Federico Pellarin
Here we characterise in a complete and explicit way the relations of algebraic dependence over $\QQ$ of complex values of Hecke-Mahler series taken at algebraic points of the multiplicative group $\GG_m^2(\CC)$. Our result contains previous theorems by Loxton and van der Poorten, Mahler, and Masser.
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Let $θ$ and $ρ$ be real numbers with $0 \le θ, ρ< 1$ and $θ$ irrational. We show that the Hecke-Mahler series $$ F_{θ, ρ} (z_1, z_2) = \sum_{k_1 \ge 1} \, \sum_{k_2 = 1}^{\lfloor k_1 θ+ ρ\rfloor} \, z_1^{k_1} z_2^{k_2}, $$ where $\lfloor \cdot \rfloor$ denotes the integer part function, takes transcendental values at any algebraic point $(β, α)$ with $0 < |β|, |βα^θ| < 1$. This extends earlier results of Mahler (1929) and Loxton and van der Poorten (1977), who settled the case $ρ=0$. Furthermore, for positive integers $b$ and $a$, with $b \ge 2$ and $a$ congruent to $1$ modulo $b-1$, we give t…
Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $δ$ and $ϕ$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.
Transcendence of Hecke-Mahler Series
本文证明了Hecke-Mahler级数在特定条件下的超越性，其中涉及非常数整数多项式、无理数参数和绝对值大于1的代数数基底。
This is the second paper in a series of two in which a global algebraic number theory of the reals is formulated with the purpose of providing a unified setting for algebraic and transcendental number theory. In this paper, to any real number $θ$ we associate its polynomial diophantine approximation ring: a tri-filtered subring of a nonstandard model of the ring $\mathbb{Z}[X]$. We characterize the filtration structure of the polynomial diophantine approximation ring according to the Mahler class and the Mahler type of $θ$. The arithmetic of polynomial diophantine approximation groups is intro…
Here we propose a survey on Mahler's theory for transcendence and algebraic independence focusing on certain applications to the arithmetic of periods of Anderson t-motives.
The arithmetic volume of hypersurfaces in toric varieties and Mahler measures
该研究确定了光滑射影环面簇中超曲面的典范算术体积，并由此证明了这类超曲面上的广义Hodge指标定理，为算术代数几何领域相关问题提供了新结果。
We express the logarithmic Mahler measure of two families of polynomials defining K3 hypersurfaces of generic Picard number 19 in terms of Eisenstein-Kronecker series. For some singular K3 surfaces (i.e of Picard number 20) we write this measure as Hecke L-series of weight 3 with "Gr\" ossencharacters".
A purity theorem for Mahler equations
本文提出了一项关于马勒方程的纯度定理，通过分析广义马勒级数的滤波结构，证明了其在解中的传播性质。
Arithmetic Rigidity of Analytic Functions and Mahler's Problem on Liouville Numbers
该研究否定解决了 Mahler 关于存在将 Liouville 数映射到 Liouville 数的超越整函数的问题，证明实解析函数若具此性质必为有理函数，并通过双高度计数估计给出定量局部结果。
Let $F$ and $G$ be Siegel cusp forms for $\Sp_4(\Z)$ and weights $k_1, k_2$ respectively. Also let $F$ and $G$ be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither $F$ nor $G$ is a Saito-Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.
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…
We study the derived Hecke action at $p$ on the ordinary $p$-adic cohomology of arithmetic subgroups of semisimple groups $\mathrm G(\mathbb Q)$, i.e., we study the derived version of Hida's theory for ordinary Hecke algebras. This is the analog at $\ell=p$ of derived Hecke actions studied by Venkatesh in the tame case. We show that properties of the derived Hecke action at $p$ are related to deep conjectures in Galois cohomology which are higher analogs of the classical Leopoldt conjecture.
The Mahler measure of a polynomial $P$ in $n$ variables is defined as the mean of $\log|P|$ over the $n$-dimensional torus. For certain polynomials with integer coefficients in two variables the Mahler measure is known to be related to special values of L-functions of arithmetic objects (e.g. Dirichlet characters and elliptic curves over $\mathbb{Q}$). Inspired by work of Deninger Boyd has investigated this relationship numerically. In this paper we reduce some conjectures of Boyd to Beilinson`s conjectures on special values of L-functions. The methods in use are widely of K-theoretical nature…
Machine learning the arithmetic of Boyd's Mahler measure conjectures
该研究利用机器学习与统计分析，探究博伊德马勒测度猜想中有理数$r_k$的算术性质，重现关键观测并发现素数2、3下的特殊算术结构。
This is an expository article that concerns the various related notions of algebraic idele class characters, the Groessencharaktere of Hecke, and cohomological automorphic representations of GL(1), all under the general title of algebraic Hecke characters. The first part of the article systematically lays the foundations of algebraic Hecke characters. The only pre-requisites are: basic algebraic number theory, familiarity with the adelic language, and basic sheaf theory. Observations that play a crucial role in the arithmetic of automorphic L-functions are also discussed. The second part of th…
We outline an algorithm for computing Hecke operators on equivariant cohomology $H^\ast_{Γ_{\text{Sp}}}(X_{\text{Sp}};ρ)$ for the symplectic group $\text{Sp}_4(\mathbb{R})$. To do this, we define a new acyclic cell complex for $\text{Sp}_4(\mathbb{R})$ and make use of the well-tempered complex.
In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an arithmetic formula for these coefficients using the ``explicit formula'' of prime number theory. In this paper, the author obtains an arithmetic formula for corresponding coefficients associated with the Euler product of Hecke polynomials, which is essentially a product of L-functions attached to weight 2 cusp forms (both newforms and oldforms) over Hecke congruence…
We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.
We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group $Γ$. Under favorable conditions, the cohomology is freely generated in a single degree over this graded Hecke algebra. From this construction we extract an action of certain $p$-adic Galois cohomology groups on $H^*(Γ, \mathbb{Q}_p)$, and formulate the central conjecture: the motivic $\mathbb{Q}$-lattice inside these Galois cohomology groups preserves $H^*(Γ,\mathbb{Q})$.
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。
Arithmetic Siegel-Weil for the spherical Hecke algebra
本文提出并部分验证了算术Siegel-Weil公式的推广，将酉Rapoport-Zink空间上环的交点重数与局部Whittaker函数的中心导数相联系，并引入Hecke对应的作用，同时推广了表示密度理论。
Let $H$ be a semisimple algebraic group, $K$ a maximal compact subgroup of $G:=H(\mathbb{R})$, and $Γ\subset H(\mathbb{Q})$ a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke-Maass forms on the locally symmetric space $Γ\backslash G/K$ to corresponding bounds on the arithmetic quotient $Γ\backslash G$ for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial $K$-types, yielding subconvex bounds for new classes of automorphic representations, a…
Densities of arithmetic Hecke triangle group orbits
研究\(q = 4\)和\(q = 6\)时算术赫克三角形群\(\Gamma_q\)轨道的渐近密度，运用初等数论技术及相关函数性质与同余条件给出新证明，发现\(q = 4\)时互质整数对集合的划分特性。
This is the second part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. From the lifting theorem obtained in the first part, we first derive a general result, showing that Mahler functions in several variables, associated with transformations having multiplicatively dependent spectral radii, take algebraic independent values at algebraic points provided that these points are sufficiently independent. Then, we focus on applications of this result and of the two main results of Part I of this work. Our…
We present an algorithm to compute the Hecke operators on the equivariant cohomology of an arithmetic subgroup $Γ$ of the general linear group $\mathrm{GL}_n$. This includes $\mathrm{GL}_n$ over a number field or a finite-dimensional division algebra. As coefficients, we may use any finite-dimensional local coefficient system. Unlike earlier methods, the algorithm works for the cohomology $H^i$ in all degrees $i$. It starts from the well-rounded retract $\tilde{W}$, a $Γ$-invariant cell complex which computes the cohomology. It extends $\tilde{W}$ to a new well-tempered complex $\tilde{W}^+$ o…
We make use of Hecke operators and arithmetic of imaginary quadratic fields to derive an explicit version of a special case of Siegel's mass formula.
In this article, we derive a sub convexity estimate of Hecke eigen cusp forms associated to certain cocompact arithmetic subgroups of SL(2,R). The main result can be considered as the holomorphic version of the estimate of Hecke eigen Maass forms, derived in a famous paper of Iwaniec and Sarnak. A stronger estimate was derived by Khayutin and Steiner in arXiv:2009.07194. However, techniques used in both the papers are very different.
We prove an approximation to an Ihara-formula for the zeta function of an arithmetic quotient of the Bruhat-Tits building of a p-adic PGL(3).
We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients in these Hecke algebras and apply this technique to sharpen recent results of P. Scholze.
Rankin-Selberg coefficients in large arithmetic progressions
本文研究 Rankin-Selberg 系数在算术级数中的分布，并在 GL_2 Maass 形式的 Ramanujan-Petersson 猜想下，将其分布水平推进至 $2/5+1/260-η$。
Arithmetic of Hecke L-functions of quadratic extensions of totally real fields
本文解决了 Shintani 方法在高维全实域上的无效性，为窄类数为1的全实域中导子为惰性有理素数的特征提供了 Shintani sets 的组合描述，简化了相关 Hecke L-函数，并将组合类数公式推广至全实域的二次扩张，有效肯定了 Hecke 猜想。
Arithmetic Fundamental Lemma for the spherical Hecke algebra
本文在酉 RZ 空间上定义 Hecke 算子，提出球面 Hecke 代数的算术基本引理及轨道积分导数恒零函数的丰富性猜想，并证明 $U(1)\ imes U(2)$ 情形。
New variants of arithmetic quantum ergodicity
本文建立了算术量子遍历性的两个新变体，分别针对自对偶 $\mathrm{GL}_2$ Hecke-Maass 新形式在级与 Laplace 特征值联合变化时的情形，以及 Hilbert 或 Bianchi 尖点形式限制到模曲面上随特征值增长而耗散的非分裂类似情形。
Let $Γ$ be a lattice in a locally compact group $G$. In earlier work, we used $KK$-theory to equip the $K$-groups of any $Γ$-$C^{*}$-algebra on which the commensurator of $Γ$ acts with Hecke operators. When $Γ$ is arithmetic, this gives Hecke operators on the $K$-theory of certain $C^{*}$-algebras that are naturally associated with $Γ$. In this paper, we first study the topological $K$-theory of the arithmetic manifold associated to $Γ$. We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Ka…
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.
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.
Non-vanishing mod $p$ of derived Hecke algebras of multiplicative groups over number fields, II
该研究针对数域上乘法群相关算术流形的上同调，在温和假设下证明其导出Hecke作用模p处处非零，且对应F_p系数全上同调环是导出Hecke代数上秩1的自由分次模，0次生成。
Non-vanishing mod $p$ of derived Hecke algebra of the multiplicative group over number field
该研究针对数域上乘法群相关算术流形的导出Hecke上同调作用，借助Grunwald--Wang定理证明温和假设下其一次部分模$p$非零，为数论中相关代数结构研究提供了新结果。
We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by É. Fouvry, S. Ganguly, E. Kowalski, and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian limiting distribution. The analogous problems for Hecke eigenvalues are also considered.
Boyd's conductor-11 Mahler measure conjecture: proof of the split-integral identity (C3), with an exact structural analysis of the family S_k
本文证明了Boyd关于导子11的Mahler测度猜想中的第三个恒等式(C3)，通过将分裂积分转化为正则积分并利用模单位公式精确计算，同时给出了族S_k的完整结构分析。
Mod $\ell$ non-vanishing of self-dual Hecke $L$-values over CM fields and applications
研究CM域上自对偶海克勒L-函数在模ℓ下的非消失性，并应用于Eisenstein同余可除性向CM伊瓦索主猜想的证明。
Let G be a reductive algebraic group associated to a self-adjoint homogeneous cone defined over Q, and let G' be an appropriate neat arithmetic subgroup of G. We present two algorithms to compute the action of the Hecke operators on the integral cohomology of G'. This simultaneously generalizes the modular algorithm of Ash-Rudolph to a larger class of groups, and provides techniques to compute the Hecke-module structure of previously inaccessible cohomology groups.
We obtain lower bounds for the number of nodal domains of Hecke eigenfunctions on the sphere. Assuming the generalized Lindelof hypothesis we prove that the number of nodal domains of any Hecke eigenfunction grows with the eigenvalue of the Laplacian. By a very different method, we show unconditionally that the average number of nodal domains of degree l Hecke eigenfunctions grows significantly faster than the uniform growth obtained under Lindelof.
A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry.
We consider the Hecke pair consisting of the group $P^+_K$ of affine transformations of a number field $K$ that preserve the orientation in every real embedding and the subgroup $P^+_O$ consisting of transformations with algebraic integer coefficients. The associated Hecke algebra $C^*(P^+_K,P^+_O)$ has a natural time evolution $σ$, and we describe the corresponding phase transition for KMS$_β$-states and for ground states. From work of Yalkinoglu and Neshveyev it is known that a Bost-Connes type system associated to $K$ has an essentially unique arithmetic subalgebra. When we import this suba…
Let $f_{\mathrm{new}}$ be a classical newform of weight $\geq 2$ and prime to $p$ level. We study the arithmetic of $f_{\mathrm{new}}$ and its unique $p$-stabilisation $f$ when $f_{\mathrm{new}}$ is $p$-irregular, that is, when its Hecke polynomial at $p$ admits a single repeated root. In particular, we study $p$-adic weight families through $f$ and its base-change to an imaginary quadratic field $F$ where $p$ splits, and prove that the respective eigencurves are both Gorenstein at $f$. We use this to construct a two-variable $p$-adic $L$-function over a Coleman family through $f$, and a three…
Explicit spectral gap for Hecke congruence covers of arithmetic Schottky surfaces
基于广义黎曼猜想，研究Hecke congruence covers的拉普拉斯算子显式谱间隙，主要贡献为在素数p接近所有情况下建立统一的谱间隙。
Intersections of Hecke correspondences on modular curves
本文计算了模曲线$\mathcal{X}_0(N)$积分模型上Hecke对应的算术交，并建立了与Siegel Eisenstein级数导数的联系，通过Rapoport--Zink空间上的精确恒等式证明。
In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use these results to define an action of the Hecke operators on the ACG of modular curves and to show that they are self-adjoint with respect to the arithmetic intersection product. The decomposition of the ACG in eigencomponents which follows allows us to define new numerical invariants, which are refined versions of the self-intersection of the dualizing sheaf. …
We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients $Γ\backslash G/K$, where $G\simeq\mathrm{PGL}_{d}(\mathbb{R})$, $K$ is a maximal compact subgroup of $G$ and $Γ<G$ is a lattice associated to a division algebra over $\mathbb{Q}$ of prime degree $d$. More generally, we introduce a new method of proving positive entropy of quantum limits, which applies to higher-rank groups. The result on AQUE is obtained by combining this with a measure-rigidity theorem due to Einsiedler-Katok, following a strategy first pioneer…
Arithmetic quantum unique ergodicity for products of hyperbolic $2$- and $3$-spaces
本文证明了双曲2-与3-空间乘积商空间上Hecke-Maass形式的算术量子唯一遍历性猜想，通过轨道维数归纳法排除了极限测度集中于真子群闭轨道的可能性。
In this paper we study two functions $F(x)$ and $J(x)$, originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Stark's conjecture. We also discuss connections with 1-cocycles for the modular group $\mathrm{PSL}(2,…
Global $GL_2$ Hecke-Baxter operator
本文为算术型可积系统构造了与$GL_2$相关的全局Hecke-Baxter算子，其特征值由全局$L$-因子给出，并推广了局部情形，推测零点满足算术Bethe ansatz方程。
We provide a Mahler/Elkies-style lower bound for the average values of dynamical Green's functions on the projective line over an arbitrary valued field, and give some dynamical and arithmetic applications.
We show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra and provide several structure results for them. We discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The scarcity of the latter was investigated recently. We explain that they provide congruences among algebraic parts of twisted central $\mathrm{L}$-values. We specialize our results to partition congruences, for which we investigate the proofs of partition congruences by Atkin and by Ono, and develop a heuris…
Arithmetic Properties of Overcolored Odd Partitions
本文研究了过彩色奇数划分计数函数$ar{a}_s(n)$在模2的幂下满足的新类型同余关系，通过生成函数操作、海克 eigenform 理论和 Newman 的结果建立这些关系。
Let M be an imaginary quadratic field, f a Hecke eigenform on GL2(Q) and πthe unitary base-change to M of the automorphic representation associated to f. Take a unitary arithmetic Hecke character χof M inducing the inverse of the central character of f. The celebrated formula of Waldspurger relates the square of an integral L(f) of f and χover the idele class group of M to the central critical value L(1/2,π\otimesχ). In this paper, we present a construction of a new p-adic L-function that interpolates L(f) over arithmetic characters for a cusp form f in the spirit of the landmark result of Kat…
Arithmetic Wu Formulas and the Generalized Hecke Theorem
本文在S-整数环（2可逆）上的分离有限型概形的修正紧支平展上同调上构造典范Steenrod平方运算，从而将绝对平展Wu类从有限域推广到远离2的算术基，并证明正则射影平坦概形的绝对Wu公式，进而得到由Hirzebruch 2-Todd级数的算术形变控制的无穷族万有模2同余式（广义Hecke定理）。
Let X be an arithmetic hyperbolic surface, ψa Hecke-Maass form, and l a geodesic segment on X. We obtain a power saving over the local bound of Burq-Gérard-Tzvetkov for the L^2 norm of ψrestricted to l, by extending the technique of arithmetic amplification developed by Iwaniec and Sarnak. We also improve the local bounds for various Fourier coefficients of ψalong l.
Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing
本文研究了虚二次域上Hecke L-值的ℓ-adic估值，并证明了Rubin p进L-函数在非分裂素数处的μ-不变量消失，填补了该领域的重要空白。
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