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arXiv 0812.3920math.AG
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动机的动机Zeta函数导论

An introduction to motivic zeta functions of motives

Yves André

摘要:

它经常发生的泰勒系数(维数正则化)费曼振幅$I$与合理的参数,扩大到一个完整的维度$D= D_0$,不仅是周期(贝尔凯尔,布鲁斯南,博格纳,Weinzierl),但实际上是多个zeta值(布罗德赫斯特,Kreimer)。为了确定(至少是理论上的)具体情况是否如此,动机哲学--更具体地说,混合泰特动机理论--提出了一种算术方法(Kontsevich):计算与$I$相关的代数簇的点数,模足够多的素数$p$,并检查点数在$p$中多项式变化。另一方面,Kapranov引入了一个新的“zeta函数”,其作用正是“插值”zeta函数之间的减少模不同的素数$p$。在本调查中,我们概述了这一思想圈及其最近的一些发展。

英文摘要:

It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes $I$ with rational parameters, expanded at an integral dimension $D= D_0$, are not only periods (Belkale, Brosnan, Bogner, Weinzierl) but actually multiple zeta values (Broadhurst, Kreimer). In order to determine, at least heuristically, whether this is the case in concrete instances, the philosophy of motives - more specifically, the theory of mixed Tate motives - suggests an arithmetic approach (Kontsevich): counting points of algebraic varieties related to $I$ modulo sufficiently many primes $p$ and checking that the number of points varies polynomially in $p$. On the other hand, Kapranov has introduced a new "zeta function", the role of which is precisely to "interpolate" between zeta functions of reductions modulo different primes $p$. In this survey, we outline this circle of ideas and some of their recent developments.

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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