An Explicit CM Type Norm Formula and Effective Nonvanishing of Class Group L-functions for CM Fields

We show that the central value of class group L-functions of CM fields can be expressed in terms of derivatives of real-analytic Hilbert Eisenstein series at CM points. Then, following an idea of Iwaniec and Kowalski we obtain a conditional explicit lower bound of class numbers of CM fields under a weaker assumption. Some results in the proof lead to an effective nonvanishing result for class group L-functions of general CM fields, generalizing the only known ineffective results.Comment: Some typos are corrected. To appear in the Pacific Journal of Mat

Relative Trace Formula, Subconvexity and Quantitative Nonvanishing of Rankin-Selberg LL-functions for GL(n+1)×GL(n)\mathrm{GL}(n+1)\times\mathrm{GL}(n)

Let $\pi'$ be a fixed unitary cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ We establish a subconvex bound in the $t$-aspect $$L(1/2+it,\pi\times\pi')\ll_{\pi,\pi',\varepsilon}(1+|t|)^{\frac{n(n+1)}{4}-\frac{1}{4\cdot (4n^2+2n-1)}+\varepsilon},$$ for any unitary pure isobaric automorphic representation $\pi$ of $\mathrm{GL}(n+1)/\mathbb{Q}.$ Moreover, the bound improves in the standard $L$-function case $$L(1/2+it, \pi')\ll_{\pi',\varepsilon}(1+|t|)^{\frac{n}{4}-\frac{1}{4(n+1)(4n-1)}+\varepsilon}.$$ We also prove an explicit lower bound for nonvanishing of central $L$-values $$\sum_{\pi\in\mathcal{A}_0}\textbf{1}_{L(1/2,\pi\times\pi')\neq 0}\gg_{\varepsilon}|\mathcal{A}_0|^{\frac{1}{n(n+1)(4n^2+2n-1)}-\varepsilon},$$ for a suitable finite family $\mathcal{A}_0$ of unitary cuspidal representations of $\mathrm{GL}(n+1)/\mathbb{Q}.$ More generally, we address the spectral side subconvexity in the case of uniform parameter growth, and a quantitative form of simultaneous nonvanishing of central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over $\mathbb{Q}$) in both level and eigenvalue aspects. Among other ingredients, our proofs employ a new relative trace formula in conjunction with P. Nelson's construction of archimedean test functions in \cite{Nel21} and volume estimates in \cite{Nel20}.Comment: 79 page

Summing Hecke Eigenvalues over Polynomials

In this paper we estimate sums of the form ∑n≤X|aSymmπ(|f(n)|)|, for symmetric power lifts of automorphic representations π attached to holomorphic forms and polynomials f(x)∈Z[x] of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when deg(f)≤4. Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves

Non-vanishing of twists of GL4(AQ)\text{GL}_4(\mathbb{A}_{\mathbb{Q}}) LL-functions

Let $\pi$ be a unitary cuspidal automorphic representation of $\text{GL}_{4}(\mathbb{A}_{\mathbb{Q}})$. Let $f \geq 1$ be given. We show that there exists infinitely many primitive even (resp. odd) Dirichlet characters $\chi$ with conductor co-prime to $f$ such that $L(s, \pi \otimes \chi)$ is non-vanishing at the central point. Our result has applications for the construction of $p$-adic $L$-functions for $\text{GSp}_{4}$ following Loeffler-Pilloni-Skinner-Zerbes, the Bloch-Kato conjecture and the Birch-Swinnerton-Dyer conjecture for abelian surfaces following Loeffler-Zerbes, strong multiplicity one results for paramodular cuspidal representations of $\text{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ and the rationality of the central values of $\text{GSp}_{4}(\mathbb{A}_{\mathbb{Q}})$ $L$-functions in the remaining non-regular weight case.Comment: 45 page

A Coarse Jacquet-Zagier Trace Formula for GL(n) with Applications

In this thesis we establish a coarse Jacquet-Zagier trace identity fo GL(n). This formula connects adjoint L-functions on GL(n) with Artin L-functions attached to certain induced Galois representations. We prove the absolute convergence when Re(s) &gt; 1, and obtain holomorphic continuation under almost all character twists. Moreover, as an application, we obtain that holomorphy of certain adjoint L-functions for GL(n) implies Dedekind conjecture of degree n. Some nonvanishing results are also proved.</p

Modular development of deep potential for complex solid solutions

The multicomponent oxide solid solution is a versatile platform to tune the delicate balance between competing spin, charge, orbital, and lattice degrees of freedom for materials design and discovery. The development of compositionally complex oxides with superior functional properties has been largely empirical and serendipitous, in part due to the exceedingly complex chemistry and structure of solid solutions that span a range of length scales. The classical molecular dynamics (MD), as a powerful statistical method to investigate materials properties over large spatial and temporal scales, often plays a secondary role in computer-aided materials discovery because of the limited availability and accuracy of classical force fields. Here, we introduce the strategy of ``modular developing deep potential" (ModDP) that enables a systematic development and improvement of deep neural network-based model potential, termed as deep potential, for complex solid solutions with minimum human intervention. The converged training database associated with an end-member material is treated as an independent module and is reused to train the deep potential of solid solutions via a concurrent learning procedure. We apply ModDP to obtain classical force fields of two technologically important solid solutions, Pb$_x$Sr$_{1-x}$TiO$_3$ and Hf$_x$Zr$_{1-x}$O$_2$. For both materials systems, a single model potential is capable of predicting various properties of solid solutions including temperature-driven and composition-driven phase transitions over a wide range of compositions. In particular, the deep potential of Pb$_x$Sr$_{1-x}$TiO$_3$ reproduces a few known topological textures such as polar vortex lattice and electric dipole waves in PbTiO$_3$/SrTiO$_3$ superlattices, paving the way for MD investigations on the dynamics of topological structures in response to external stimuli.Comment: 32 pages, 9 figure