算术群的一个简单迹公式

Growth of cohomology of arithmetic groups and the stable trace formula: the case of $U(2,1)$

We present a strategy to use the stable trace formula to compute growth in the cohomology of cocompact arithmetic lattices in a reductive group $G$. We then implement it in the case where $G = U(2,1)$, using results of Rogwaski. This recovers a result of Marshall.

算术曲面的Grothendieck迹映射通过留数和高阶代数

Grothendieck's trace map for arithmetic surfaces via residues and higher adeles

建立了算术曲面上微分形式留数沿竖曲线的倒易律，并将曲面的Grothendieck迹图描述为一个留数之和。无穷远点，然后纳入理论和互惠法扩展到所有的曲线上的表面。讨论了算术曲面的广义对偶性的应用。

Arithmetic properties of the Frobenius traces defined by a rational abelian variety (with two appendices by J-P. Serre)

Let $A$ be an abelian variety over $\mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $ρ_A$ is open in $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. We investigate the arithmetic of the traces $a_{1, p}$ of the Frobenius at $p$ in $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ under $ρ_A$, modulo varying primes $p$. In particular, we obtain upper bounds for the counting function $\#\{p \leq x: a_{1, p} = t\}$ and we prove an Erdös-Kac type theorem for the number of prime factors of $a_{1, p}$. We also formulate a conjecture about the asymp…

On arithmetic of the superspecial locus

We develop a method for describing the Galois action on the superspecial locus of the Siegel moduli space in characteristic $p$. Using this description, we give a modern treatment for the results of Ibukiyama and Katsura [Compos. Math., 1994] concerning the $\mathbb{F}_p$-rational points and the trace of a Hecke operator of Atkin-Lehner type. This also leads to analogues with level-$N$ structure. The trace of the Hecke operator can be reduced into one term (instead of finitely many terms a priori) by the simple trace formula when $N$ is large.

Quasimodular moonshine and arithmetic connections

We prove the existence of a module for the largest Mathieu group, whose trace functions are weight two quasimodular forms. Restricting to the subgroup fixing a point, we see that the integrality of these functions is equivalent to certain divisibility conditions on the number of $\mathbb{F}_p$ points on Jacobians of modular curves. Extending such expressions to arbitrary primes, we find trace functions for modules of cyclic groups of prime order with similar connections. Moreover, for cyclic groups, we give an explicit vertex operator algebra construction whose trace functions are given only i…

Geometric Satake, categorical traces, and arithmetic of Shimura varieties

We survey some recent work on the geometric Satake of p-adic groups and its applications to some arithmetic problems of Shimura varieties. We reformulate a few constructions appeared in the previous works more conceptually.

Fourier-Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)

In this article, we develop an arithmetic analogue of Fourier--Jacobi period integrals for a pair of unitary groups of equal rank. We construct the so-called Fourier--Jacobi cycles, which are algebraic cycles on the product of unitary Shimura varieties and abelian varieties. We propose the arithmetic Gan--Gross--Prasad conjecture for these cycles, which is related to central derivatives of certain Rankin--Selberg $L$-functions, and develop a relative trace formula approach toward this conjecture. As a necessary ingredient, we propose the conjecture of the corresponding arithmetic fundamental l…

某些算术函数的归纳公式

Inductive Formulas for some Arithmetic Functions

我们证明了递归公式涉及的除数和三角数的总和，并给出了各种身份有关的算术函数，除数函数提供归纳的身份，这样的算术函数。

Arithmetic formulas for the Fourier coefficients of Hauptmoduln of level 2, 3, and 5

We give arithmetic formulas for the coefficients of Hauptmoduln of higher levels as analogues of Kaneko's result. We also obtain their asymptotic formulas by employing Murty-Sampath's method.

Arithmetic Siegel-Weil formula on $X_{0}(N)$

In this paper, we proved an arithmetic Siegel-Weil formula and the modularity of some arithmetic theta function on the modular curve $X_0(N)$ when $N$ is square free. In the process, we also construct some generalized Delta function for $Γ_0(N)$ and proved some explicit Kronecker limit formula for Eisenstein series on $X_0(N)$.

Periods, cycles, and $L$-functions: a relative trace formula approach

This is a report for the author's talk in ICM-2018. Motivated by the formulas of Gross--Zagier and Waldspurger, we review conjectures and theorems on automorphic period integrals, special cycles on Shimura varieties, and their connection to central values of L-functions and their derivatives. We focus on the global Gan--Gross--Prasad conjectures, their arithmetic versions and some variants in the author's joint work with Rapoport and Smithling. We discuss the approach of relative trace formulas and the arithmetic fundamental lemma conjecture. In the function field setting, Z. Yun and the autho…

Formulas for moments of class numbers in arithmetic progressions

In this paper, we obtain explicit formulas for the second moments for Hurwitz class numbers $H(4n-t^2)$ with $t$ running through a fixed congruence class modulo $3$.

四元数群的算术

Arithmetic of Quaternion Groups

设$p$是素数，$a$是二次非剩余$\bmod p$ .则丢番图方程$x_0^2 - ax_1^2 -px_2^2 + apx_3^2=1$的整解集构成一个余紧离散子群$Γ_{p,a}\subset SL(2,\mathbb{R})$，并与$\mathbb{Q}$上的四元数代数中的一阶单位群可交换.本文讨论的问题是$Γ_{p,a}$生成元集迹的估计。总结在几个表中的经验结果表明，该迹线具有显着的和不规则的波动，这是令人想起的行为的发电机的大小为佩尔方程的解决方案。四元数代数中的一阶单位群的几何和算术在本文代码的开发中起着关键作用。

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…

Hilbert模曲面上的算术交与Faltings高度

Arithmetic Intersection on a Hilbert Modular Surface and the Faltings Height

本文证明了$\mathbb Z$上Hilbert模曲面中算术Hirzebruch-Zagier因子与算术CM圈之间的一个显式算术交公式.作为应用，我们得到了第一个“非交换”的Chowla-Selberg公式，它是Colmez猜想的一个特例;算术Humbert曲面与亏格为2的算术Siegel模簇中CM圈的一个显式算术交公式; Lauter关于Igusa不变量CM值的约化子的猜想;以及CM亏格为2的曲线的一个坏约化结果.

Twisted arithmetic Siegel Weil formula on X0(N)

In this paper, we study twisted arithmetic divisors on the modular curve X_0(N) with N square-free. For each pair (Δ, r) where Δ>0 and Δ\equiv r^2 \mod 4N, we constructed a twisted arithmetic theta function ϕ_{Δ, r}(τ) which is a generating function of arithmetic twisted Heegner divisors. We prove the modularity of ϕ_{Δ, r}(τ), along the way, we also identify the arithmetic pairing \langle ϕ_{Δ, r}(τ),\widehatω_N \rangle with special value of some Eisenstein series, where \widehatω_N is a normalized metric Hodge line bundle.

阿基米德算术西格尔-韦尔公式：关于 Shimura 曲线上的一般权重特异除子的平均公式

Archimedean arithmetic Siegel--Weil formula for general weights over Shimura curves

本文基于阿基米德算术理论，研究Shimura曲线上的特异除子权重问题，通过Whittaker函数的导数推导出平均公式，揭示了特殊除子与算术高度配对的关系。

算术四元数群的生成元与一个丢番图问题

Generators of Arithmetic Quaternion Groups and a Diophantine Problem

设$p$是素数，$a$是二次非剩余$\bmod p$ .则丢番图方程$x_0^2 - ax_1^2 -px_2^2 + apx_3^2=1$的整解集构成一个余紧离散子群$Γ_{p,a}\subset SL(2,\mathbb{R})$，并与$\mathbb{Q}$上的四元数代数中的一阶单位群可交换.本文讨论的问题是$Γ_{p,a}$生成元集迹的估计。总结在几个表中的经验结果表明，该迹线具有显着的和不规则的波动，这是令人想起的行为的发电机的大小为佩尔方程的解决方案。四元数代数中的一阶单位群的几何和算术在本文代码的开发中起着关键作用。

Hecke多项式的Euler积的某些系数的计算公式

An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials

在1997年，作者发现了一个标准的黎曼假设的黎曼ζ函数，涉及非负的某些系数与黎曼ζ函数。在1999年，Bubieri和Lagarias利用素数理论的“显式公式”得到了这些系数的算术公式。本文给出了Hecke多项式的Euler积的相应系数的一个计算公式，它实质上是Hecke同余子群上的L-函数与权为2的尖点形式（新形式和旧形式）的乘积.这些系数的非负性同时给出了所有这些L-函数的黎曼假设的判据。

Hilbert模曲面上的一个算术交公式

An arithmetic intersection formula on Hilbert modular surfaces

本文给出了Hilbert模曲面上的模曲线的对角嵌入与非双二次CM四次域上的CM圈的算术交公式.这证实了作者与J. Bruinier在\cite{BY}中提出的猜想的一个特例，并且是Gross和Zagier关于奇异模的美丽因子分解公式的推广.作为应用，我们证明了第一个非平凡的非交换Chowla-Selberg公式，它是Colmez猜想的一个特例.

Asymptotics for Hecke eigenvalues of automorphic forms on compact arithmetic quotients

In this paper, we describe the asymptotic distribution of Hecke eigenvalues in the Laplace eigenvalue aspect for certain families of Hecke-Maass forms on compact arithmetic quotients. Instead of relying on the trace formula, which was the primary tool in preceding studies on the subject, we use Fourier integral operator methods. This allows us to treat not only spherical, but also non-spherical Hecke-Maass forms with corresponding remainder estimates. Our asymptotic formulas are available for arbitrary simple and connected algebraic groups over number fields with cocompact arithmetic subgroups…

交比运算与广义Mertens公式

On the arithmetic of crossratios and generalised Mertens' formulas

我们发展双曲几何和算术equidistribution问题之间的关系，所产生的行动的算术群对真正的双曲空间，特别是在维高达5。我们证明了二次虚数域和有理数上的定四元数代数的Mertens公式的推广，关于两种不同自然复杂度的二次无理数的计数结果，以及具有误差界的二进制二次、Hermitian和Hamilton形式的（代数）整数表示的计数结果。对于每一个这样的陈述，我们证明了相应的算术定义点的等分布结果。进一步，我们研究了这类点的交比的渐近性质，推广了Pollicott关于Schottky-Klein素函数的最新结果.

一种双变量的算术函数

An arithmetic function of two variables

定义了两个变量的算术函数。函数的一些性质给出了与所谓的莫比乌斯反演公式类似的公式。一个启发式的声明建议。

交替幂的多变量算术级数的多项式公式

Polynomial formulations of Multivariable Arithmetic Progressions of Alternating Powers

本文受 Lanphier 工作启发，建立交替幂和的广义多变量多项式公式，并借助 Euler-Maclaurin 求和公式的类比给出多变量型 Lerch-Hurwitz zeta 函数的渐近公式。

Kuroda's formula and arithmetic statistics

Kuroda's formula relates the class number of a multi-quadratic number field $K$ to the class numbers of its quadratic subfields $k_i$. A key component in this formula is the unit group index $Q(K) = [\mathcal{O}_{K}^{\times}: \prod_i\mathcal{O}_{k_i}^{\times}]$. We study how $Q(K)$ behaves on average in certain natural families of totally real biquadratic fields $K$ parametrized by prime numbers.

On summatory arithmetic functions and a Volterra Integral equation

We obtain asymptotic results for well known summatory arithmetic functions, such as $ψ(x),$ and establish connections to new summatory functions. A new Volterra integral equation is offered, which is solved by summatory arithmetic functions. We conclude with some further integral formulas and provide number theoretic formulas as applications.

Weak Siegel-Weil formula for M_2(Q) and arithmetic on quaternions

In this paper, we prove the weak Siegel-Weil formula for the space M_2(Q) . We study the Hecke correspondence and representation numbers associated to Eichler orders, and give the explicit formula for degree of Hecke correspondence and average representation numbers over genus. This formula could recover the main results in [DuYang]. We could identify these numbers with the Fourier coefficients of Eisenstein series via Siegel-Weil formula(weak Siegel-Weil formula for M_2(Q). In the last part of this article, we reprove four square sum Theorem via Siegel-Weil formula and Kudla's matching.

论算术内积公式与 L 函数的中心导数

On the arithmetic inner product formula and central derivatives of L-functions

本文定义权 3/2 尖点形式的算术 theta 提升并证明算术内积公式，完成 Kudla 纲领、给出 Gross–Zagier 公式新证，同时刻画高权 L 函数中心导数并提出相关消失猜想。

模n正则整数上算术函数的加权平均

Weighted averages of arithmetic functions over regular integers modulo $n$

该研究推广了Kiuchi与Matsuoka的恒等式，建立了算术函数部分和的一般变换公式，将其应用于经典乘性函数的加权平均，得到了显式渐近公式，为推导渐近结果提供了统一框架。

The Selberg trace formula as a Dirichlet series

We explore an idea of Conrey and Li of expressing the Selberg trace formula as a Dirichlet series. We describe two applications, including an interpretation of the Selberg eigenvalue conjecture in terms of quadratic twists of certain Dirichlet series, and a formula for an arithmetically weighted sum of the complete symmetric square L-functions associated to cuspidal Maass newforms of squarefree level N>1.

The function field Sathé-Selberg formula in arithmetic progressions and `short intervals'

We use a function field analogue of a method of Selberg to derive an asymptotic formula for the number of (square-free) monic polynomials in $\mathbb{F}_q[X]$ of degree $n$ with precisely $k$ irreducible factors, in the limit as $n$ tends to infinity. We then adapt this method to count such polynomials in arithmetic progressions and short intervals, and by making use of Weil's `Riemann hypothesis' for curves over $\mathbb{F}_q$, obtain better ranges for these formulae than are currently known for their analogues in the number field setting. Finally, we briefly discuss the regime in which $q$ t…

正交Shimura簇上算术闭链的拉回公式

Pullback formulas for arithmetic cycles on orthogonal Shimura varieties

本文研究正交Shimura簇上Gillet-Soule算术周群中的特殊闭链在低维嵌入正交Shimura簇上的拉回行为，针对非正常相交情形构造了法丛形变上Green流的对数展开式。

The Legendre Approximation and Arithmetic Bias

An interesting episode in the history of the prime number theorem concerns a formula proposed by Legendre for counting the primes below a given bound. We point out that arithmetic bias likely played an important role in arriving at that formula and in its subsequent widespread, decades-long recognition. We also show that the Legendre constant 1.08366 satisfies a certain simple and natural criterion, and conjecture that this criterion is how Legendre arrived at that erroneous constant in his formula.

实二次域的算术 Dijkgraaf-Witten 不变量、二次剩余图与密度公式

Arithmetic Dijkgraaf-Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas

本文针对实二次域整数环，利用素数间的二次剩余图简化了 Hirano 的模 2 算术 Dijkgraaf-Witten 不变量公式，回答了 Ken Ono 的问题，并进一步给出了相应不变量的密度公式。

冯·诺依曼维数与迹公式 II：GL(2) 中算术群代数的 Jacquet-Langlands 对应

Von Neumann Dimensions and Trace Formulas II: A Jacquet-Langlands correspondence for Arithmetic Group Algebras in $\rm{GL}(2)$

本文为 GL(2) 与四元数代数的 S-算术群冯·诺依曼代数模建立全局 Jacquet-Langlands 对应，并给出由 zeta 函数零值表示的维数比公式。

等差数列中素数的显式公式I

Explicit formulae for primes in arithmetic progressions, I

在$q<\log^{α_1} x$是非例外的条件下，我们将给出一个具有$C/\log^αx$形式的误差项的$ψ(x, q, a)$的显式公式，对于$α$和$α_1$的各种值。我们也将给出一个显式公式$ψ(x, q, a)$与误差项$C/\log^A x$工作$q$是例外或非例外，但条件是$\frac{0.4923A}πq^{1/2}\log^2 q<\log x/\log\log x$。此外，我们还将给出非例外模上的Bjueri-Vinogradov定理的一个显式形式。

全球面Hecke代数的半Lie算术基本引理

Semi-Lie arithmetic fundamental lemma for the full spherical Hecke algebra

本文针对Yifeng Liu提出的半Lie算术基本引理，将其推广到全球面Hecke代数并表述相应猜想，同时给出两类猜想中加权轨道积分的显式公式，证明了第一个非平凡情形。

关于叶空间中L函数与Atiyah-Bott-Lefschetz迹公式的相似性

On the analogy between L-functions and Atiyah-Bott-Lefschetz trace formulas for foliated spaces

本文的动机Deninger的程序。首先，利用Alvarez Lopez-Kordyukov的结果，证明了黎曼叶理上分支的叶平坦线丛的上同调群的Atiyah-Bott-Lefschetz迹公式.然后，我们认为，通过精确的计算，上同调群相关的分歧叶平坦向量丛在一个合适的叶状空间（其存在性仍然是未知的）可能是有用的研究算术L函数通过Atiyah-Bott-Lefschetz迹公式。

模曲线 $\mathcal{X}_{0}(N)$ 上的算术 Siegel-Weil 公式

Arithmetic Siegel-Weil formula on $\mathcal{X}_{0}(N)$

本文在任意水平 N 的模曲线 X_0(N) 上建立了算术 Siegel-Weil 公式，通过 Rapoport-Zink 空间上的局部算术相交数与二次型局部表示密度导数之间的精确恒等式完成证明，并为奇数无平方因子情形提供了新证法。

The arithmetic Kuznetsov formula on $GL(3)$, I: The Whittaker case

The original formulae of Kuznetsov for $SL(2,\mathbb{Z})$ allowed one to study either a spectral average via Kloosterman sums or to study an average of Kloosterman sums via a spectral interpretation. In previous papers, we have developed the spectral Kuznetsov formulae at the minimal weights for $SL(3,\mathbb{Z}))$, and in these formulae, the big-cell Kloosterman sums occur with weight functions attached to four different integral kernels, according to the choice of signs of the indices. These correspond to the $J$- and $K$-Bessel functions in the case of $GL(2)$. In this paper, we demonstrate…

算术簇的 $\mathbb R$-除子的数值等价性与 Shioda-Tate 公式

Numerical equivalence of $\mathbb R$-divisors and Shioda-Tate formula for arithmetic varieties

本文证明算术簇上数值平凡的算术 $\mathbb R$-除子恰为主除子的线性组合，并建立联系 Arakelov-Chow 空间、Mordell-Weil 秩与 Néron-Severi 秩的高维算术 Shioda-Tate 公式。

关于等差数列幂和系数的一个注记

A note on the coefficients of power sums of arithmetic progressions

本文提出了一个仅含二项式系数和二项式幂的双重求和公式，用于计算任意等差数列幂和相关多项式的系数，并给出了涉及第一类和第二类 Stirling 数的 Bernoulli 多项式的显式公式。

等差数列中的平方数II

Squarefull numbers in arithmetic progression II

本文改进了前一篇关于等差数列中平方数的渐近公式的误差项。

关于等差数列中圆问题的余项

On the remainder term in the circle problem in an arithmetic progression

本文改进了关于等差数列中圆问题的渐近公式中余项的估计。

Systoles of Arithmetic Hyperbolic Surfaces and 3-manifolds

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with invariant trace field $k$. The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic fiel…

算术级数与二元三次多项式上的除数函数

The divisor function along arithmetic progressions and binary cubic polynomials

该研究提出带两个小因子模的算术级数中除数函数的新等分布估计，给出两项应用：指数2/3的几乎所有模的算术级数上除数函数的渐近公式，以及沿非齐次二元三次多项式XY²+1的除数函数渐近公式。

Divisor problem in arithmetic progressions modulo a prime power

We obtain an asymptotic formula for the average value of the divisor function over the integers $n \le x$ in an arithmetic progression $n \equiv a \pmod q$, where $q=p^k$ for a prime $p\ge 3$ and a sufficiently large integer $k$. In particular, we break the classical barrier $q \le x^{2/3}$ for such formulas, and generalise a recent result of R.~Khan (2015), making it uniform in $k$.

函数域运算中Carlitz基与数字导数基的比较

A comparison of the Carlitz and digit derivatives bases in function field arithmetic

本文比较了$\F_q[[T]].$上连续函数的Carlitz多项式和数位导数的几个性质和构造，特别是指出了它们作为规范正交基的密切关系。与Carlitz系数公式相平行，我们给出了展开系数的数字导数的闭公式。

有限域上的算术傅里叶变换：一般性消没、卷积与等分布

Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution

本文研究一般连通交换代数群上迹函数的算术傅里叶变换，通过证明通用消没定理并构造Tannakian范畴，利用Deligne的黎曼假设导出等分布定理，推广了Katz的工作，并给出应用实例与开放问题。

Exponential sums over Piatetski-Shapiro primes in arithmetic progressions

Let $γ<1<c$ and $19(c-1)+171(1-γ)<9$. In this paper, we establish an asymptotic formula for exponential sums over Piatetski-Shapiro primes $p=[n^{1/γ}]$ in arithmetic progressions.

算术Tori的Brauer-Siegel与特殊点的Galois轨道的下界

Brauer-Siegel for Arithmetic Tori and lower bounds for Galois orbits of special points

在\cite{S}中，Shyr推导出一个类似于Dirichlet的类数公式的算术环面。我们使用这个公式推导出一个Brauer-Siegel公式的环面，其监管机构和类数的产品有关的判别环面。我们应用这个公式来获得渐近和下界的伽罗瓦轨道的CM点的西格尔模品种$A_{g,1}$。具体来说，我们要求这些轨道的大小增长像一个权力的判别式的基本自同态代数。本文在CM域的广义黎曼假设下，对$g\leq 5$和所有的$g$，无条件地证明了这一点.在此过程中，我们推导出了数域的理想类群中挠率的一般传递原理。

Igusa类多项式的算子的一个算术交公式

An arithmetic intersection formula for denominators of Igusa class polynomials

本文证明了交换曲面Siegel模空间上算术相交数（CM（K）. G1）_{ \ell }的一个显式公式，推广了Bruinier-Yang和Yang的工作.利用这些相交数可以计算Igusa类多项式的交子，这对于构造密码学中亏格为2的曲线有重要的应用. Bruinier和Yang给出了算术Hilbert模曲面上相交数的一个公式，并由此在关于本原四次CM域K的分歧的强假设下得到了相交数（CM（K）. G 1）_{ \ell }的一个代数公式.杨后来证明了这个猜想假设O_K是由一个元素自由生成的整数环的实二次子域。本文证明了更一般的本原四次CM域的（CM（K）.G1）_{ \ell }的一个公式，并使用了与杨文中不同的证明方法.我们证明了这个相交数的紧界，它适用于所有的原始四次CM域。作为结果，我们得到了一个公式的倍数的Igusa类多项式的本原四次CM域。我们的证明需要研究Goren和Lauter提出的嵌入问题，并使用我们以前的文章将Gross-Zagier和Dorman的工作推广到任意判别式。

An average theorem for tuples of k-free numbers in arithmetic progressions

We obtain an asymptotic formula, in the spirit of the Montgomery-Hooley refinement of the Barban-Davenport-Halberstam Theorem, for the variance associated with tuples of k-free numbers in arithmetic progressions.

Igusa类多项式算子算术交公式的比较

Comparing arithmetic intersection formulas for denominators of Igusa class polynomials

Bruinier和Yang给出了算术Hilbert模曲面上的相交数的一个公式，并由此在K中的分支的强假设下得到了CM（K）·G_1的一个代数公式.杨后来证明了这一猜想略强假设的分歧。Lauter和Viray在最近的工作中，用一种独立于Yang的证明方法，在较温和的假设下证明了本原四次CM域的CM（K）.G_1的一个不同公式。在本文中，我们表明，这两个公式同意，一类原始四次CM领域，这是略大于由杨和Lauter和Viray考虑的领域的交集。此外，证明这些公式一致并不依赖于杨或劳特和Viray的结果。作为我们证明的结果，我们得出结论，Bruinier-Yang公式对于比Yang证明的稍大的一类四次CM域K成立，因为它与Lauter-Viray公式一致，Lauter-Viray公式在这些情况下被证明。这些相交数的因式分解在密码学中有应用：它们的精确公式允许人们计算Igusa类多项式的分解子，这对密码学中使用的亏格2曲线的构造有重要的应用。