all positive odd square-free positive $d$. Here $\widetilde{ϕ'}$ is dual form of $ϕ'$ and $χ_{8d}$ is the quadratic character $(\frac{8d}{\cdot})$. To prove this, we obtain asymptotic formulas for twisted first moment of central values of quadratic twisted $L$-functions on $\textrm{GL}(3)$, which will have many other applications.