Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives On the analytic class number formula for Selberg zeta functions Gerard Freixas i Montplet C.N.R.S. – Institut de Math´ ematiques de Jussieu - Paris Rive Gauche Shimura varieties and hyperbolicity of moduli spaces UQAM, Montr´ eal, May 2018 1 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives Dedekind zeta function Let K / Q a number field and ζ K ( s ) its Dedekind zeta function: � � � 1 − ( N p ) − s � − 1 . 1 ζ K ( s ) = ( N a ) s = 0 � = a ⊆O K p ⊂O K maximal ideal ◮ Absolute convergence for Re( s ) > 1 and meromorphic continuation to C . ◮ Simple pole at s = 1. ◮ Functional equation. ◮ Riemann hypothesis. 2 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives Analytic class number formula Theorem The residue of ζ K ( s ) at s = 1 is given by h K R K Res s =1 ζ K ( s ) = 2 r 1 (2 π ) r 2 � , w K | ∆ K / Q | where ◮ r 1 (resp. r 2 ) number of real (resp. complex) embeddings of K. ◮ h K = # Cl ( K ) is the class number. ◮ R K is the regulator. ◮ w K = # O K , tors . ◮ ∆ K / Q is the absolute discriminant. 3 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives Selberg zeta function The Selberg zeta function is a dynamical zeta function. Primitive closed geodesics play the role of prime numbers. ◮ Γ ⊂ PSL 2 ( R ) fuchsian group of the first kind. ◮ Γ acts on the upper half plane H by isometries, with respect to the hyperbolic metric. ◮ Y := Γ \ H has the structure of a Riemann surface. ◮ Y becomes compact after adding finitely many cusps. ◮ Correspondences: closed geodesics in Y ↔ free homotopy classes of curves ↔ conjugacy classes of hyperbolic elements in Γ 4 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives Definition (Selberg zeta function of Γ) The Selberg zeta function is defined by the absolutely convergent double product � 1 − e − ( s + k ) ℓ ( γ ) � ∞ � � Z ( s , Γ) = , Re( s ) > 1 , γ k =0 where ◮ γ runs over oriented primitive closed geodesics. ◮ ℓ ( γ ) is the length of γ . 5 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives Introduced by Selberg in connection with the trace formula, applied to a suitable test function h (resolvent trace formula). For Γ co-compact and torsion free, it reads: � + ∞ � h ( t k ) =vol hyp (Γ \ H ) h ( r ) tanh( π r ) dr 4 π −∞ k ∞ � � ℓ ( γ ) / 2 � + h ( k ℓ ( γ )) sinh( ℓ ( γ ) / 2) , γ k =1 where: k form the spectrum of − y 2 � � ∂ x 2 + ∂ 2 ∂ 2 ◮ the λ k = 1 4 + t 2 . ∂ y 2 ◮ � h is the Fourier transform of h . ◮ “curiosity”: appearance of the � A genus... 6 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives Example For Γ = PSL 2 ( Z ), Sarnak shows: ∞ � � (1 − ε − 2( s + k ) ) h ( d ) , Z ( s , PSL 2 ( Z )) = d d > 0 k =0 d ≡ 0 or 1 (4) where: ◮ ε d > 1 is the fundamental solution of the Pell equation x 2 − dy 2 = 4. ◮ h ( d ) is the class number of binary integral quadratic forms of discriminant d . 7 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives The Selberg zeta has some analogies with the Dedekind zeta: ◮ meromorphic extension to C (trace formula). ◮ functional equation. ◮ simple zero at s = 1. ◮ for Γ co-compact (also PSL 2 ( Z )), the non-trivial zeroes are located on Re( s ) = 1 2 , and correspond to the (discrete) spectrum of the hyperbolic Laplacian � ∂ 2 � ∂ x 2 + ∂ 2 ∆ hyp = − y 2 . ∂ y 2 8 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives The Selberg zeta has some analogies with the Dedekind zeta: ◮ meromorphic extension to C (trace formula). ◮ functional equation. ◮ simple zero at s = 1. ◮ for Γ co-compact (also PSL 2 ( Z )), the non-trivial zeroes are located on Re( s ) = 1 2 , and correspond to the (discrete) spectrum of the hyperbolic Laplacian � ∂ 2 � ∂ x 2 + ∂ 2 ∆ hyp = − y 2 . ∂ y 2 Missing: ◮ Analytic class number formula: expression for Z ′ (1 , Γ)? 8 / 38
� � Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives If Γ is co-compact and torsion free: d’Hoker–Phong, Sarnak � det ∆ hyp Z ′ (1 , Γ) � trace formula Gillet–Soul´ e Arithmetic Riemann–Roch (cohomological side) where det ∆ hyp is the zeta regularized determinant of the Laplacian ∆ hyp . 9 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives The Grothendieck–Riemann–Roch theorem Let f : X → Y be a projective morphism of smooth, complex algebraic varieties and E a vector bundle on X . Grothendieck–Riemann–Roch is the relation of characteristic classes: ch( Rf ∗ E ) = f ∗ (ch( E ) td( T • in CH • ( Y ) Q . X / Y )) In particular, if Y = Spec C (Hirzebruch–Riemann–Roch): � � ( − 1) p dim H p ( X , E ) = χ ( X , E ) = ch( E ) td( T X ) ∈ Z . X p 10 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives The Riemann–Roch theorem in Arakelov geometry ◮ π : X → Spec Z a regular projective arithmetic variety. ◮ E = ( E , h ) a C ∞ hermitian vector bundle. ◮ ω a K¨ ahler metric on X ( C ), invariant under F ∞ . ◮ A 0 , p ( X ( C ) , E C ) carries a hermitian L 2 scalar product. ◮ ∆ 0 , p Laplacian acting on A 0 , p ( X ( C ) , E C ). ∂ ◮ Flat base change, Dolbeault isomorphism and Hodge theory give H p ( X , E ) C ≃ H 0 , p ∂ ( X ( C ) , E C ) ≃ ker ∆ 0 , p ⊂ A 0 , p ( X ( C ) , E C ) . ∂ Hence H p ( X , E ) C inherits the L 2 scalar product. 11 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives The arithmetic Riemann–Roch theorem of Gillet–Soul´ e computes the arithmetic degree of the cohomology : � ( − 1) p log # H p ( X , E ) tor � deg H • ( X , E ) L 2 = � ( − 1) p log vol( H p ( X , E ) free , h L 2 ) , − p corrected by the holomorphic analytic torsion : � ( − 1) p p log det ∆ 0 , p T = T ( E C , h , ω ) = ∂ . p Hence it catches the whole spectrum of the Laplacians: harmonic forms and non-trival eigenvalues. 12 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives There is a theory of arithmetic characteristic classes for hermitian vector bundles, or more generally finite complexes of those. They simultaneously refine the characteristic classes in CH • ( X ) Q and the Chern–Weil representatives of their de Rham realizations in H • dR ( X ( C ) , C ). • ( X ) Q . They land in the arithmetic Chow groups � CH For instance: ◮ � ch( E ) arithmetic Chern class. ◮ � td( T • X / Z , ω ) arithmetic Todd genus of the tangent complex. 13 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives There is a morphism � top ( X ) X : � � R CH � [ � ′ n P · P , g ] ✤ � � ′ n P log # k ( P ) + 1 X ( C ) g , 2 where P denotes a closed point in X with residue field k ( P ), and g is a top degree current on X ( C ). 14 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives Theorem (Gillet–Soul´ e) There is an equality of real numbers � deg H • ( X , E ) L 2 − 1 � ch( E ) � � td( T • 2 T = X / Z , ω ) X � − 1 ch( E C ) td( T X C ) R ( T X C ) , 2 X ( C ) where R is the R-genus of Gillet–Soul´ e. The R -genus is the additive characteristic class determined by the power series: � � x m � ζ ′ ( − m ) + ζ ( − m )(1 + 1 2 + . . . + 1 R ( x ) = m ) m ! . m ≥ 1 odd 15 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives ◮ There is a generalization to projective morphisms between arithmetic varieties. It holds in any degree. It involves the holomorphic analytic torsion forms of Bismut–K¨ ohler (Gillet–R¨ ossler–Soul´ e, Burgos–F.–Litcanu). ◮ The arithmetic Riemann–Roch theorem refines the Grothendieck–Riemann–Roch theorem in the context of arithmetic varieties. ◮ One of the key points of the proof is the behaviour of holomorphic analytic torsion with respect to closed embeddings (Bismut–Lebeau). 16 / 38
Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives ◮ There is a generalization to projective morphisms between arithmetic varieties. It holds in any degree. It involves the holomorphic analytic torsion forms of Bismut–K¨ ohler (Gillet–R¨ ossler–Soul´ e, Burgos–F.–Litcanu). ◮ The arithmetic Riemann–Roch theorem refines the Grothendieck–Riemann–Roch theorem in the context of arithmetic varieties. ◮ One of the key points of the proof is the behaviour of holomorphic analytic torsion with respect to closed embeddings (Bismut–Lebeau). ◮ All the terms are difficult to compute! 16 / 38
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