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**Twisted Kloosterman sums and \(p\)-adic Bessel functions.**
*(English)*
Zbl 0552.12010

One of the key points of Dwork’s cohomology theory is that it establishes a bridge between the \(p\)-adic theory of linear differential equations “of deformation” (i.e., admitting a nice integral formula for their solutions) and the \(p\)-adic theory of \(L\)-functions associated to character sums.

In the present paper the authors construct a 1-parameter analytic family of \(p\)-adic analytic cohomology spaces \(\{W_x\}_x\) whose deformation differential equation is essentially a direct sum of the equations satisfied by the Bessel functions \(J_{-j/d}(x)\), with \(d\in\mathbb N\) fixed, \((p,d)=1\), and \(j=0,1,\dots,d-1\). This family is also endowed with a Frobenius map \(W_x\to W_{x^p}\), horizontal with respect to the deformation equation, whose eigenvalues are related to some twisted Kloosterman sums. From this construction and the detailed study of the Frobenius map, the authors derive (for certain values of \(j\)) precise results on the growth of solutions of the equation for \(J_{-j/d}\) at the boundary of their disks of convergence. They also evaluate the \(p\)-adic size of the reciprocal roots of the \(L\)-functions connected to the above mentioned Kloosterman sums and derive formulas for them in terms of Gauss sums and ratios of Bessel functions.

The authors finally discuss the variation of the Frobenius map with the character parameter \(a=-j/d\) in line with the so-called “Boyarski principle”. They define a 2-variable generalization of the \(p\)-adic gamma function and obtain in terms of that function a 2-variable result in \(x\) and a which specializes at \(x=0\) to a formula which implies the famous Gross-Koblitz result on Gauss sums.

In the present paper the authors construct a 1-parameter analytic family of \(p\)-adic analytic cohomology spaces \(\{W_x\}_x\) whose deformation differential equation is essentially a direct sum of the equations satisfied by the Bessel functions \(J_{-j/d}(x)\), with \(d\in\mathbb N\) fixed, \((p,d)=1\), and \(j=0,1,\dots,d-1\). This family is also endowed with a Frobenius map \(W_x\to W_{x^p}\), horizontal with respect to the deformation equation, whose eigenvalues are related to some twisted Kloosterman sums. From this construction and the detailed study of the Frobenius map, the authors derive (for certain values of \(j\)) precise results on the growth of solutions of the equation for \(J_{-j/d}\) at the boundary of their disks of convergence. They also evaluate the \(p\)-adic size of the reciprocal roots of the \(L\)-functions connected to the above mentioned Kloosterman sums and derive formulas for them in terms of Gauss sums and ratios of Bessel functions.

The authors finally discuss the variation of the Frobenius map with the character parameter \(a=-j/d\) in line with the so-called “Boyarski principle”. They define a 2-variable generalization of the \(p\)-adic gamma function and obtain in terms of that function a 2-variable result in \(x\) and a which specializes at \(x=0\) to a formula which implies the famous Gross-Koblitz result on Gauss sums.

Reviewer: Francesco Baldassarri (Padova)

### MSC:

12H25 | \(p\)-adic differential equations |

11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

11S40 | Zeta functions and \(L\)-functions |

11L05 | Gauss and Kloosterman sums; generalizations |