On the analytic class number formula for Selberg zeta functions - - PowerPoint PPT Presentation

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

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

Dedekind zeta function

Let K/Q a number field and ζK(s) its Dedekind zeta function: ζK(s) =

• 0=a⊆OK

1 (Na)s =

• p⊂OK

maximal ideal

• 1 − (Np)−s−1 .

◮ Absolute convergence for Re(s) > 1 and meromorphic

continuation to C.

◮ Simple pole at s = 1. ◮ Functional equation. ◮ Riemann hypothesis.

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

Analytic class number formula

Theorem The residue of ζK(s) at s = 1 is given by Ress=1 ζK(s) = 2r1(2π)r2 hKRK wK

• |∆K/Q|

, where

◮ r1 (resp. r2) number of real (resp. complex) embeddings of K. ◮ hK = #Cl(K) is the class number. ◮ RK is the regulator. ◮ wK = #OK,tors. ◮ ∆K/Q is the absolute discriminant.

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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 ⊂ PSL2(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 inY ↔ free homotopy classes of curves ↔ conjugacy classes of hyperbolic elements in Γ

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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 Z(s, Γ) =

• γ

∞

• k=0
• 1 − e−(s+k)ℓ(γ)

, Re(s) > 1, where

◮ γ runs over oriented primitive closed geodesics. ◮ ℓ(γ) is the length of γ.

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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:

• k

h(tk) =volhyp(Γ\H) 4π +∞

−∞

h(r) tanh(πr)dr +

• γ

∞

• k=1
• h(kℓ(γ))

ℓ(γ)/2 sinh(ℓ(γ)/2), where:

◮ the λk = 1 4 + t2 k form the spectrum of −y2 ∂2 ∂x2 + ∂2 ∂y2

• .

◮

h is the Fourier transform of h.

◮ “curiosity”: appearance of the

A genus...

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

Example For Γ = PSL2(Z), Sarnak shows: Z(s, PSL2(Z)) =

• d>0

d≡0 or 1 (4) ∞

• k=0

(1 − ε−2(s+k)

d

)h(d), where:

◮ εd > 1 is the fundamental solution of the Pell equation

x2 − dy2 = 4.

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

discriminant d.

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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 PSL2(Z)), the non-trivial zeroes are

located on Re(s) = 1

2, and correspond to the (discrete)

spectrum of the hyperbolic Laplacian ∆hyp = −y2 ∂2 ∂x2 + ∂2 ∂y2

• .

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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 PSL2(Z)), the non-trivial zeroes are

located on Re(s) = 1

2, and correspond to the (discrete)

spectrum of the hyperbolic Laplacian ∆hyp = −y2 ∂2 ∂x2 + ∂2 ∂y2

• .

Missing:

◮ Analytic class number formula: expression for Z ′(1, Γ)?

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

If Γ is co-compact and torsion free: Z ′(1, Γ)

trace formula d’Hoker–Phong, Sarnak det ∆hyp

• Gillet–Soul´

e

• Arithmetic Riemann–Roch

(cohomological side)

where det ∆hyp is the zeta regularized determinant of the Laplacian ∆hyp.

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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 •

X/Y ))

in CH•(Y )Q. In particular, if Y = Spec C (Hirzebruch–Riemann–Roch): χ(X, E) =

• p

(−1)p dim Hp(X, E) =

• X

ch(E) td(TX) ∈ Z.

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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∞.

◮ A0,p(X(C), EC) carries a hermitian L2 scalar product. ◮ ∆0,p ∂

Laplacian acting on A0,p(X(C), EC).

◮ Flat base change, Dolbeault isomorphism and Hodge theory

give Hp(X, E)C ≃ H0,p

∂ (X(C), EC) ≃ ker ∆0,p ∂

⊂ A0,p(X(C), EC). Hence Hp(X, E)C inherits the L2 scalar product.

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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:

• deg H•(X, E)L2 =
• (−1)p log #Hp(X, E)tor

−

• p

(−1)p log vol(Hp(X, E)free, hL2), corrected by the holomorphic analytic torsion: T = T(EC, h, ω) =

• p

(−1)pp log det ∆0,p

∂ .

Hence it catches the whole spectrum of the Laplacians: harmonic forms and non-trival eigenvalues.

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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).

They land in the arithmetic Chow groups CH

• (X)Q.

For instance:

◮

ch(E) arithmetic Chern class.

◮

td(T •

X/Z, ω) arithmetic Todd genus of the tangent complex.

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

There is a morphism

• X :

CH

top(X)

R

[′ nP · P, g] ✤

′ nP log #k(P) + 1

2

• X(C) g,

where P denotes a closed point in X with residue field k(P), and g is a top degree current on X(C).

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Theorem (Gillet–Soul´ e) There is an equality of real numbers

• deg H•(X, E)L2 − 1

2T =

• X
• ch(E)

td(T •

X/Z, ω)

− 1 2

• X(C)

ch(EC) td(TXC)R(TXC), where R is the R-genus of Gillet–Soul´ e. The R-genus is the additive characteristic class determined by the power series: R(x) =

• m≥1
• dd
• ζ′(−m) + ζ(−m)(1 + 1

2 + . . . + 1 m) xm m! .

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◮ 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¨

• hler

(Gillet–R¨

• ssler–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).

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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¨

• hler

(Gillet–R¨

• ssler–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!

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

Non-example

◮ P1 Z → Spec Z seen as an integral model of PSL2(Z)\H ∪ {∞}. ◮ E = O the trivial line bundle. ◮ K¨

ahler metric induced from the hyperbolic metric on H.

◮ The hyperbolic metric is singular at the elliptic fixed points

and ∞.

◮ ∆hyp has continuous spectrum (Eisenstein series). The

analytic torsion is not defined.

◮ The arithmetic Riemann–Roch theorem does not apply. ◮ In a conjectural formula, log Z ′(1, PSL2(Z)) should replace

the undefined analytic torsion.

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

Some inspiration: Noether formula on Hilbert modular surfaces

Let ( X, D) be a toroidal compactification of a connected Hilbert modular surface Γ\H2, Γ ⊂ SL2(OF) sufficiently small. By the Noether formula (or Hirzebruch–Riemann–Roch): χ(O

X) =

• X

td(Ω

X).

The formula computes the dimension of the space of cusp forms of parallel weight 2: since H1( X, O

X) = 0, we have

χ(O

X) = 1 + dim S(2,2)(Γ, C).

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Instead of td(Ω

X), it is more natural to consider td(Ω X(log D)):

by a theorem of Mumford, it has a Chern–Weil representative using the dual of the hyperbolic metric. Then

• X

td(Ω

X(log D)) = 1

6 volhyp(Γ\H2) (2π)2 . = [Γ: SL2(OF)] · ζF(−1). Here . = means equality up to fudge factors. By the residue exact sequence and the multiplicativity of Todd,

• ne finds an expression
• X

td(Ω

X) =

• X

td(Ω

X(log D)) + boundary,

where boundary is some explicit intersection number supported on D.

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Hirzebruch computes the boundary contribution from an explicit description of the toroidal compactification and Meyer’s theorem: boundary . =

• p cusp

Lp(0), where Lp(s) is the Shimizu L-function naturally attached to the cusp p. All in all χ(O

X) .

= [Γ: SL2(OF)]ζF(−1) +

• p cusp

Lp(0). In the arithmetic setting, for P1

Z we expect a similar formula.

Instead of values of L-functions (over F), we expect logarithmic derivatives of L-functions (over Q).

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An arithmetic Noether formula

◮ π: X → S := Spec OK an arithmetic surface. ◮ σ1, . . . , σn : S → X generically disjoint sections. ◮ We assume that

X(C) =

• ν : K֒

→C

Γν\H ∪ {cusps}, where the Γν are fuchsian groups of the first kind and same type, and σ1(ν), . . . , σn(ν) is the set of elliptic fixed points and cusps of the ν component. We suppose their orders are 2 ≤ m1, . . . , mn ≤ ∞.

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We define a hermitian Q-line bundle: ωX/S(D)hyp, D =

• i
• 1 − 1

mi

• σi,

where hyp indicates we endow it with the dual of the hyperbolic

• metric. It is a Mumford good hermitian line bundle.

After Bost and K¨ uhn, its arithmetic self-intersection number is well-defined: (ωX/S(D)2

hyp) ∈ R.

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We define the arithmetic ψ Q-line bundle

• ψW =
• i
• 1 − 1

m2

i

• (σ∗

i ωX/S)W ∈

Pic(S) ⊗ Q, where W indicates the Wolpert metric: if z is a holomorphic coordinate in a neighborhood of σi(ν) ∈ X(C), such that the hyperbolic metric reads

|dz|2 |z|2(log |z|)2 (cusp)

• r

4|dz|2 |z|2−2/m(1 − |z|2/m)2 (fixed point),

then we declare dz |z=0 W = 1.

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Finally, the L2 metric on H•(X, O) is already well-defined (the hyperbolic metric has finite volume). Theorem (F.–von Pippich) There is an equality of real numbers 12 deg H•(X, O) + 6

• ν : K֒

→C

log Z ′(1, Γν) + deg ψW − δ = (ωX/S(D)2

hyp) − 1

2

• i=j
• 1 − 1

mi 1 − 1 mj

• (σi · σj)fin

+[K : Q]C(Γ), where δ is a measure of bad reduction (Artin conductor) and C(Γ) is an explicit constant depending only on the type of the fuchsian groups Γν.

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Example: the case of PSL2(Z)

Back to P1

Z → Spec Z seen as a model of PSL2(Z)\H ∪ {∞}.

Sections σ∞, σi, σρ corresponding to the cusp ∞ and the elliptic fixed points i = √−1 and ρ = eπi/3.

◮

deg H•(P1

Z, O) .

= 0.

◮ δ = 0. ◮ (ωP1

Z/Z(D)2

hyp) .

= ζ′(−1)

ζ(−1) (Bost, K¨

uhn).

◮

deg(σ∗

∞ωP1

Z/Z)W = 0 .

= ζ′(0)

ζ(0) . ◮

deg(σ∗

i ωP1

Z/Z)W .

= hF(Ei) . = L′(0,χi)

L(0,χi) (Lerch–Chowla–Selberg). ◮

deg(σ∗

ρωP1

Z/Z)W .

= hF(Eρ) . = L′(0,χρ)

L(0,χρ) .

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

Corollary (F.–von Pippich) The quantity log Z ′(1, PSL2(Z)) is an explicit rational linear combination of ζ′(−1) ζ(−1) , ζ′(0) ζ(0) , L′(0, χi) L(0, χi) , L′(0, χρ) L(0, χρ) , log 2, log 3, γ, 1.

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Corollary (F.–von Pippich) The quantity log Z ′(1, PSL2(Z)) is an explicit rational linear combination of ζ′(−1) ζ(−1) , ζ′(0) ζ(0) , L′(0, χi) L(0, χi) , L′(0, χρ) L(0, χρ) , log 2, log 3, γ, 1. Compare to the Noether formula for a toroidal compactification of a Hilbert modular surface: χ(O

X) .

= [Γ: SL2(OF)]ζF(−1) +

• p cusp

Lp(0).

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

Some keywords on the proof:

◮ Glueing formula (Mayer–Vietoris) for determinants of

Laplacians on compact Riemannian surfaces (Burghelea–Friedlander–Kappeler).

◮ Variant of the glueing formula for non-compact complete

Riemannian manifolds (Carron).

◮ Conformal invariance of Dirichlet-to-Neumann operators. ◮ Explicit computations of determinants of (pseudo-)Laplacians

• n hyperbolic model cusps or cones (brane cosmology).

◮ Selberg trace formula for fuchsian groups. ◮ For the corollary: moduli interpretation.

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

What’s next?

◮ One would like an arithmetic Riemann–Roch formula that

applies to arithmetic toroidal compactifications of Shimura varieties, and extensions of automorphic vector bundles with “invariant” metrics.

◮ A general theorem, i.e. for arbitrary arithmetic varieties with

singular K¨ ahler metrics and hermitian vector bundles with singular metrics (say Mumford good metrics), seems out of reach.

◮ Even in the Shimura variety setting, where we have

automorphic techniques at our disposal, the analytical difficulties are huge.

◮ An asymptotic version (arithmetic Hilbert–Samuel) is however

known (F.–Berman).

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

Some experiments

With D. Eriksson and S. Sankaran we did some “experiments” with a conjectural formula for Hilbert modular surfaces and the Jacquet–Langlands correspondence. Let H → S := Spec Z[1/N] be an arithmetic toroidal compactification of a Hilbert modular surface, of suitable level. Let D be the boundary divisor. We endow H(C) with the singular (along D(C)) K¨ ahler metric induced on each component from the uniformization by H2. The cotangent sheaf ΩH/S carries the singular dual metric.

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Formally, arithmetic Riemann–Roch applied to the trivial sheaf reads

• deg H•(H, O)L2 − 1

2T = − 1 24

• H
• c1(ΩH/S)

c2(ΩH/S) + TOP. In the formula, T is an undefined holomorphic analytic torsion. The characteristic classes are not defined neither, but they are if we work instead with the logarithmic cotangent sheaf:

• H
• c1(ΩH/S(log D))

c2(ΩH/S(log D)) ∈ R/

• p|N

Q log p. Indeed, ΩH/S(log D) is Mumford good and Burgos–Kramer–K¨ uhn defined arithmetic characteristic classes for those.

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Study the Dolbeault complex 0 → A0,0(H(C)) ∂ → A0,1(H(C)) ∂ → A0,2(H(C)) → 0, using H(C) = ⊔jΓj\H2. The non-trivial discrete spectrum of the Laplacian on each factor corresponds to cuspidal irreducible automorphic representations of GL2(AF). Each such representation decomposes as π = ⊗νπν. We are interested in the possible types at archimedean places ν1, ν2.

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Three possibilities: (πν1, πν2) =      (discrete highest wt -2, principal) (principal, discrete highest wt -2) (principal, principal). Playing with the raising and lowering operators in each variable on H2, we see that in the undefined expression T = − log det ∆0,1

∂

+ 2 log det ∆0,2

∂ ,

• nly the contribution of the mixed types should survive.

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There is also the continuous spectrum. Playing with the raising and lowering operators, we similarly see that each cusp contributes to A0,1 with multiplicity two, and A0,2 with multiplicity one. The intertwining operators are however the same, hence these contributions should cancel out. All in all we should have T = log det ∆0,1

∂

|V , where V ⊂ A0,1(H(C)) is the space generated by the cuspidal representations π with mixed archimedean type. From the Selberg trace formula one sees that ∆0,1

∂

| V behaves like the Laplacian on the disjoint union of two compact Riemann surfaces, and its determinant is well-defined!

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

Conjecture (Eriksson–F.–Sankaran) There should be an equality in R/

p|N Q log p

• deg H•(H, O)L2 − 1

2T . = − 1 24

• H
• c1(ΩH/S(log D))

c2(ΩH/S(log D)) +

• p cusp

L′

p(0)

Lp(0) + TOP, where Lp(s) is the Shimizu L-function attached to the cusp p.

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We can prove: Theorem (Eriksson–F.–Sankaran)

• H

c1(ΩH/S(log D)) c2(ΩH/S(log D))

• H(C) c1(ΩH(C))2

. = ζ′

F(−1)

ζF(−1). This uses the moduli interpretation and the theory of Borcherds products. Therefore, we expect:

• deg H•(H, O)L2 − 1

2T . =

?

ζ′

F(−1)

ζF(−1) +

• p cusp

L′

p(0)

Lp(0) + TOP.

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

What can be said about the conjecture?

◮ If we work with a level of the form Γ0(p) ∩ Γ1(n) for p some

auxiliary inert prime, the formula holds after taking p-new parts and modulo log |Q

×|:

(i) Jacquet–Langlands to reduce to a compact Shimura curve /F. (ii) For the Shimura curve, the arithmetic self-intersection of the canonical sheaf has been computed by X. Yuan and F.–Sankaran. (iii) The Shimizu L values have some functoriality behaviour, and they go away when we take p-new parts.

◮ The boundary contribution should afford an inconditional

arithmetic intersection definition. It should pop up when we pass from the undefined td(ΩH/S) to td(ΩH/S(log D)) (approximation of the hyperbolic metric by smooth metrics). Being studied by Mathieu Dutour.

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Introduction Arithmetic Riemann–Roch Analytic class number formula for Selberg zetas Perspectives

◮ Following the strategy of proof of the arithmetic

Riemann–Roch theorem, a possible approach would be to reduce to a modular curve embedded in H. For a modular curve, we already know the theorem. In this procedure one needs the analogue of the Bismut–Lebeau immersion formula. Base change in the theory of automorphic forms should play a role.

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With S. Sankaran we are also considering the case of Picard modular surfaces:

◮ The boundary is much simpler, given by elliptic curves with

complex multiplication.

◮ We can compute the integral of the arithmetic Todd class

(work in progress).

◮ We can see that the boundary contribution in a conjectural

arithmetic Riemann–Roch formula should be given by the Faltings heights of the corresponding elliptic curves.

◮ The spectral theory methods developed with A. von Pippich

seem easier to adapt than for Hilbert modular surfaces, thanks to the simple structure of the boundary (but still very hard!).

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