2020/08/25 by Connes, Alain, Consani, Caterina · 1 citation
#Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT) #Quantum Algebra (math.QA) #\MSC[2008] 11M55 11M06 46L87 58B34
paper · doi:10.48550/arxiv.2008.10974
We introduce the notion of \it quasi-inner function and show that the product u=ρ_∞∏ ρv of m+1 ratios of local L-factors ρv(z)=γv(z)/γv(1-z) over a finite set F of places of the field of rational numbers inclusive of the archimedean place is quasi-inner on the left of the critical line \Re(z)= \frac 12 in the following sense. The off diagonal part u21 of the matrix of the multiplication by u in the orthogonal decomposition of the Hilbert space L2 of square integrable functions on the critical line into the Hardy space H2 and its orthogonal complement is a compact operator. When interpreted on the unit disk, the quasi-inner condition means that the associated Haenkel matrix is compact. We show that none of the individual non-archimedean ratios ρv is quasi-inner and, in order to prove our main result we use Gauss multiplication theorem to factor the archimedean ratio ρ_∞ into a product of m quasi-inner functions whose product with each ρv retains the property to be quasi-inner. Finally we prove that Sonin's space is simply the kernel of the diagonal part u22 for the quasi-inner function u=ρ_∞, and when u(F)=∏v∈ F ρv the kernels of the u(F)22 form an inductive system of infinite dimensional spaces which are the semi-local analogues of (classical) Sonin's spaces.