A p-adic Multivariate Version of Igusa's Stationary Phase Method
英文摘要:
This paper has been withdrawn by the author. The proof of the main result has a gap that the author does not know how to fix it.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
W. A. Zuniga-Galindo
This paper has been withdrawn by the author. The proof of the main result has a gap that the author does not know how to fix it.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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Igusa and Denef-Sperber Conjectures on nondegenerate p-adic exponential sums
我们证明了[Igusa,J.,“高等学位形式讲座”,Lect. math. phys.,Springer-Verlag,59(1978)]和[Denef,J.和Sperber,S.,“Exponential sums mod p^n and Newton polyhedra”,Bull. Belg.数学会,Suppl.(2001)55-63]关于模p^m的非退化指数和。
On the smallest poles of Igusa's p-adic zeta functions
设K是一个p-adic域。我们探讨了Igusa的p-adic zeta函数,它与K^n的开紧子集上的K-解析函数相关联。首先推导出该亚纯函数在候选极点处的洛朗级数中一个重要系数的计算公式。然后,我们用这个公式来确定所有的值,当n=2时,这些值小于-1/2;当n=3时,这些值小于-1,它们是极点的实部。
This thesis contains some new results on the uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sums
Lower bound for the poles of Igusa's p-adic zeta functions
设K是p-adic域,R是K的赋值环,P是R的极大理想,q是剩余域R/P的基数,f是R上n>1元的多项式,χ是R^{ \times }的特征标.设M_i(u)为f=u在(R/P^i)^n中的解的个数,其中i为\in \mathbb{Z}_{\geq 0},u为\in R/P^i.这些数与f的Igusa p-adic zeta函数Z_{f,χ}(s)有关。我们解释了M_i(u)与Z_{f,χ}(s)的极点的最小实部之间的关系。我们还证明了M_i(u)可被q^{ \ulcorner(n/2)(i-1)\urcorner }整除,其中角点表示我们必须向上舍入.这将暗示我们的主要结果:Z_{f,χ}(s)没有实部小于-n/2的极点。我们还将考虑任意K-解析函数f。
For p prime, we give an explicit formula for Igusa's local zeta function associated to a polynomial mapping f=(f_1,...,f_t): Q_p^n -> Q_p^t, with f_1,...,f_t in Z_p[x_1,...,x_n], and an integration measure on Z_p^n of the form |g(x)||dx|, with g another polynomial in Z_p[x_1,...,x_n]. We treat the special cases of a single polynomial and a monomial ideal separately. The formula is in terms of Newton polyhedra and will be valid for f and g sufficiently non-degenerated over F_p with respect to their Newton polyhedra. The formula is based on, and is a generalization of results of Denef - Hoornaer…
$p$-adic Maass--Shimura operators on $μ$-ordinary Igusa varieties
该研究针对霍奇型志村簇,将p进整近全纯自守形式上的代数马阿斯-志村微分算子扩展到μ-普通曼托万伊古萨簇,通过p进傅里叶理论等方法得到相关代数作用,扩展、澄清并恢复了先前构造。
In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerated surface singularity. We start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f in Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerated with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of the complex zero locus of f close to the origin. Essentially the entire paper is…
A PEL-type Igusa Stack and the $p$-adic Geometry of Shimura Varieties
本文通过构造一个称为 Igusa 叠的 v-叠,将无限 p-级 Shimura 簇的良好约化区域描述为与仿射格拉斯曼流形的 Schubert 胞腔在 Fargues-Fontaine 曲线上的纤维积,并推广到极小紧化,同时当 Schubert 胞腔替换为有界子叠时恢复积分模型。
One mentions in a lot of papers that the poles of Igusa's p-adic zeta function determine the asymptotic behavior of the number of solutions of polynomial congruences. However, no publication clarifies this connection precisely. We try to get rid of this gap.
Monodromy representations of $p$-adic differential equations in families
本文推导了混合特征非阿基米德域上环带上的常微分方程局部单值定理的相对版本,并应用于$p$-进上同调与$p$-进Hodge理论,包括简化半稳定约化定理证明、Berger定理的相对化及多变量局部单值定理。
We generalise Dwork's theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely, with $\mathbf z=(z_1,z_2,...,z_d)$, we show that the Taylor coefficients of the multi-variable series $q(\mathbf z)=z_i\exp(G(\mathbf z)/F(\mathbf z))$ are integers, where $F(\mathbf z)$ and $G(\mathbf z)+\log(z_i) F(\mathbf z)$, $i=1,2,...,d$, are specific solutions of certain GKZ systems. This result implies the integrality of the Taylor coefficients of numer…
Conjectures of J. Igusa for p-adic local zeta functions and of J. Denef and F. Loeser for topological local zeta functions assert that (the real part of) the poles of these local zeta functions are roots of the Bernstein-Sato polynomials (i.e. the b-functions). We prove these conjectures for certain hyperplane arrangements, including the case of reduced hyperplane arrangements in three-dimensional affine space.
We present a version of Smale's $α$-theory for ultrametric fields, such as the $p$-adics and their extensions, which gives us a multivariate version of Hensel's lemma.
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