Weil Formula Research Articles

Zero-Dimensional Shimura Varieties and Central Derivatives of Eisenstein Series

Abstract We formulate and prove a version of the arithmetic Siegel–Weil formula for (zero dimensional) Shimura varieties attached to tori, equipped with some additional data. More precisely, we define a family of “special” divisors in terms of Green functions at archimedean and non-archimedean places and prove that their degrees coincide with the Fourier coefficients of the central derivative of an Eisenstein series. The proof relies on the usual Siegel–Weil formula to provide a direct link between both sides of the identity, and in some sense, offers a more conceptual point of view on prior results in the literature.

Higher Siegel–Weil formula for unitary groups: the non-singular terms

We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the rth\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r^{\\mathrm{th}}$\\end{document} central derivative of non-singular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of “virtual dimension 0” on the moduli stack of hermitian shtukas with r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r$\\end{document} legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series.

Diagonal restriction of Eisenstein series and Kudla–Millson theta lift

Abstract We consider the Kudla–Millson theta series associated to a quadratic space of signature ( N , N ) {(N,N)} . By combining a “see-saw” argument with the Siegel–Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree N is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon, Pozzi and Vonk to totally real fields.

Kudla–Rapoport conjecture for Krämer models

In this paper, we propose a modified Kudla–Rapoport conjecture for the Krämer model of unitary Rapoport–Zink space at a ramified prime, which is a precise identity relating intersection numbers of special cycles to derivatives of Hermitian local density polynomials. We also introduce the notion of special difference cycles, which has surprisingly simple description. Combining this with induction formulas of Hermitian local density polynomials, we prove the modified Kudla–Rapoport conjecture when $n=3$. Our conjecture, combining with known results at inert and infinite primes, implies the arithmetic Siegel–Weil formula for all non-singular coefficients when the level structure of the corresponding unitary Shimura variety is defined by a self-dual lattice.

Phytochemical Analysis of Bulung Boni (Caulerpa cylindracea S.) N- Hexane Extract with Gc-Ms Method and Toxicity Test on Mice (Mus musculus L.)

One of the marine resource commodities owned by Indonesia is seaweed. Bulung boni is one type of seaweed that grown in Indonesia. Bulung boni is the regional name of Caulerpa cylindracea. Bulung boni is usually used as a food and medicine for most people in Bali. The purpose of this research is to study the content of phytochemical compounds in n-hexane extract of bulung boni and determine the toxicity effect on mice (Mus musculus). The methods used in this study were maceration, GC-MS (Gas Chromatography-Mass Spectrometry), observation, and calculation with the Thomson and Weil formula. This study used 24 mice as experimental animals. Mice were divided into 6 treatment groups, group 1 as control was given 1% Na-CMC solution, groups 2-6 were given test extracts with doses, 100, 200, 300, 400, 500 mg/30 g BW mice. The results showed the highest compound contained in the n-hexane extract of bulung boni was Hexadecanoic acid, methyl ester with an AUC value of 35.21%. Toxicity tests on mice showed no deaths experienced by mice, so the LD50 value determined was pseudo at 16.6 g/ kg BW and fell into the practically non-toxic category

Calabi–Yau metrics with conical singularities along line arrangements

Given a finite collection of lines $L_j \subset \mathbb{CP}^2$ together with real numbers $0 \lt \beta_j \lt 1$ satisfying natural constraint conditions, we show the existence of a Ricci–flat Kähler metric $g_{RF}$ with cone angle $2\pi\beta_j$ along each line $L_j$ asymptotic to a polyhedral Kähler cone at each multiple point. Moreover, we discuss a Chern–Weil formula that expresses the energy of $g_{RF}$ as a logarithmic Euler characteristic with points weighted according to the volume density of the metric.

From sum of two squares to arithmetic Siegel–Weil formulas

The main goal of this expository article is to survey recent progress on the arithmetic Siegel–Weil formula and its applications. We begin with the classical sum of two squares problem and put it in the context of the Siegel–Weil formula. We then motivate the geometric and arithmetic Siegel–Weil formula using the classical example of the product of modular curves. After explaining the recent result on the arithmetic Siegel–Weil formula for Shimura varieties of arbitrary dimension, we discuss some aspects of the proof and its application to the arithmetic inner product formula and the Beilinson–Bloch conjecture. Rather than being intended as a complete survey of this vast field, this article focuses more on examples and background to provide easier access to several recent works by the author with W. Zhang and Y. Liu.

Residues of connections and the Chevalley–Weil formula for curves

Given a finite group of automorphisms of a compact Riemann surface, the Chevalley– Weil formula computes the character valued Euler characteristic of an equivariant line bundle. The goal of this article is to give a proof by computing using residues of a Gauss–Manin connection.

On the subring of special cycles

Abstract By old results with Millson, the generating series for the cohomology classes of special cycles on orthogonal Shimura varieties over a totally real field are Hilbert–Siegel modular forms. These forms arise via theta series. Using this result and the Siegel–Weil formula, we show that the products in the subring of cohomology generated by the special cycles are controlled by the Fourier coefficients of triple pullbacks of certain Siegel–Eisenstein series. As a consequence, there are comparison isomorphisms between special subrings for different Shimura varieties. In the case in which the signature of the quadratic space V is ( m , 2 ) (m,2) at an even number d + d_{+} of archimedean places, the comparison gives a “combinatorial model” for the special cycle ring in terms of the associated totally positive definite space.

On the arithmetic Siegel–Weil formula for GSpin Shimura varieties

We formulate and prove a local arithmetic Siegel–Weil formula for GSpin Rapoport–Zink spaces, which is a precise identity between the arithmetic intersection numbers of special cycles on GSpin Rapoport–Zink spaces and the derivatives of local representation densities of quadratic forms. As a first application, we prove a semi-global arithmetic Siegel–Weil formula as conjectured by Kudla, which relates the arithmetic intersection numbers of special cycles on GSpin Shimura varieties at a place of good reduction and the central derivatives of nonsingular Fourier coefficients of incoherent Siegel Eisenstein series.

New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces

Given a domain D in \mathbb{C}^n and a compact subset K of D , the set \mathcal{A}_K^D of all restrictions of functions holomorphic on D the modulus of which is bounded by 1 is a compact subset of the Banach space C(K) of continuous functions on K . The sequence (d_m(\mathcal{A}_K^D))_{m\in \mathbb{N}} of Kolmogorov m -widths of \mathcal{A}_K^D provides a measure of the degree of compactness of the set \mathcal{A}_K^D in C(K) and the study of its asymptotics has a long history, essentially going back to Kolmogorov’s work on \epsilon -entropy of compact sets in the 1950s. In the 1980s Zakharyuta showed that for suitable D and K the asymptotics \lim_{m\to \infty}\frac{- \log d_m(\mathcal{A}_K^D)}{m^{1/n}} = 2\pi \bigg( \frac{n!}{C(K,D)} \bigg) ^{1/n}, where C(K,D) is the Bedford–Taylor relative capacity of K in D , is implied by a conjecture, now known as Zakharyuta's Conjecture, concerning the approximability of the regularised relative extremal function of K and D by certain pluricomplex Green functions. Zakharyuta’s Conjecture was proved by Nivoche in 2004 thus settling (1) at the same time. We shall give a new proof of the asymptotics (1) for D strictly hyperconvex and K non-pluripolar which does not rely on Zakharyuta’s Conjecture. Instead we proceed more directly by a two-pronged approach establishing sharp upper and lower bounds for the Kolmogorov widths. The lower bounds follow from concentration results of independent interest for the eigenvalues of a certain family of Toeplitz operators, while the upper bounds follow from an application of the Bergman–Weil formula together with an exhaustion procedure by special holomorphic polyhedra.

Kudla–Rapoport cycles and derivatives of local densities

We prove the local Kudla–Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport–Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla–Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia–Sankaran, we also prove cases of the arithmetic Siegel–Weil formula in any dimension.

Components and acute toxicity of nanoherbal haramonting (Rhodomyrtus tomentosa)

Introduction: Rhodomyrtus tomentosa (haramonting), a typical plant of North Sumatera, Indonesia, contains important medicinal ingredients. Nano sized drugs have high loading capacities and can be given at high concentrations. This study aimed to determine the components and toxicity of nanoherbal haramonting. It also aimed to determine the effect of nanoherbal haramonting on the histology of the liver, kidneys, lungs, heart and brain. Methods: High-energy milling was performed to produce nanoherbal haramonting. Thin-layer chromatography was utilised to determine the chemical components of the nanoherb. Antioxidant tests were performed by using the 1,1-diphenyl-2-picryhydrazil method. The three-stage of lethal dose 50 (LD50) which comprised the dose orientation test, preliminary test and actual phase test/LD50 determination, and the Thomson–Weil formula was applied to measure the lethal concentration 50 (LC50) of nanoherbal haramonting. Organs were collected for histological investigation after 14 days of the lethality test. Results: Nanoherbal haramonting had an average diameter distribution of 600.1 nm ± 135.8. It contained flavonoids, steroids, glycosides, saponins and tannins. Its LC50 and LD50 values were 2961.535 ppm and 10.4 ± 0.135 mg/kg BW, respectively. The histology of the heart, kidney, lungs, heart and brain were changed and affected by nanoherbal haramonting treatment at each dose level. Conclusion: Nanoherbal haramonting has strong antioxidative activity and small size, can be effectively used as medicine in the future because it contains secondary metabolite compounds that can be developed as drugs. However, it has mild toxicity.

The $$\mathrm {PGSp}_{1,1}({\mathbb {A}})$$-distinguished representations

In this paper, we use the regularized quaternionic Siegel–Weil formula to deal with the $$\mathrm {PGSp}_{1,1}({\mathbb {A}})$$ -period problem for a cuspidal representation $$\tau $$ of $$\mathrm {PGSp}_4({\mathbb {A}}_E)$$ , where E/F is a quadratic extension of number fields and E is contained in the 4-dimensional quaternion division algebra D of F with $$s_D=2$$ .

A Chern–Weil formula for the Chern character of a perfect curved module

Let $k$ be a field of characteristic 0 and $\mathcal{A}$ a curved $k$-algebra. We obtain a Chern–Weil-type formula for the Chern character of a perfect $\mathcal{A}$-module taking values in $HN^{II}\_0(\mathcal{A})$, the negative cyclic homology of the second kind associated to $\mathcal{A}$, when $\mathcal{A}$ satisfies a certain smoothness condition.

An exceptional Siegel–Weil formula and poles of the Spin L-function of

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of$\text{PGSp}_{6}$discovered by Pollack, we prove that a cuspidal representation of$\text{PGSp}_{6}$is a (weak) functorial lift from the exceptional group$G_{2}$if its (partial) Spin L-function has a pole at$s=1$.

The Weil formula and the Rouche principle in $C^n[m\times m

The Weil formula and the Rouche principle in $C^n[m\times m

On spaces associated with invariant divisors on Galois covers of Riemann surfaces and their applications

Let $f:X \to S$ be a Galois cover of Riemann surfaces, with Galois group $G$. In this paper we analyze the $G$-invariant divisors on $X$, and their associated spaces of meromorphic functions, differentials, and $q$-differentials. We generalize the trace formula for non-trivial elements of $G$ on $q$-differentials, as well as the Chevalley--Weil Formula. When $G$ is Abelian or when the genus of $S$ is 0 we prove additional results, and we also determine the non-special $G$-invariant divisors when both conditions are satisfied.

Arithmetic Siegel–Weil formula on X0(N)

In this paper, we proved an arithmetic Siegel–Weil formula and the modularity of some arithmetic theta function on the modular curve X0(N) when N is square free. In the process, we also constructed some generalized Delta function for Γ0(N) and proved some explicit Kronecker limit formula for Eisenstein series on X0(N).

Green forms and the arithmetic Siegel–Weil formula

We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p,2) and U(p,1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.