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Quadratic relations between periods of Kloosterman connections
Claude Sabbah  1@  
1 : Université Paris-Saclay
Ecole Polytechnique Université Paris Saclay

In recent works, Broadhurst and Roberts have conjectured, and checked numerically, various properties related to symmetric moments of Kloosterman sums and Kloosterman connections. In particular, they conjectured that, for k1, the integrals involving the Bessel functions I0,K0:

0I0(t)iK0k-it2j-1dt, 

i,j=1,…,[(k-1)/2]

should satisfy quadratic relations. In a joint work with J. Fresán and J.-D. Yu, we establish a general framework for quadratic relations between periods of bundles with flat connection on smooth quasi-projective varieties, extending the work of Matsumoto et al. in the nineties, and we prove quadratic relations for the Bessel moments in the form suggested by Broadhurst and Roberts. However, we do not get the same precise form for the coefficients as conjectured by B-R, but both match numerically up to k=22.


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