Emmanuel Kowalski
We prove a large sieve statement for the average distribution of Frobenius conjugacy classes in arithmetic monodromy groups over finite fields. As a first application we prove a stronger version of a result of Chavdarov on the ``generic'' irreducibility of the numerator of the zeta functions in a family of curves with large monodromy.
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Using the sieve for Frobenius, we show that, in a certain sense, the roots of the L-functions of "most" algebraic curves over finite fields do not satisfy any non-trivial (linear or multiplicative) rational dependency relations. This can be seen as an analogue of conjectures of linear independence among ordinates of zeros of L-functions over number fields. As a corollary, we find, for "most" pairs of distinct algebraic curves over a finite field, the limiting distribution of the (suitably normalized) difference between the number of rational points over extensions of the ground field. The meth…
These notes give a basic introduction to the theory of $p$-adic and motivic zeta functions, motivic integration, and the monodromy conjecture.
The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser motivic zeta function $Z_{\text{DL}}(h,T)$ of a quasi-ordinary power series $h$ of arbitrary dimension over an algebraically closed field of characteristic zero from its characteristic exponents without using embedded resolution of singularities. This allows us to effectively represent $Z_{\text{DL}}(h,T)=P(T)/Q(T)$ such that almost all the candidate poles g…
By using a variant of Kowalski's large sieve for Frobenius in compatible systems, we obtain zero-density estimates for arguments of $\ell$-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.
We initiate the study of Selberg zeta functions $Z_{Γ,χ}$ for geometrically finite Fuchsian groups $Γ$ and finite-dimensional representations $χ$ with non-expanding cusp monodromy. We show that for all choices of $(Γ,χ)$, the Selberg zeta function $Z_{Γ,χ}$ converges on some half-plane in $\mathbb{C}$. In addition, under the assumption that $Γ$ admits a strict transfer operator approach, we show that $Z_{Γ,χ}$ extends meromorphically to all of $\mathbb{C}$.
We study in detail the family of Calabi-Yau hypersurfaces defined by x_1+...+x_n+1/(x_1...x_n)=t over a finite field k. We determine its local and global monodromy and the trivial factors of its moment zeta function and Dwork's unit root zeta function.
Cohen-Lenstra heuristics and vanishing of zeta functions for superelliptic curves over finite fields
该研究针对有限域上超椭圆曲线的函数域，证明了Cohen-Lenstra型类群分布结果与ζ函数消失的稀疏性结果，通过推广的Hurwitz概形点计数技术及酉大单值群定理完成证明。
In this note we give two small results concerning the correlations of the Selberg sieve weights. We then use these estimates to derive a new (conditional) lower bound on the variance of the primes in short intervals, and also on the so-called `form factor' for the pair correlations of the zeros of the Riemann zeta function. Our bounds ultimately rely on the estimates of Bettin--Chandee for trilinear Kloosterman fractions.
Euler Product Sieve
本文研究了基于Euler乘积的Riemann zeta函数逼近，证明了Euler Product Sieve生成素数的合理性，并指出Bounded Riemann Hypothesis等价于Riemann Hypothesis与零点简单性的合取。
We compute the Z/\ell and \ell-adic monodromy of every irreducible component of the moduli space M_g^f of curves of genus and and p-rank f. In particular, we prove that the Z/\ell-monodromy of every component of M_g^f is the symplectic group Sp_{2g}(Z/\ell) if g>=3 and \ell is a prime distinct from p. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.
The holomorphy conjecture states roughly that Igusa's zeta function associated to a hypersurface and a character is holomorphic on $\mathbb{C}$ whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article we prove the holomorphy conjecture for surface singularities which are nondegenerate over $\mathbb{C}$ with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volume (which appears in the formula of Varchenko for the zeta functi…
The aim of this paper is to prove the weight-monodromy conjecture (Deligne's conjecture on the purity of monodromy filtration) for varieties p-adically uniformized by the Drinfeld upper half spaces of any dimension. The ingredients of the proof are to prove a special case of the Hodge standard conjecture, and apply a positivity argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink. As an application, by combining our results with the results of Schneider-Stuhler, we compute the local zeta functions of p-adically uniformized varieties in terms of representation theor…
This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent $Φ(s, z, c)$ is obtained from the Lerch zeta function $ζ(s, a, c)$ by the change of variable $z=e^{2 πi a}$. We show that it analytically continues to a maximal domain of holomorphy in three complex variables $(s, z, c)$ as a multivalued function defined over the base manifold ${\bf C} \times P^1({\bf C} \smallsetminus \{0, 1, \infty\}) \times ({\bf C}\smallsetminus {\bf Z})$. and compute the monodromy functions definin…
We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field $\mathbb{F}_q$, for almost all couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds on differences. Using similar ideas, we then study the biases in the distribution of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick--Waxman and Keating--Rudnick, building on cohomological interpretations and determinations of monodromy groups by Katz. Our results are based on…
This is the second of four papers that study algebraic and analytic structures associated with the Lerch zeta function. In this paper we analytically continue it as a function of three complex variables. We that it is well defined as a multivalued function on the manifold M equal to C^3 with the hyperplanes corresponding to integer values of the two variables a and c removed. We show that it becomes single valued on the maximal abelian cover of M. We compute the monodromy functions describing the multivalued nature of this function on M, and determine various of their properties.
The Chen series map giving the universal monodromy representation of $\mathbb{P}^1\backslash\{0,1,\infty\}$ is extended to an injective 1-cocycle of $PSL(2, \mathbb{Z})$ into power series with complex coefficients in two non-commuting variables, twisted by an action of $S_3.$ The definition of the 1-cocycle is effected by parallel transport of flat sections of the bundle, also with an $S_3$ twisting, along paths in $\mathbb{P}^1\backslash\{0,1,\infty\}$ which are explicitly associated to elements of $PSL(2, \mathbb{Z})$. The resulting action of the modular group on the polylogarithm generating…
We give a quantitative version of a result due to N. Katz about L-functions of elliptic curves over function fields over finite fields. Roughly speaking, Katz's Theorem states that, on average over a suitably chosen algebraic family, the L-function of an elliptic curve over a function field becomes "as irreducible as possible" when seen as a polynomial with rational coefficients, as the cardinality of the field of constants grows. A quantitative refinement is obtained as a corollary of our main result which gives an estimate for the proportion of elliptic curves studied whose L-functions have …
Equidistribution and arithmetic $Λ$-distributions
本文提出了容许$\mathbb{Z}$-集参数化概率空间族的抽象等分布概念，证明其矩母函数可表为动机Euler乘积，并结合Poonen筛法推广计算了函数域$L$-函数和zeta函数的渐近$Λ$-分布。
On Igusa's conjectures for weighted homogeneous polynomials with isolated singularities
本文推广p进McKay对应，将Denef的Igusa局部zeta函数公式拓展至Q-嵌入奇点消解情形，证明了相关强单值群猜想与指数和猜想对特定加权齐次多项式成立。
The Carlson-type zero-density theorem for the Beurling zeta function
本文利用 Pintz 的新方法结合经典零点检测和与 Halász 方法，在无需额外条件的情况下，证明了 Beurling zeta 函数的 Carlson 型零点密度定理。
The discrete second moment of the Riemann zeta-function along powers of Gram points
该研究证明了沿Gram点幂的Riemann ζ函数离散二阶矩的极限公式，通过双大筛法改进了导数检验边界，结合Lindelöf假设拓展了适用区域，并得到离散泛函定理的应用结果。
A Non-Sieving Application of the Euler Zeta Function
本文探讨了在无法用筛法从乘积项中剔除合数时 Euler Zeta 函数的应用，提出通过调整分子中与 1 相加或相减的值使方程两边相等，以期预测多项式输出中的素数数量。
Frobenius constants for families of elliptic curves
本文研究代数几何中描述 Frobenius 解单值性的 Frobenius 常数，将其表示为模形式的迭代积分，并利用模形式周期理论以 zeta 值表达了部分常数。
Chen's iterated integrals may be generalized by interpolation of functions of the positive integer number of times which particular forms are iterated in integrals along specific paths, to certain complex values. These generalized iterated integrals satisfy both an additive and a (non-classical) multiplicative iterative property, in addition to a comultiplication formula. This theory is developed in the first part of the paper, after which various applications are discussed, including the expression of certain zeta functions as complex iterated integrals (from which an obstruction to the exist…
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
本文提出一种加速振荡级数收敛的算法，并将其应用于多对数函数和Hurwitz zeta函数的高效计算，扩展了Borwein的Riemann zeta函数算法。
Complexity of counting points on curves and the factor $P_1(T)$ of the zeta function of surfaces
本文研究有限域上曲线和曲面点计数的计算复杂性，提出了首个高效的 Arthur-Merlin 协议证明曲线点数及相关结构，并扩展至曲面 zeta 函数因子 $P_1(T)$ 的计算，给出了首个非平凡的计算复杂性上界。
We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\ell-monodromy of every irreducible component of the stratum H_g^f of hyperelliptic curves of genus g and p-rank f is the symplectic group Sp_{2g}(Z/\ell) if g > 2, f > 0 and \ell is an odd prime distinct from p. These results yield applications about the generic behavior of hyperelliptic curves of given genus and p-rank. The first application is that a generic hyperelliptic curve of genus g > 2 and p-rank 0 is not su…
More than 83.9% of the zeros of the Riemann zeta function are distinct and more than 67.35% are simple and on the critical line
本研究通过计算机辅助不等式与Lean 4形式化证明，改进了黎曼zeta函数零点分布的下界：超过83.9%的零点互异，超过67.35%为临界线上的简单零点，并给出相应上界。
Averages of Artin characters and their relation to the Davenport--Heilbronn theorem
本文通过Artin特征平均值改进了三次扩张计数函数的误差项界，在GRH下达到$X^{2/3-1/51+\epsilon}$，并证明至少75%的L-函数在中心点非零，且函数域结果无条件。
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