On the analytic class number formula for Selberg zeta functionsgerard.freixas/Slides/Freixas-Montreal-2018.pdf · IntroductionArithmetic Riemann{RochAnalytic class number formula

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

On the analytic class number formula for Selbergzeta functions

Gerard Freixas i Montplet

C.N.R.S. – Institut de Mathematiques de Jussieu - Paris Rive Gauche

Shimura varieties and hyperbolicity of moduli spaces

UQAM, Montreal, May 2018

1 / 38


Dedekind zeta function

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

ζK (s) =∑

06=a⊆OK

1

(Na)s=

∏p⊂OK

maximal ideal

(1− (Np)−s

)−1.

I Absolute convergence for Re(s) > 1 and meromorphiccontinuation to C.

I Simple pole at s = 1.

I Functional equation.

I Riemann hypothesis.

2 / 38


Analytic class number formula

Theorem

The residue of ζK (s) at s = 1 is given by

Ress=1 ζK (s) = 2r1(2π)r2hKRK

wK

√|∆K/Q|

,

where

I r1 (resp. r2) number of real (resp. complex) embeddings of K .

I hK = #Cl(K) is the class number.

I RK is the regulator.

I wK = #OK ,tors.

I ∆K/Q is the absolute discriminant.

3 / 38


Selberg zeta function

The Selberg zeta function is a dynamical zeta function. Primitiveclosed geodesics play the role of prime numbers.

I Γ ⊂ PSL2(R) fuchsian group of the first kind.

I Γ acts on the upper half plane H by isometries, with respectto the hyperbolic metric.

I Y := Γ\H has the structure of a Riemann surface.

I Y becomes compact after adding finitely many cusps.I Correspondences:

closed geodesics inY ↔ free homotopy classes of curves

↔ conjugacy classes of hyperbolic elements in Γ

4 / 38


Definition (Selberg zeta function of Γ)

The Selberg zeta function is defined by the absolutely convergentdouble product

Z (s, Γ) =∏γ

∞∏k=0

(1− e−(s+k)`(γ)

), Re(s) > 1,

where

I γ runs over oriented primitive closed geodesics.

I `(γ) is the length of γ.

5 / 38


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(k`(γ))`(γ)/2

sinh(`(γ)/2),

where:

I the λk = 14 + t2

k form the spectrum of −y2(∂2

∂x2 + ∂2

∂y2

).

I h is the Fourier transform of h.

I “curiosity”: appearance of the A genus...

6 / 38


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:

I εd > 1 is the fundamental solution of the Pell equationx2 − dy2 = 4.

I h(d) is the class number of binary integral quadratic forms ofdiscriminant d .

7 / 38


The Selberg zeta has some analogies with the Dedekind zeta:

I meromorphic extension to C (trace formula).

I functional equation.

I simple zero at s = 1.

I for Γ co-compact (also PSL2(Z)), the non-trivial zeroes arelocated on Re(s) = 1

2 , and correspond to the (discrete)spectrum of the hyperbolic Laplacian

∆hyp = −y2

(∂2

∂x2+

∂2

∂y2

).

Missing:

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

8 / 38


The Selberg zeta has some analogies with the Dedekind zeta:

I meromorphic extension to C (trace formula).

I functional equation.

I simple zero at s = 1.

I for Γ co-compact (also PSL2(Z)), the non-trivial zeroes arelocated on Re(s) = 1

2 , and correspond to the (discrete)spectrum of the hyperbolic Laplacian

∆hyp = −y2

(∂2

∂x2+

∂2

∂y2

).

Missing:

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

8 / 38


If Γ is co-compact and torsion free:

Z ′(1, Γ) ootrace formula

d’Hoker–Phong, Sarnak// det ∆hypOO

Gillet–Soule

Arithmetic Riemann–Roch

(cohomological side)

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

9 / 38


The Grothendieck–Riemann–Roch theorem

Let f : X → Y be a projective morphism of smooth, complexalgebraic varieties and E a vector bundle on X .

Grothendieck–Riemann–Roch is the relation of characteristicclasses:

ch(Rf∗E ) = f∗(ch(E ) td(T •X/Y )) in CH•(Y )Q.

In particular, if Y = SpecC (Hirzebruch–Riemann–Roch):

χ(X ,E ) =∑p

(−1)p dimHp(X ,E ) =

∫X

ch(E ) td(TX ) ∈ Z.

10 / 38


The Riemann–Roch theorem in Arakelov geometry

I π : X → SpecZ a regular projective arithmetic variety.

I E = (E , h) a C∞ hermitian vector bundle.

I ω a Kahler metric on X (C), invariant under F∞.

I A0,p(X (C), EC) carries a hermitian L2 scalar product.

I ∆0,p

∂Laplacian acting on A0,p(X (C), EC).

I Flat base change, Dolbeault isomorphism and Hodge theorygive

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.

11 / 38


The arithmetic Riemann–Roch theorem of Gillet–Soule computesthe arithmetic degree of the cohomology:

degH•(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: harmonicforms and non-trival eigenvalues.

12 / 38


There is a theory of arithmetic characteristic classes for hermitianvector bundles, or more generally finite complexes of those.

They simultaneously refine the characteristic classes in CH•(X )Qand the Chern–Weil representatives of their de Rham realizationsin H•dR(X (C),C).

They land in the arithmetic Chow groups CH•(X )Q.

For instance:

I ch(E) arithmetic Chern class.

I td(T •X/Z, ω) arithmetic Todd genus of the tangent complex.

13 / 38


There is a morphism

∫X : CH

top(X ) // R

[∑′ nP · P, g ] //

∑′ nP log #k(P) + 12

∫X (C) g ,

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

14 / 38


Theorem (Gillet–Soule)

There is an equality of real numbers

degH•(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–Soule.

The R-genus is the additive characteristic class determined by thepower series:

R(x) =∑m≥1odd

(ζ ′(−m) + ζ(−m)(1 +

1

2+ . . .+

1

m)

)xm

m!.

15 / 38


I There is a generalization to projective morphisms betweenarithmetic varieties. It holds in any degree. It involves theholomorphic analytic torsion forms of Bismut–Kohler(Gillet–Rossler–Soule, Burgos–F.–Litcanu).

I The arithmetic Riemann–Roch theorem refines theGrothendieck–Riemann–Roch theorem in the context ofarithmetic varieties.

I One of the key points of the proof is the behaviour ofholomorphic analytic torsion with respect to closedembeddings (Bismut–Lebeau).

I All the terms are difficult to compute!

16 / 38


I There is a generalization to projective morphisms betweenarithmetic varieties. It holds in any degree. It involves theholomorphic analytic torsion forms of Bismut–Kohler(Gillet–Rossler–Soule, Burgos–F.–Litcanu).

I The arithmetic Riemann–Roch theorem refines theGrothendieck–Riemann–Roch theorem in the context ofarithmetic varieties.

I One of the key points of the proof is the behaviour ofholomorphic analytic torsion with respect to closedembeddings (Bismut–Lebeau).

I All the terms are difficult to compute!

16 / 38


Non-example

I P1Z → SpecZ seen as an integral model of PSL2(Z)\H∪ ∞.

I E = O the trivial line bundle.

I Kahler metric induced from the hyperbolic metric on H.

I The hyperbolic metric is singular at the elliptic fixed pointsand ∞.

I ∆hyp has continuous spectrum (Eisenstein series). Theanalytic torsion is not defined.

I The arithmetic Riemann–Roch theorem does not apply.

I In a conjectural formula, logZ ′(1,PSL2(Z)) should replacethe undefined analytic torsion.

17 / 38


Some inspiration: Noether formula on Hilbert modularsurfaces

Let (X ,D) be a toroidal compactification of a connected Hilbertmodular surface Γ\H2, Γ ⊂ SL2(OF ) sufficiently small.

By the Noether formula (or Hirzebruch–Riemann–Roch):

χ(OX

) =

∫X

td(ΩX

).

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

X) = 0, we have

χ(OX

) = 1 + dimS(2,2)(Γ,C).

18 / 38


Instead of td(ΩX

), it is more natural to consider td(ΩX

(logD)):by a theorem of Mumford, it has a Chern–Weil representative usingthe dual of the hyperbolic metric. Then∫

Xtd(Ω

X(logD)) =

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,one finds an expression∫

Xtd(Ω

X) =

∫X

td(ΩX

(logD)) + boundary ,

where boundary is some explicit intersection number supported onD.

19 / 38


Hirzebruch computes the boundary contribution from an explicitdescription of the toroidal compactification and Meyer’s theorem:

boundary.

=∑p cusp

Lp(0),

where Lp(s) is the Shimizu L-function naturally attached to thecusp p. All in all

χ(OX

).

= [Γ: SL2(OF )]ζF (−1) +∑p cusp

Lp(0).

In the arithmetic setting, for P1Z we expect a similar formula.

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

20 / 38


An arithmetic Noether formula

I π : X → S := SpecOK an arithmetic surface.

I σ1, . . . , σn : S → X generically disjoint sections.

I We assume that

X (C) =⊔

ν : K →C

Γν\H ∪ cusps,

where the Γν are fuchsian groups of the first kind and sametype, and σ1(ν), . . . , σn(ν) is the set of elliptic fixed pointsand cusps of the ν component. We suppose their orders are2 ≤ m1, . . . ,mn ≤ ∞.

21 / 38


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 hyperbolicmetric. It is a Mumford good hermitian line bundle.

After Bost and Kuhn, its arithmetic self-intersection number iswell-defined:

(ωX/S(D)2hyp) ∈ R.

22 / 38


We define the arithmetic ψ Q-line bundle

ψW =∑i

(1− 1

m2i

)(σ∗i ωX/S)W ∈ Pic(S)⊗Q,

where W indicates the Wolpert metric: if z is a holomorphiccoordinate in a neighborhood of σi (ν) ∈ X (C), such that thehyperbolic metric reads

|dz |2

|z |2(log |z |)2(cusp) or

4|dz |2

|z |2−2/m(1− |z |2/m)2(fixed point),

then we declare‖dz |z=0 ‖W = 1.

23 / 38


Finally, the L2 metric on H•(X ,O) is already well-defined (thehyperbolic metric has finite volume).

Theorem (F.–von Pippich)

There is an equality of real numbers

12 degH•(X ,O) + 6∑

ν : K →C

logZ ′(1, Γν) + deg ψW − δ =

(ωX/S(D)2hyp)− 1

2

∑i 6=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 fuchsiangroups Γν .

24 / 38


Example: the case of PSL2(Z)

Back to P1Z → SpecZ seen as a model of PSL2(Z)\H ∪ ∞.

Sections σ∞, σi , σρ corresponding to the cusp ∞ and the ellipticfixed points i =

√−1 and ρ = eπi/3.

I degH•(P1Z,O)

.= 0.

I δ = 0.

I (ωP1Z/Z

(D)2hyp)

.= ζ′(−1)

ζ(−1) (Bost, Kuhn).

I deg(σ∗∞ωP1Z/Z

)W = 0.

= ζ′(0)ζ(0) .

I deg(σ∗i ωP1Z/Z

)W.

= hF (Ei ).

= L′(0,χi )L(0,χi )

(Lerch–Chowla–Selberg).

I deg(σ∗ρωP1Z/Z

)W.

= hF (Eρ).

=L′(0,χρ)L(0,χρ) .

25 / 38


Corollary (F.–von Pippich)

The quantity logZ ′(1,PSL2(Z)) is an explicit rational linearcombination 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 ofa Hilbert modular surface:

χ(OX

).

= [Γ: SL2(OF )]ζF (−1) +∑p cusp

Lp(0).

26 / 38


Corollary (F.–von Pippich)

The quantity logZ ′(1,PSL2(Z)) is an explicit rational linearcombination 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 ofa Hilbert modular surface:

χ(OX

).

= [Γ: SL2(OF )]ζF (−1) +∑p cusp

Lp(0).

26 / 38


Some keywords on the proof:

I Glueing formula (Mayer–Vietoris) for determinants ofLaplacians on compact Riemannian surfaces(Burghelea–Friedlander–Kappeler).

I Variant of the glueing formula for non-compact completeRiemannian manifolds (Carron).

I Conformal invariance of Dirichlet-to-Neumann operators.

I Explicit computations of determinants of (pseudo-)Laplacianson hyperbolic model cusps or cones (brane cosmology).

I Selberg trace formula for fuchsian groups.

I For the corollary: moduli interpretation.

27 / 38


What’s next?

I One would like an arithmetic Riemann–Roch formula thatapplies to arithmetic toroidal compactifications of Shimuravarieties, and extensions of automorphic vector bundles with“invariant” metrics.

I A general theorem, i.e. for arbitrary arithmetic varieties withsingular Kahler metrics and hermitian vector bundles withsingular metrics (say Mumford good metrics), seems out ofreach.

I Even in the Shimura variety setting, where we haveautomorphic techniques at our disposal, the analyticaldifficulties are huge.

I An asymptotic version (arithmetic Hilbert–Samuel) is howeverknown (F.–Berman).

28 / 38


Some experiments

With D. Eriksson and S. Sankaran we did some “experiments”with a conjectural formula for Hilbert modular surfaces and theJacquet–Langlands correspondence.

Let H → S := SpecZ[1/N] be an arithmetic toroidalcompactification of a Hilbert modular surface, of suitable level.

Let D be the boundary divisor.

We endow H(C) with the singular (along D(C)) Kahler metricinduced on each component from the uniformization by H2. Thecotangent sheaf ΩH/S carries the singular dual metric.

29 / 38


Formally, arithmetic Riemann–Roch applied to the trivial sheafreads

degH•(H,O)L2 −1

2T = − 1

24

∫Hc1(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 ifwe work instead with the logarithmic cotangent sheaf:∫

Hc1(ΩH/S(logD))c2(ΩH/S(logD)) ∈ R/

∑p|N

Q log p.

Indeed, ΩH/S(logD) is Mumford good and Burgos–Kramer–Kuhndefined arithmetic characteristic classes for those.

30 / 38


Study the Dolbeault complex

0→ A0,0(H(C))∂→ A0,1(H(C))

∂→ A0,2(H(C))→ 0,

using H(C) = tjΓj\H2.

The non-trivial discrete spectrum of the Laplacian on each factorcorresponds to cuspidal irreducible automorphic representations ofGL2(AF ).

Each such representation decomposes as π = ⊗νπν .

We are interested in the possible types at archimedean placesν1, ν2.

31 / 38


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 onH2, we see that in the undefined expression

T = − log det ∆0,1

∂+ 2 log det ∆0,2

∂,

only the contribution of the mixed types should survive.

32 / 38


There is also the continuous spectrum. Playing with the raisingand lowering operators, we similarly see that each cusp contributesto A0,1 with multiplicity two, and A0,2 with multiplicity one. Theintertwining operators are however the same, hence thesecontributions 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 cuspidalrepresentations 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 Riemannsurfaces, and its determinant is well-defined!

33 / 38


Conjecture (Eriksson–F.–Sankaran)

There should be an equality in R/∑

p|N Q log p

degH•(H,O)L2 −1

2T

.=− 1

24

∫Hc1(ΩH/S(logD))c2(ΩH/S(logD))

+∑p cusp

L′p(0)

Lp(0)+ TOP,

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

34 / 38


We can prove:

Theorem (Eriksson–F.–Sankaran)∫H c1(ΩH/S(logD))c2(ΩH/S(logD))∫

H(C) c1(ΩH(C))2

.=ζ ′F (−1)

ζF (−1).

This uses the moduli interpretation and the theory of Borcherdsproducts.

Therefore, we expect:

degH•(H,O)L2 −1

2T

.=?

ζ ′F (−1)

ζF (−1)+∑p cusp

L′p(0)

Lp(0)+ TOP.

35 / 38


What can be said about the conjecture?I 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 andF.–Sankaran.

(iii) The Shimizu L values have some functoriality behaviour, andthey go away when we take p-new parts.

I The boundary contribution should afford an inconditionalarithmetic intersection definition. It should pop up when wepass from the undefined td(ΩH/S) to td(ΩH/S(logD))(approximation of the hyperbolic metric by smooth metrics).Being studied by Mathieu Dutour.

36 / 38


I Following the strategy of proof of the arithmeticRiemann–Roch theorem, a possible approach would be toreduce to a modular curve embedded in H. For a modularcurve, we already know the theorem. In this procedure oneneeds the analogue of the Bismut–Lebeau immersion formula.Base change in the theory of automorphic forms should play arole.

37 / 38


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

I The boundary is much simpler, given by elliptic curves withcomplex multiplication.

I We can compute the integral of the arithmetic Todd class(work in progress).

I We can see that the boundary contribution in a conjecturalarithmetic Riemann–Roch formula should be given by theFaltings heights of the corresponding elliptic curves.

I The spectral theory methods developed with A. von Pippichseem easier to adapt than for Hilbert modular surfaces, thanksto the simple structure of the boundary (but still very hard!).

38 / 38

Documents

On the analytic class number formula for Selberg zeta functionsgerard.freixas/Slides/Freixas-Montreal-2018.pdf · IntroductionArithmetic Riemann{RochAnalytic class number formula