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?

Local to global formulas in geometry and number theory

Gerard Freixas i Montplet

C.N.R.S. – Institut de Math´ ematiques de Jussieu - Paris Rive Gauche

Kolloquium ¨ uber Reine Mathematik Universit¨ at Hamburg January 2019

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The Gauss–Bonet theorem

• (M, gM) a compact connected Riemannian surface.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The Gauss–Bonet theorem

• (M, gM) a compact connected Riemannian surface.
• Local invariants:

◮ the gaussian curvature K. ◮ the area element dA.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The Gauss–Bonet theorem

• (M, gM) a compact connected Riemannian surface.
• Local invariants:

◮ the gaussian curvature K. ◮ the area element dA.

• Global invariant: the Euler–Poincar´

e characteristic of M χ(M) = dim H0(M, R) − dim H1(M, R) + dim H2(M, R)

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The Gauss–Bonet theorem

Theorem (Gauss–Bonet) χ(M) = 1 2π

• M

KdA.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The Gauss–Bonet theorem

Theorem (Gauss–Bonet) χ(M) = 1 2π

• M

KdA. Recall that the Euler–Poincar´ e characteristic is computed in terms

• f the genus g of M (“number of holes”):

χ(M) = 2 − 2g.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The Gauss–Bonet theorem

Theorem (Gauss–Bonet) χ(M) = 1 2π

• M

KdA. Recall that the Euler–Poincar´ e characteristic is computed in terms

• f the genus g of M (“number of holes”):

χ(M) = 2 − 2g. Hence 1 2π

• M

KdA = 2 − 2g.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The Gauss–Bonet formula takes the form

• local = global.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Riemann–Roch formula

• X a compact connected Riemann surface:

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Riemann–Roch formula

• X a compact connected Riemann surface:

◮ compact differentiable surface.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Riemann–Roch formula

• X a compact connected Riemann surface:

◮ compact differentiable surface. ◮ local charts valued in C.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Riemann–Roch formula

• X a compact connected Riemann surface:

◮ compact differentiable surface. ◮ local charts valued in C. ◮ holomorphic change of charts.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Riemann–Roch formula

• X a compact connected Riemann surface:

◮ compact differentiable surface. ◮ local charts valued in C. ◮ holomorphic change of charts. ◮ meromorphic functions f ; they form a field C(X). It is a finite

extension of C(t).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Riemann–Roch formula

• X a compact connected Riemann surface:

◮ compact differentiable surface. ◮ local charts valued in C. ◮ holomorphic change of charts. ◮ meromorphic functions f ; they form a field C(X). It is a finite

extension of C(t).

◮ meromorphic differential forms ω: locally f (z)dz with f

meromorphic.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Divisor (local input): D =

k nkPk a finite formal combination

• f points Pk ∈ X with integer coefficients nk.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Divisor (local input): D =

k nkPk a finite formal combination

• f points Pk ∈ X with integer coefficients nk. For example:

◮ div f = P∈X(ordP f )P, for f ∈ C(X)×. ◮ KX = P∈X(ordP ω)P (canonical divisor). Independent of ω

modulo divisors of meromorphic functions div f .

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Associated to a divisor D = nkPk we have two global numerical invariants:

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Associated to a divisor D = nkPk we have two global numerical invariants:

◮ degree: deg D = k nk. For instance, deg(div f ) = 0.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Associated to a divisor D = nkPk we have two global numerical invariants:

◮ degree: deg D = k nk. For instance, deg(div f ) = 0. ◮ holomorphic Euler–Poincar´

e characteristic: χ(X, D) = dim H0(X, O(D)) − dim H0(X, O(KX − D)), where H0(X, O(D)) = {f ∈ C(X)× | div f + D ≥ 0} ∪ {0} is the C-vector space of meromorphic functions with divisor “controlled” by D (coherent cohomology).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Theorem (Riemann–Roch) The degree and Euler–Poincar´ e characteristic are related by χ(X, D) = deg D − 1 2 deg KX.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Theorem (Riemann–Roch) The degree and Euler–Poincar´ e characteristic are related by χ(X, D) = deg D − 1 2 deg KX. Main ingredients in the proof:

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Theorem (Riemann–Roch) The degree and Euler–Poincar´ e characteristic are related by χ(X, D) = deg D − 1 2 deg KX. Main ingredients in the proof:

◮ a bit of Hodge theory:

H1(X, R) ⊗ C ≃ H0(X, KX) ⊕ H0(X, KX). de Rham coh. ≃ harmonic forms

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Theorem (Riemann–Roch) The degree and Euler–Poincar´ e characteristic are related by χ(X, D) = deg D − 1 2 deg KX. Main ingredients in the proof:

◮ a bit of Hodge theory:

H1(X, R) ⊗ C ≃ H0(X, KX) ⊕ H0(X, KX). de Rham coh. ≃ harmonic forms

◮ the Gauss–Bonet theorem stated in the form

deg KX = 2g − 2.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The Riemann–Roch formula takes the form deg local = global.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

An asymptotic form of Riemann–Roch is sometimes enough:

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

An asymptotic form of Riemann–Roch is sometimes enough: Corollary (Hilbert–Samuel) If deg D > 0, then dim H0(X, O(kD)) = k deg D + o(k), as k → +∞. In particular the dimension grows linearly in k.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Minkowski’s geometry of numbers

• K a number field: finite field extension of Q, of degree

[K : Q] = dimQ K = #{σ: K ֒ → C} < ∞.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Minkowski’s geometry of numbers

• K a number field: finite field extension of Q, of degree

[K : Q] = dimQ K = #{σ: K ֒ → C} < ∞.

• OK ⊂ K its ring of integers: those x ∈ K satisfying a

polynomial equation xn + an−1xn−1 + . . . + a0 = 0, ak ∈ Z. OK is a free Z-module of rank [K : Q].

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Minkowski’s geometry of numbers

• K a number field: finite field extension of Q, of degree

[K : Q] = dimQ K = #{σ: K ֒ → C} < ∞.

• OK ⊂ K its ring of integers: those x ∈ K satisfying a

polynomial equation xn + an−1xn−1 + . . . + a0 = 0, ak ∈ Z. OK is a free Z-module of rank [K : Q].

• OK ⊂ K is the biggest subring with these properties.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• ∆K/Q ∈ Z the discriminant of K.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• ∆K/Q ∈ Z the discriminant of K.
• ∆K/Q is a measure of “bad reduction”. In particular it satisfies:

p (prime) | ∆K/Q ⇐ ⇒ OK ⊗Z Fp has nilpotents. We say that p ramifies in K.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• ∆K/Q ∈ Z the discriminant of K.
• ∆K/Q is a measure of “bad reduction”. In particular it satisfies:

p (prime) | ∆K/Q ⇐ ⇒ OK ⊗Z Fp has nilpotents. We say that p ramifies in K.

• If Q K, then |∆K/Q| > 1.

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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 OK =

• Z[ 1+

√ D 2

] if D ≡ 1 mod 4, Z[ √ D]

• therwise,

and ∆K/Q =

• D

if D ≡ 1 mod 4, 4D

• therwise.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Local input: a ⊆ OK a non-trivial ideal.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Local input: a ⊆ OK a non-trivial ideal.
• Two global numerical invariants associated to a:

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Local input: a ⊆ OK a non-trivial ideal.
• Two global numerical invariants associated to a:

◮ the norm N(a) = #(OK/a).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Local input: a ⊆ OK a non-trivial ideal.
• Two global numerical invariants associated to a:

◮ the norm N(a) = #(OK/a). ◮ the (co)volume: a embeds diagonally as a lattice in an

Euclidean space KR: a ֒ → KR := ⊥

• σ : K֒

→C(C, | · |)

complex conjugation KR/a is a compact Riemannian torus of volume vol(KR/a).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Theorem (Minkowski) The norm and the volume are related by vol(KR/a) = N(a)

• |∆K/Q|,

where ∆K/Q is the discriminant of the number field.

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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(KR/a) = − log N(a) − 1 2 log |∆K/Q|, and we have notations for these quantities:

• χL2(a) =

deg a − 1 2 log |∆K/Q|.

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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(KR/a) = − log N(a) − 1 2 log |∆K/Q|, and we have notations for these quantities:

• χL2(a) =

deg a − 1 2 log |∆K/Q|.

◮

χL2 is a sort of Euler characteristic.

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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(KR/a) = − log N(a) − 1 2 log |∆K/Q|, and we have notations for these quantities:

• χL2(a) =

deg a − 1 2 log |∆K/Q|.

◮

χL2 is a sort of Euler characteristic.

◮

deg is called arithmetic degree.

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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(KR/a) = − log N(a) − 1 2 log |∆K/Q|, and we have notations for these quantities:

• χL2(a) =

deg a − 1 2 log |∆K/Q|.

◮

χL2 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.

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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

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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.

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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) > 2dim V vol(V /Γ). Then Ω ∩ (Γ \ {0}) = ∅.

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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) > 2dim V vol(V /Γ). Then Ω ∩ (Γ \ {0}) = ∅.

• Typically V = KR and Γ = a ⊂ OK a non-trivial ideal.

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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) > 2dim V vol(V /Γ). Then Ω ∩ (Γ \ {0}) = ∅.

• Typically V = KR and Γ = a ⊂ OK a non-trivial ideal.
• One derives |∆K/Q| > 1 and the finiteness of the class number:

Cl(K) = {projective OK − modules of rank 1}/iso. hK = class number of K = #Cl(K) < +∞.

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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

ζK(s) =

• 0=a⊆OK

1 N(a)s , Re(s) > 1, its Dedekind zeta function.

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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

ζK(s) =

• 0=a⊆OK

1 N(a)s , Re(s) > 1, its Dedekind zeta function.

• Euler product factorisation in “local factors”:

ζK(s) =

• p
• 1 − (Np)−s−1 ,

Re(s) > 1, where p ⊂ OK are the maximal ideals.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The analytic class number formula

• ζK(s) has a meromorphic continuation to s ∈ C, with a simple

pole at s = 1.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The analytic class number formula

• ζK(s) has a meromorphic continuation to s ∈ C, with a simple

pole at s = 1.

• Functonial equation relating ζK(s) and ζK(1 − s).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The analytic class number formula

• ζK(s) has a meromorphic continuation to s ∈ C, with a simple

pole at s = 1.

• Functonial equation relating ζK(s) and ζK(1 − s).
• Generalized Riemann Hypothesis: non-trivial zeros of ζK(s) are

located on Re(s) = 1/2.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The analytic class number formula

• ζK(s) has a meromorphic continuation to s ∈ C, with a simple

pole at s = 1.

• Functonial equation relating ζK(s) and ζK(1 − s).
• Generalized Riemann Hypothesis: non-trivial zeros of ζK(s) are

located on Re(s) = 1/2.

• Hilbert–P´
• lya conjecture: there exists an unbounded self-adjoint
• perator H, such that

ζK 1 2 + it

• = 0
• t ∈ Spec H.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Theorem (Dirichlet, Dedekind) The residue of ζK(s) at s = 1 is given by Ress=1 ζK(s) = 2r1(2π)r2 hK · RK |O×

K tor| ·

• |∆K/Q|

.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Theorem (Dirichlet, Dedekind) The residue of ζK(s) at s = 1 is given by Ress=1 ζK(s) = 2r1(2π)r2 hK · RK |O×

K tor| ·

• |∆K/Q|

.

◮ r1 (resp. 2r2) is the number of real (resp. complex non-real)

embeddings of K.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Theorem (Dirichlet, Dedekind) The residue of ζK(s) at s = 1 is given by Ress=1 ζK(s) = 2r1(2π)r2 hK · RK |O×

K tor| ·

• |∆K/Q|

.

◮ r1 (resp. 2r2) is the number of real (resp. complex non-real)

embeddings of K.

◮ hK is the class number of K (number of isomorphism classes

• f projective OK-modules of rank 1).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Theorem (Dirichlet, Dedekind) The residue of ζK(s) at s = 1 is given by Ress=1 ζK(s) = 2r1(2π)r2 hK · RK |O×

K tor| ·

• |∆K/Q|

.

◮ r1 (resp. 2r2) is the number of real (resp. complex non-real)

embeddings of K.

◮ hK is the class number of K (number of isomorphism classes

• f projective OK-modules of rank 1).

◮ RK is called the regulator of K.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The analytic class number formula takes the form Ress=1 ◦ Mer. Cont. local = global.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Asymptotic volumes of cusp forms

• Γ = SL2(Z).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Asymptotic volumes of cusp forms

• Γ = SL2(Z).
• H the upper half plane with coordinate τ = x + iy.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Asymptotic volumes of cusp forms

• Γ = SL2(Z).
• H the upper half plane with coordinate τ = x + iy.
• Action of Γ on H ∪ P1(Q) by fractional linear transformations:

a b c d

• τ = aτ + b

cτ + d .

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Asymptotic volumes of cusp forms

• Γ = SL2(Z).
• H the upper half plane with coordinate τ = x + iy.
• Action of Γ on H ∪ P1(Q) by fractional linear transformations:

a b c d

• τ = aτ + b

cτ + d .

• Open modular curve of level 1:

Y (1) := H/SL2(Z)

∼

− → C.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Asymptotic volumes of cusp forms

• Γ = SL2(Z).
• H the upper half plane with coordinate τ = x + iy.
• Action of Γ on H ∪ P1(Q) by fractional linear transformations:

a b c d

• τ = aτ + b

cτ + d .

• Open modular curve of level 1:

Y (1) := H/SL2(Z)

∼

− → C.

• Compactified modular curve of level 1:

X(1) := (H ∪ P1(Q))/SL2(Z) = Y (1) ∪ {∞}

∼

− → P1(C).

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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 2k if f (γτ) = (cτ + d)2kf (τ), γ = a b c d

• ∈ SL2(Z)

and is holomorphic at ∞: Fourier expansion f (τ) =

• n≥0

anqn, q = e2πiτ.

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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 2k if f (γτ) = (cτ + d)2kf (τ), γ = a b c d

• ∈ SL2(Z)

and is holomorphic at ∞: Fourier expansion f (τ) =

• n≥0

anqn, q = e2πiτ.

• We say that f is a cusp form if moreover a0 = 0.

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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 2k if f (γτ) = (cτ + d)2kf (τ), γ = a b c d

• ∈ SL2(Z)

and is holomorphic at ∞: Fourier expansion f (τ) =

• n≥0

anqn, q = e2πiτ.

• We say that f is a cusp form if moreover a0 = 0.
• Cusp forms of weight 2k constitute a finite dimensional vector

space S2k(Γ, C). The dimension is linear in k: Riemann–Roch.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• On the space of cusp forms of weight 2k there is a real structure

(resp. integral structure):

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• On the space of cusp forms of weight 2k there is a real structure

(resp. integral structure):

• A cusp form f (τ) is real (resp. integral) if

f (τ) =

• n≥1

anqn, an ∈ R (resp. an ∈ Z).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• On the space of cusp forms of weight 2k there is a real structure

(resp. integral structure):

• A cusp form f (τ) is real (resp. integral) if

f (τ) =

• n≥1

anqn, an ∈ R (resp. an ∈ Z).

• Notations: S2k(Γ, R), resp. S2k(Γ, Z).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• On the space of cusp forms of weight 2k there is a real structure

(resp. integral structure):

• A cusp form f (τ) is real (resp. integral) if

f (τ) =

• n≥1

anqn, an ∈ R (resp. an ∈ Z).

• Notations: S2k(Γ, R), resp. S2k(Γ, Z).
• S2k(Γ, Z) ⊂ S2k(Γ, R) is a lattice.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• On the space of cusp forms of weight 2k there is a real structure

(resp. integral structure):

• A cusp form f (τ) is real (resp. integral) if

f (τ) =

• n≥1

anqn, an ∈ R (resp. an ∈ Z).

• Notations: S2k(Γ, R), resp. S2k(Γ, Z).
• S2k(Γ, Z) ⊂ S2k(Γ, R) is a lattice.

Is there a natural Euclidean structure? Can we compute the volume of the lattice S2k(Γ, Z)?

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• An Euclidean structure on S2k(Γ, R) is induced by the Petersson

scalar product: f , g =

• H/SL2(Z)

f (τ)g(τ)(4πy)2k−2dx ∧ dy. The integral is computed over a fundamental domain for the action

• f SL2(Z) on H.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• An Euclidean structure on S2k(Γ, R) is induced by the Petersson

scalar product: f , g =

• H/SL2(Z)

f (τ)g(τ)(4πy)2k−2dx ∧ dy. The integral is computed over a fundamental domain for the action

• f SL2(Z) on H.
• Similar to Minkowski’s theorem, we rather consider
• χL2(S2k(Γ)) := − log vol(S2k(Γ, R)/S2k(Γ, Z)).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Exact formulas for χL2(S2k(Γ)) are hard to obtain. An asymptotic estimate is easier.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Exact formulas for χL2(S2k(Γ)) are hard to obtain. An asymptotic estimate is easier. Theorem (Berman–F.) The volume of S12k(Γ, Z) obeys the following asymptotics:

• χL2(S12k(Γ)) = −6

ζ′(−1) ζ(−1) + 1 2

• k2 + o(k2),

as k → +∞, 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.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• These kind of statements are called of arithmetic

Hilbert–Samuel type.

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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.

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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¨

• ssler.

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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.

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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.

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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!

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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.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Kronecker’s first limit formula

Definition (Real analytic Eisenstein series for SL2(Z)) E(τ, s) =

• Γ∞\Γ

Im(γτ)s = 1 2

• c,d∈Z

(c,d)=1

y2 |cτ + d|2s , Re(s) > 1, where Γ∞ is the stabiliser of ∞ in SL2(Z): Γ∞ = ± 1 Z 1

• .

It is real analytic in τ and holomorphic in s for Re(s) > 1.

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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) = ys + Φ(s)y1−s + . . . , where the rest is L2 in a neighborhood of ∞ with respect to the hyperbolic volume form.

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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) = ys + Φ(s)y1−s + . . . , where the rest is L2 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).

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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) = ys + Φ(s)y1−s + . . . , where the rest is L2 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 Φ(s) = 3 π 1 s − 1 + 3 π

• 2 − 2 log(4π) + 2ζ′(−1)

ζ(−1)

• + O(s − 1).

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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 ds

• s=0E(τ, s) = 1

12 log(|∆(τ)|2y12), where ∆(τ) = q

n≥1(1 − qn)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.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The Laplace operator ∆ = −(Im τ) ∂2 ∂u2 + ∂2 ∂v2

• acting on

C∞(C)Z+τZ has spectrum λm,n = (2π)2 |mτ + n|2 Im τ , (m, n) ∈ Z2.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

The Laplace operator ∆ = −(Im τ) ∂2 ∂u2 + ∂2 ∂v2

• acting on

C∞(C)Z+τZ has spectrum λm,n = (2π)2 |mτ + n|2 Im τ , (m, n) ∈ Z2. The associated spectral zeta function is ζsp(s) :=

• (m,n)=0

1 λs

m,n

= 2(2π)−2sζ(s)E(τ, s). The corresponding analytic torsion is T := − d ds

• s=0ζsp(s) = log det ∆.

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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 S2(Γ) of cusp forms of weight 2 for SL2(Z) vanishes, hence there is an exact formula for the volume :-)

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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 S2(Γ) of cusp forms of weight 2 for SL2(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.

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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 S2(Γ) of cusp forms of weight 2 for SL2(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 PSL2(Z). It arises in the geometric side of the Selberg trace formulal. It was introduced by Selberg 60 years ago.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Definition (Selberg zeta function of PSL2(Z)) The Selberg zeta function of PSL2(Z) is Z(s, PSL2(Z)) =

• d
• k≥0
• 1 − ǫ−2(s+k)

d

h(d) , Re(s) > 1. Here:

◮ d > 0 is a discriminant with d ≡ 0 or 1 mod 4. ◮ ǫd = (x0 +

√ dy0)/2 is the fundamental solution of the Pell equation x2 − dy2 = 4 (smallest possible with ǫd > 1).

◮ h(d) is the class number of binary integral quadratic forms of

discriminant d (i.e. up to SL2(Z) equivalence).

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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/SL2(Z). These are in bijection with binary quadratic forms of positive discriminant, and the lengths are given by the logarithms of the fundamental units.

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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/SL2(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).

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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/SL2(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.

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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/SL2(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.
• Z(s, PSL2(Z)) vanishes at s = 1

2 + it, for λ = 1 4 + t2 in the

discrete spectrum of the hyperbolic Laplacian ∆hyp on H/SL2(Z).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Theorem (F. – von Pippich) The special value log Z ′(1, PSL2(Z)) is an explicit rational linear combination of the quantities ζ′(−1) ζ(−1) , ζ′

Q(√−1)(0)

ζQ(√−1)(0), ζ′

Q(√−3)(0)

ζQ(√−3)(0), log 2, log 3, log π, γ, 1. Here:

◮ ζQ(√−1)(s) is the Dedekind zeta function of Q(√−1). ◮ ζQ(√−3)(s) is the Dedekind zeta function of Q(√−3). ◮ γ is the Euler constant.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Observe the (conjectural) analogy between Z(s, PSL2(Z)) and

Dedeking zeta functions ζK(s):

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Observe the (conjectural) analogy between Z(s, PSL2(Z)) and

Dedeking zeta functions ζK(s):

◮ meromorphic continuation and functional equation s ↔ 1 − s.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Observe the (conjectural) analogy between Z(s, PSL2(Z)) and

Dedeking zeta functions ζK(s):

◮ meromorphic continuation and functional equation s ↔ 1 − s. ◮ non-trivial zeros located on Re(s) = 1 2.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Observe the (conjectural) analogy between Z(s, PSL2(Z)) and

Dedeking zeta functions ζK(s):

◮ meromorphic continuation and functional equation s ↔ 1 − s. ◮ non-trivial zeros located on Re(s) = 1 2. ◮ non-trivial zeros spectrum of a self-adjoint operator.

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

• Observe the (conjectural) analogy between Z(s, PSL2(Z)) and

Dedeking zeta functions ζK(s):

◮ meromorphic continuation and functional equation s ↔ 1 − s. ◮ non-trivial zeros located on Re(s) = 1 2. ◮ non-trivial zeros spectrum of a self-adjoint operator.

• In this analogy:

Z ′(1, PSL2(Z))

• Ress=1 ζK(s).

Hence, the computation of Z ′(1, PSL2(Z)) is the analytic class number formula PSL2(Z).

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Local to global formulas Asymptotic volumes of cusp forms What about exact formulas?

Question Is their a formalism that explains the analytic class number formula

• f Dedekind zeta functions as a sort of arithmetic Riemann–Roch

theorem?

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