论文标题
贝塞尔时刻之间的二次关系
Quadratic relations between Bessel moments
论文作者
论文摘要
Broadhurst和Roberts的计算是通过计算某些Feynman振幅的计算，最近猜想并以数值为单位检查了Bessel Momments \ [\ Int_0^\ int_0^\ int_0^\ int_0^\ infty I_0（t） \ qquad（i，j = 1，\ ldots，\ lfloor（k-1）/2 \ rfloor），\ rfloor），\ rfloor \] $ k \ geq 1 $是固定整数，$ i_0 $和$ k_0 $表示修改后的贝塞尔功能。在本文中，我们将这些积分及其变体解释为Kloosterman Connection的中间DE RHAM共同体和对称能力的扭曲同源性之间的时期配对系数。在Arxiv：2005.11525中开发的一般框架的基础上，这使我们能够证明Broadhurst和Roberts建议的形式的二次关系，这些形式构成了这些数字之间的所有代数关系。我们还对Deligne的猜想进行了明确的猜测，从而解释了Kloosterman总和$ L $ functions的临界值的许多评估，从贝塞尔矩的决定因素来看。
Motivated by the computation of certain Feynman amplitudes, Broadhurst and Roberts recently conjectured and checked numerically to high precision a set of remarkable quadratic relations between the Bessel moments \[ \int_0^\infty I_0(t)^i K_0(t)^{k-i}t^{2j-1}\,\mathrm{d}t \qquad (i, j=1, \ldots, \lfloor (k-1)/2\rfloor), \] where $k \geq 1$ is a fixed integer and $I_0$ and $K_0$ denote the modified Bessel functions. In this paper, we interpret these integrals and variants thereof as coefficients of the period pairing between middle de Rham cohomology and twisted homology of symmetric powers of the Kloosterman connection. Building on the general framework developed in arXiv:2005.11525, this enables us to prove quadratic relations of the form suggested by Broadhurst and Roberts, which conjecturally comprise all algebraic relations between these numbers. We also make Deligne's conjecture explicit, thus explaining many evaluations of critical values of $L$-functions of symmetric power moments of Kloosterman sums in terms of determinants of Bessel moments.
