Alain Connes is Professor Emeritus at the Collège de France and IHES.
Awards include the Fields Medal (1982), the Crafoord Prize (2001) and the CNRS Gold Medal (2004). More information.
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Quasi-inner functions and local factors Alain Connes, Caterina Consani Journal of Number Theory, 226 , pp. 139-167, 2021. Journal ArticleAbstract @article{connes-consani, title = {Quasi-inner functions and local factors}, author = {Alain Connes and Caterina Consani}, editor = {Elsevier}, url = {/wp-content/uploads/quasi-inner.pdf}, year = {2021}, date = {2021-09-06}, journal = {Journal of Number Theory}, volume = {226}, pages = {139-167}, abstract = {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 R(z)=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∈Fρv 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.}, keywords = {}, pubstate = {published}, tppubtype = {article} } 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 R(z)=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∈Fρv 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. |
Weil positivity and trace formula, the archimedean placeForthcoming Alain Connes, Caterina Consani Selecta Mathematica, Forthcoming. Journal ArticleAbstract @article{Connes2021bb, title = {Weil positivity and trace formula, the archimedean place}, author = {Alain Connes and Caterina Consani}, editor = {Birkhauser}, url = {https://alainconnes.org/wp-content/uploads/Selecta.pdf}, year = {2021}, date = {2021-07-20}, journal = {Selecta Mathematica}, abstract = {We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula. We explore in great details the simplest case of the single archimedean place. The root of this result is the positivity of the trace of the scaling action compressed onto the orthogonal complement of the range of the cutoff projections associated to the cutoff in phase space, for Lambda=1. We express the difference between the Weil distribution and the trace associated to the above compression of the scaling action, in terms of prolate spheroidal wave functions, and use, as a key device, the theory of hermitian Toeplitz matrices to control that difference. All the concepts and tools used in this paper make sense in the general semi-local case, where Weil positivity implies RH.}, keywords = {}, pubstate = {forthcoming}, tppubtype = {article} } We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula. We explore in great details the simplest case of the single archimedean place. The root of this result is the positivity of the trace of the scaling action compressed onto the orthogonal complement of the range of the cutoff projections associated to the cutoff in phase space, for Lambda=1. We express the difference between the Weil distribution and the trace associated to the above compression of the scaling action, in terms of prolate spheroidal wave functions, and use, as a key device, the theory of hermitian Toeplitz matrices to control that difference. All the concepts and tools used in this paper make sense in the general semi-local case, where Weil positivity implies RH. |
On absolute algebraic geometry, the affine case Forthcoming Alain Connes, Caterina Consani Advances in Mathematics, pp. 1–45, Forthcoming. Journal ArticleAbstract @article{Connes2021bb, title = {On absolute algebraic geometry, the affine case }, author = {Alain Connes and Caterina Consani}, editor = {Elsevier}, url = {https://alainconnes.org/wp-content/uploads/JAG2020-1.pdf}, year = {2021}, date = {2021-07-20}, journal = {Advances in Mathematics}, pages = {1--45}, abstract = {We develop algebraic geometry for general Segal’s Γ-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under SpecZ).}, keywords = {}, pubstate = {forthcoming}, tppubtype = {article} } We develop algebraic geometry for general Segal’s Γ-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under SpecZ). |
