Local to global formulas in geometry and number theory Gerard - PowerPoint PPT Presentation

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Example (Quadratic fields) √ Let D be square-free integer and K = Q ( D ). Then � √ Z [ 1+ D ] if D ≡ 1 mod 4 , 2 O K = √ Z [ D ] otherwise , and � if D ≡ 1 mod 4 , D ∆ K / Q = 4 D otherwise . 13 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • Local input: a ⊆ O K a non-trivial ideal. 14 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • Local input: a ⊆ O K a non-trivial ideal. • Two global numerical invariants associated to a : 14 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • Local input: a ⊆ O K a non-trivial ideal. • Two global numerical invariants associated to a : ◮ the norm N ( a ) = #( O K / a ). 14 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • Local input: a ⊆ O K a non-trivial ideal. • Two global numerical invariants associated to a : ◮ the norm N ( a ) = #( O K / a ). ◮ the (co)volume: a embeds diagonally as a lattice in an Euclidean space K R : � ⊥ � complex conjugation � a ֒ → K R := → C ( C , | · | ) σ : K ֒ K R / a is a compact Riemannian torus of volume vol( K R / a ). 14 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Theorem (Minkowski) The norm and the volume are related by � | ∆ K / Q | , vol( K R / a ) = N ( a ) where ∆ K / Q is the discriminant of the number field. 15 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? In Arakelov geometry we like to write Minkowski’s formula in additive form: − log vol( K R / a ) = − log N ( a ) − 1 2 log | ∆ K / Q | , and we have notations for these quantities: deg a − 1 χ L 2 ( a ) = � � 2 log | ∆ K / Q | . 16 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? In Arakelov geometry we like to write Minkowski’s formula in additive form: − log vol( K R / a ) = − log N ( a ) − 1 2 log | ∆ K / Q | , and we have notations for these quantities: deg a − 1 χ L 2 ( a ) = � � 2 log | ∆ K / Q | . ◮ � χ L 2 is a sort of Euler characteristic. 16 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? In Arakelov geometry we like to write Minkowski’s formula in additive form: − log vol( K R / a ) = − log N ( a ) − 1 2 log | ∆ K / Q | , and we have notations for these quantities: deg a − 1 χ L 2 ( a ) = � � 2 log | ∆ K / Q | . ◮ � χ L 2 is a sort of Euler characteristic. ◮ � deg is called arithmetic degree. 16 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? In Arakelov geometry we like to write Minkowski’s formula in additive form: − log vol( K R / a ) = − log N ( a ) − 1 2 log | ∆ K / Q | , and we have notations for these quantities: deg a − 1 χ L 2 ( a ) = � � 2 log | ∆ K / Q | . ◮ � χ L 2 is a sort of Euler characteristic. ◮ � deg is called arithmetic degree. Minkowski’s formula is the simplest instance of the Riemann–Roch formula in Arakelov geometry. We will come back to this later. 16 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Actually, | ∆ K / Q | is the norm of an ideal (different), so that 17 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Actually, | ∆ K / Q | is the norm of an ideal (different), so that Minkowski’s formula takes the form � deg local = global . 17 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Minkowski’s formula is combined with: Theorem (Minkowski) Let Γ ⊂ V be a lattice in an Euclidean vector space. Let Ω ⊆ V be a non-empty convex set, closed under x �→ − x. Assume vol(Ω) > 2 dim V vol( V / Γ) . Then Ω ∩ (Γ \ { 0 } ) � = ∅ . 18 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Minkowski’s formula is combined with: Theorem (Minkowski) Let Γ ⊂ V be a lattice in an Euclidean vector space. Let Ω ⊆ V be a non-empty convex set, closed under x �→ − x. Assume vol(Ω) > 2 dim V vol( V / Γ) . Then Ω ∩ (Γ \ { 0 } ) � = ∅ . • Typically V = K R and Γ = a ⊂ O K a non-trivial ideal. 18 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Minkowski’s formula is combined with: Theorem (Minkowski) Let Γ ⊂ V be a lattice in an Euclidean vector space. Let Ω ⊆ V be a non-empty convex set, closed under x �→ − x. Assume vol(Ω) > 2 dim V vol( V / Γ) . Then Ω ∩ (Γ \ { 0 } ) � = ∅ . • Typically V = K R and Γ = a ⊂ O K a non-trivial ideal. • One derives | ∆ K / Q | > 1 and the finiteness of the class number: Cl ( K ) = { projective O K − modules of rank 1 } / iso . h K = class number of K = # Cl ( K ) < + ∞ . 18 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? The analytic class number formula • Let K be a number field and � 1 ζ K ( s ) = N ( a ) s , Re( s ) > 1 , 0 � = a ⊆O K its Dedekind zeta function. 19 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? The analytic class number formula • Let K be a number field and � 1 ζ K ( s ) = N ( a ) s , Re( s ) > 1 , 0 � = a ⊆O K its Dedekind zeta function. • Euler product factorisation in “local factors”: � � 1 − ( N p ) − s � − 1 , ζ K ( s ) = Re( s ) > 1 , p where p ⊂ O K are the maximal ideals. 19 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? The analytic class number formula • ζ K ( s ) has a meromorphic continuation to s ∈ C , with a simple pole at s = 1. 20 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? The analytic class number formula • ζ K ( s ) has a meromorphic continuation to s ∈ C , with a simple pole at s = 1. • Functonial equation relating ζ K ( s ) and ζ K (1 − s ). 20 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? The analytic class number formula • ζ K ( s ) has a meromorphic continuation to s ∈ C , with a simple pole at s = 1. • Functonial equation relating ζ K ( s ) and ζ K (1 − s ). • Generalized Riemann Hypothesis: non-trivial zeros of ζ K ( s ) are located on Re( s ) = 1 / 2. 20 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? The analytic class number formula • ζ K ( s ) has a meromorphic continuation to s ∈ C , with a simple pole at s = 1. • Functonial equation relating ζ K ( s ) and ζ K (1 − s ). • Generalized Riemann Hypothesis: non-trivial zeros of ζ K ( s ) are located on Re( s ) = 1 / 2. • Hilbert–P´ olya conjecture: there exists an unbounded self-adjoint operator H , such that � 1 � ζ K 2 + it = 0 t ∈ Spec H . � 20 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Theorem (Dirichlet, Dedekind) 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 � . |O × K tor | · | ∆ K / Q | 21 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Theorem (Dirichlet, Dedekind) 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 � . |O × K tor | · | ∆ K / Q | ◮ r 1 (resp. 2 r 2 ) is the number of real (resp. complex non-real) embeddings of K . 21 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Theorem (Dirichlet, Dedekind) 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 � . |O × K tor | · | ∆ K / Q | ◮ r 1 (resp. 2 r 2 ) is the number of real (resp. complex non-real) embeddings of K . ◮ h K is the class number of K (number of isomorphism classes of projective O K -modules of rank 1). 21 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Theorem (Dirichlet, Dedekind) 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 � . |O × K tor | · | ∆ K / Q | ◮ r 1 (resp. 2 r 2 ) is the number of real (resp. complex non-real) embeddings of K . ◮ h K is the class number of K (number of isomorphism classes of projective O K -modules of rank 1). ◮ R K is called the regulator of K . 21 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? The analytic class number formula takes the form Res s =1 ◦ Mer. Cont. local = global . 22 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Asymptotic volumes of cusp forms • Γ = SL 2 ( Z ). 23 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Asymptotic volumes of cusp forms • Γ = SL 2 ( Z ). • H the upper half plane with coordinate τ = x + iy . 23 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Asymptotic volumes of cusp forms • Γ = SL 2 ( Z ). • H the upper half plane with coordinate τ = x + iy . • Action of Γ on H ∪ P 1 ( Q ) by fractional linear transformations: � a � τ = a τ + b b c τ + d . c d 23 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Asymptotic volumes of cusp forms • Γ = SL 2 ( Z ). • H the upper half plane with coordinate τ = x + iy . • Action of Γ on H ∪ P 1 ( Q ) by fractional linear transformations: � a � τ = a τ + b b c τ + d . c d • Open modular curve of level 1: ∼ Y (1) := H / SL 2 ( Z ) − → C . 23 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Asymptotic volumes of cusp forms • Γ = SL 2 ( Z ). • H the upper half plane with coordinate τ = x + iy . • Action of Γ on H ∪ P 1 ( Q ) by fractional linear transformations: � a � τ = a τ + b b c τ + d . c d • Open modular curve of level 1: ∼ Y (1) := H / SL 2 ( Z ) − → C . • Compactified modular curve of level 1: ∼ X (1) := ( H ∪ P 1 ( Q )) / SL 2 ( Z ) = Y (1) ∪ {∞} → P 1 ( C ) . − 23 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • A holomorphic map f : H → C is called a modular form of weight 2 k if � a � b f ( γτ ) = ( c τ + d ) 2 k f ( τ ) , γ = ∈ SL 2 ( Z ) c d and is holomorphic at ∞ : Fourier expansion � a n q n , q = e 2 π i τ . f ( τ ) = n ≥ 0 24 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • A holomorphic map f : H → C is called a modular form of weight 2 k if � a � b f ( γτ ) = ( c τ + d ) 2 k f ( τ ) , γ = ∈ SL 2 ( Z ) c d and is holomorphic at ∞ : Fourier expansion � a n q n , q = e 2 π i τ . f ( τ ) = n ≥ 0 • We say that f is a cusp form if moreover a 0 = 0. 24 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • A holomorphic map f : H → C is called a modular form of weight 2 k if � a � b f ( γτ ) = ( c τ + d ) 2 k f ( τ ) , γ = ∈ SL 2 ( Z ) c d and is holomorphic at ∞ : Fourier expansion � a n q n , q = e 2 π i τ . f ( τ ) = n ≥ 0 • We say that f is a cusp form if moreover a 0 = 0. • Cusp forms of weight 2 k constitute a finite dimensional vector space S 2 k (Γ , C ). The dimension is linear in k : Riemann–Roch. 24 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • On the space of cusp forms of weight 2 k there is a real structure (resp. integral structure): 25 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • On the space of cusp forms of weight 2 k there is a real structure (resp. integral structure): • A cusp form f ( τ ) is real (resp. integral) if � a n q n , f ( τ ) = a n ∈ R (resp. a n ∈ Z ) . n ≥ 1 25 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • On the space of cusp forms of weight 2 k there is a real structure (resp. integral structure): • A cusp form f ( τ ) is real (resp. integral) if � a n q n , f ( τ ) = a n ∈ R (resp. a n ∈ Z ) . n ≥ 1 • Notations: S 2 k (Γ , R ), resp. S 2 k (Γ , Z ). 25 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • On the space of cusp forms of weight 2 k there is a real structure (resp. integral structure): • A cusp form f ( τ ) is real (resp. integral) if � a n q n , f ( τ ) = a n ∈ R (resp. a n ∈ Z ) . n ≥ 1 • Notations: S 2 k (Γ , R ), resp. S 2 k (Γ , Z ). • S 2 k (Γ , Z ) ⊂ S 2 k (Γ , R ) is a lattice. 25 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • On the space of cusp forms of weight 2 k there is a real structure (resp. integral structure): • A cusp form f ( τ ) is real (resp. integral) if � a n q n , f ( τ ) = a n ∈ R (resp. a n ∈ Z ) . n ≥ 1 • Notations: S 2 k (Γ , R ), resp. S 2 k (Γ , Z ). • S 2 k (Γ , Z ) ⊂ S 2 k (Γ , R ) is a lattice. Is there a natural Euclidean structure? Can we compute the volume of the lattice S 2 k (Γ , Z ) ? 25 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • An Euclidean structure on S 2 k (Γ , R ) is induced by the Petersson scalar product: � f ( τ ) g ( τ )(4 π y ) 2 k − 2 dx ∧ dy . � f , g � = H / SL 2 ( Z ) The integral is computed over a fundamental domain for the action of SL 2 ( Z ) on H . 26 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • An Euclidean structure on S 2 k (Γ , R ) is induced by the Petersson scalar product: � f ( τ ) g ( τ )(4 π y ) 2 k − 2 dx ∧ dy . � f , g � = H / SL 2 ( Z ) The integral is computed over a fundamental domain for the action of SL 2 ( Z ) on H . • Similar to Minkowski’s theorem, we rather consider χ L 2 ( S 2 k (Γ)) := − log vol( S 2 k (Γ , R ) / S 2 k (Γ , Z )) . � 26 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Exact formulas for � χ L 2 ( S 2 k (Γ)) are hard to obtain. An asymptotic estimate is easier. 27 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Exact formulas for � χ L 2 ( S 2 k (Γ)) are hard to obtain. An asymptotic estimate is easier. Theorem (Berman–F.) The volume of S 12 k (Γ , Z ) obeys the following asymptotics: � ζ ′ ( − 1) � ζ ( − 1) + 1 k 2 + o ( k 2 ) , χ L 2 ( S 12 k (Γ)) = − 6 � as k → + ∞ , 2 where ζ ( s ) is the Riemann zeta function. Observe that the growth is quadratic in k , as opposed to the dimension, which is linear in k . This is typical of the arithmetic setting: asymptotics of volumes grow one order faster than dimensions. 27 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • These kind of statements are called of arithmetic Hilbert–Samuel type. 28 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • These kind of statements are called of arithmetic Hilbert–Samuel type. • The significance of ζ ′ ( − 1) /ζ ( − 1) in Arakelov geometry was discovered by Bost and K¨ uhn. 28 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • These kind of statements are called of arithmetic Hilbert–Samuel type. • The significance of ζ ′ ( − 1) /ζ ( − 1) in Arakelov geometry was discovered by Bost and K¨ uhn. • The appearance of these quantities is nowadays a tiny part of the Kudla programme and conjectures of Maillot–R¨ ossler. 28 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? What about exact formulas? • Exact formulas (as opposed to asymptotic) are the content of the Riemann–Roch theorem in Arakelov geometry. 29 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? What about exact formulas? • Exact formulas (as opposed to asymptotic) are the content of the Riemann–Roch theorem in Arakelov geometry. • The Riemann–Roch theorem in Arakelov geometry involves a spectral invariant called holomorphic analytic torsion (goes back to Ray–Singer). It is a global invariant. 29 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? What about exact formulas? • Exact formulas (as opposed to asymptotic) are the content of the Riemann–Roch theorem in Arakelov geometry. • The Riemann–Roch theorem in Arakelov geometry involves a spectral invariant called holomorphic analytic torsion (goes back to Ray–Singer). It is a global invariant. • Holomorphic analytic torsion is typical of ”higher dimensions”, and is thus 0 in Minkowski’s case! 29 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? What about exact formulas? • Exact formulas (as opposed to asymptotic) are the content of the Riemann–Roch theorem in Arakelov geometry. • The Riemann–Roch theorem in Arakelov geometry involves a spectral invariant called holomorphic analytic torsion (goes back to Ray–Singer). It is a global invariant. • Holomorphic analytic torsion is typical of ”higher dimensions”, and is thus 0 in Minkowski’s case! • The simplest example of arithmetic Riemann–Roch involving non-trivial analytic torsion is Kronecker’s first limit formula. 29 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Kronecker’s first limit formula Definition (Real analytic Eisenstein series for SL 2 ( Z )) � � y 2 Im ( γτ ) s = 1 E ( τ, s ) = | c τ + d | 2 s , Re( s ) > 1 , 2 c , d ∈ Z Γ ∞ \ Γ ( c , d )=1 � 1 � Z where Γ ∞ is the stabiliser of ∞ in SL 2 ( Z ): Γ ∞ = ± . 0 1 It is real analytic in τ and holomorphic in s for Re( s ) > 1. 30 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • The Eisenstein series E ( τ, s ) has a Fourier expansion of the form E ( τ, s ) = y s + Φ( s ) y 1 − s + . . . , where the rest is L 2 in a neighborhood of ∞ with respect to the hyperbolic volume form. 31 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • The Eisenstein series E ( τ, s ) has a Fourier expansion of the form E ( τ, s ) = y s + Φ( s ) y 1 − s + . . . , where the rest is L 2 in a neighborhood of ∞ with respect to the hyperbolic volume form. • E ( τ, s ) has a meromorphic continuation to s ∈ C , with a simple pole at s = 1 (scattering theory). It satisfies the functional equation E ( τ, s ) = Φ( s ) E ( τ, 1 − s ) . 31 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • The Eisenstein series E ( τ, s ) has a Fourier expansion of the form E ( τ, s ) = y s + Φ( s ) y 1 − s + . . . , where the rest is L 2 in a neighborhood of ∞ with respect to the hyperbolic volume form. • E ( τ, s ) has a meromorphic continuation to s ∈ C , with a simple pole at s = 1 (scattering theory). It satisfies the functional equation E ( τ, s ) = Φ( s ) E ( τ, 1 − s ) . • The function Φ( s ) is an example of scattering matrix (as in quantum mechanics), and has a Laurent expansion at s = 1 � � 2 − 2 log(4 π ) + 2 ζ ′ ( − 1) Φ( s ) = 3 s − 1 + 3 1 + O ( s − 1) . ζ ( − 1) π π 31 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Theorem (Kronecker) The derivative of E ( τ, s ) at s = 0 is given by � d s =0 E ( τ, s ) = 1 � 12 log( | ∆( τ ) | 2 y 12 ) , � ds where ∆( τ ) = q � n ≥ 1 (1 − q n ) 24 is Ramanujan’s weight 12 cusp form. This is equivalent to arithmetic Riemann–Roch applied to the trivial hermitian line bundle on an elliptic curve, equipped with its flat invariant metric of volume 1. The quantity ( d / ds ) | s =0 E ( τ, s ) is (essentially) the contribution of the holomorphic analytic torsion. 32 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? The Laplace operator � ∂ 2 � ∂ u 2 + ∂ 2 C ∞ ( C ) Z + τ Z ∆ = − (Im τ ) acting on ∂ v 2 has spectrum λ m , n = (2 π ) 2 | m τ + n | 2 ( m , n ) ∈ Z 2 . , Im τ 33 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? The Laplace operator � ∂ 2 � ∂ u 2 + ∂ 2 C ∞ ( C ) Z + τ Z ∆ = − (Im τ ) acting on ∂ v 2 has spectrum λ m , n = (2 π ) 2 | m τ + n | 2 ( m , n ) ∈ Z 2 . , Im τ The associated spectral zeta function is � 1 = 2(2 π ) − 2 s ζ ( s ) E ( τ, s ) . ζ sp ( s ) := λ s m , n ( m , n ) � =0 The corresponding analytic torsion is � T := − d � � s =0 ζ sp ( s ) = log det ∆ . ds 33 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? An exotic example We go back to the case of modular forms. The space S 2 (Γ) of cusp forms of weight 2 for SL 2 ( Z ) vanishes, hence there is an exact formula for the volume :-) 34 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? An exotic example We go back to the case of modular forms. The space S 2 (Γ) of cusp forms of weight 2 for SL 2 ( Z ) vanishes, hence there is an exact formula for the volume :-) With Anna von Pippich we proved an arithmetic Riemann–Roch theorem that applies to this context. Because of the vanishing of the space of cusp forms above, our theorem reduces to an exact evaluation of the analytic torsion. 34 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? An exotic example We go back to the case of modular forms. The space S 2 (Γ) of cusp forms of weight 2 for SL 2 ( Z ) vanishes, hence there is an exact formula for the volume :-) With Anna von Pippich we proved an arithmetic Riemann–Roch theorem that applies to this context. Because of the vanishing of the space of cusp forms above, our theorem reduces to an exact evaluation of the analytic torsion. The analytic torsion in this example is given by a special value of an exotic zeta function: the Selberg zeta function of PSL 2 ( Z ). It arises in the geometric side of the Selberg trace formulal. It was introduced by Selberg 60 years ago. 34 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Definition (Selberg zeta function of PSL 2 ( Z )) The Selberg zeta function of PSL 2 ( Z ) is � � h ( d ) � � 1 − ǫ − 2( s + k ) Z ( s , PSL 2 ( Z )) = , Re( s ) > 1 . d d k ≥ 0 Here: ◮ d > 0 is a discriminant with d ≡ 0 or 1 mod 4. √ ◮ ǫ d = ( x 0 + dy 0 ) / 2 is the fundamental solution of the Pell equation x 2 − dy 2 = 4 (smallest possible with ǫ d > 1). ◮ h ( d ) is the class number of binary integral quadratic forms of discriminant d (i.e. up to SL 2 ( Z ) equivalence). 35 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • The actual definition of the Selberg zeta function is in terms of lengths of closed hyperbolic geodesics on H / SL 2 ( Z ). These are in bijection with binary quadratic forms of positive discriminant, and the lengths are given by the logarithms of the fundamental units. 36 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • The actual definition of the Selberg zeta function is in terms of lengths of closed hyperbolic geodesics on H / SL 2 ( Z ). These are in bijection with binary quadratic forms of positive discriminant, and the lengths are given by the logarithms of the fundamental units. • The Selberg zeta function has a meromorphic continuation to s ∈ C , with a simple zero at s = 1 (Selberg trace formula). 36 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • The actual definition of the Selberg zeta function is in terms of lengths of closed hyperbolic geodesics on H / SL 2 ( Z ). These are in bijection with binary quadratic forms of positive discriminant, and the lengths are given by the logarithms of the fundamental units. • The Selberg zeta function has a meromorphic continuation to s ∈ C , with a simple zero at s = 1 (Selberg trace formula). • It has a functional equation relating the values at s and 1 − s . 36 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • The actual definition of the Selberg zeta function is in terms of lengths of closed hyperbolic geodesics on H / SL 2 ( Z ). These are in bijection with binary quadratic forms of positive discriminant, and the lengths are given by the logarithms of the fundamental units. • The Selberg zeta function has a meromorphic continuation to s ∈ C , with a simple zero at s = 1 (Selberg trace formula). • It has a functional equation relating the values at s and 1 − s . 4 + t 2 in the • Z ( s , PSL 2 ( Z )) vanishes at s = 1 2 + it , for λ = 1 discrete spectrum of the hyperbolic Laplacian ∆ hyp on H / SL 2 ( Z ). 36 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? Theorem (F. – von Pippich) The special value log Z ′ (1 , PSL 2 ( Z )) is an explicit rational linear combination of the quantities ζ ′ ζ ′ Q ( √− 1) (0) Q ( √− 3) (0) ζ ′ ( − 1) ζ ( − 1) , ζ Q ( √− 1) (0) , ζ Q ( √− 3) (0) , log 2 , log 3 , log π, γ, 1 . Here: ◮ ζ Q ( √− 1) ( s ) is the Dedekind zeta function of Q ( √− 1) . ◮ ζ Q ( √− 3) ( s ) is the Dedekind zeta function of Q ( √− 3) . ◮ γ is the Euler constant. 37 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • Observe the (conjectural) analogy between Z ( s , PSL 2 ( Z )) and Dedeking zeta functions ζ K ( s ): 38 / 39

Local to global formulas Asymptotic volumes of cusp forms What about exact formulas? • Observe the (conjectural) analogy between Z ( s , PSL 2 ( Z )) and Dedeking zeta functions ζ K ( s ): ◮ meromorphic continuation and functional equation s ↔ 1 − s . 38 / 39