Special values of Motivic L-functions I Cambridge, 14 January 2020

Matthias Flach

I Talk 1 I Some history I The example of number fields I Determinant functors I Talk 2 I General formulation of the Tamagawa number conjecture (Deligne,Beilinson,Bloch,Kato,Fontaine,Perrin-Riou,....) I Proofs of known cases: Iwasawa Theory and p-adic L-functions I Detailed proof for Dirichlet L-functions I Talk 3 I Zeta functions of arithmetic schemes I Special values in terms of Weil-Arakelov cohomology groups and (variants of) cyclic homology X → Spec(Q) smooth, projective Motive: Direct Summand M of h(X )(n) = L hi (X )(n)[−i] For M = hi (X )(n) one has realisations

i i i M = H (X ( ), ), M = H (X / ), M = H (X¯ , )(n) B C Q dR dR Q l et Q Ql and motivic cohomology H0(M), H1(M) and an L-function L(M, s)

Arithmetic schemes vs. motives over Q

X = Spec Z[X1,..., Xn]/(f1,..., fm) Zeta function Y 1 ζ(X, s) = 1 − Nx −s x∈X closed L-functions : ”Decompose under correspondences” Arithmetic schemes vs. motives over Q

X = Spec Z[X1,..., Xn]/(f1,..., fm) Zeta function Y 1 ζ(X, s) = 1 − Nx −s x∈X closed L-functions : ”Decompose under correspondences”

X → Spec(Q) smooth, projective Motive: Direct Summand M of h(X )(n) = L hi (X )(n)[−i] For M = hi (X )(n) one has realisations

i i i M = H (X ( ), ), M = H (X / ), M = H (X¯ , )(n) B C Q dR dR Q l et Q Ql and motivic cohomology H0(M), H1(M) and an L-function L(M, s) Basic example: Dirichlet L-functions

Cyclotomic fields / GL1/Q 8

( x Degree one

L-functions

Motivic L-functions Reciprocity Law Automorphic L-functions / (Arithmetic geometry) Langlands Correspondene (Representation Theory) 6

? ' v Selberg class (Analysis) L-functions

Motivic L-functions Reciprocity Law Automorphic L-functions / (Arithmetic geometry) Langlands Correspondene (Representation Theory) 6

? ' v Selberg class (Analysis)

Basic example: Dirichlet L-functions

Cyclotomic fields / GL1/Q 8

( x Degree one Euler 1734

1 1 1 π2 1 + + + + ··· = 22 32 42 6 1 1 1 π4 1 + + + + ··· = 24 34 44 90 1 1 1 π3 1 − + − + ··· = 33 53 73 32

∞ X 1 = π2k · rational number n2k n=1 ∞ X 1 =? n2k+1 n=1

Leibniz 1673

1 1 1 π 1 − + − + ··· = 3 5 7 4 Leibniz 1673

1 1 1 π 1 − + − + ··· = 3 5 7 4 Euler 1734

1 1 1 π2 1 + + + + ··· = 22 32 42 6 1 1 1 π4 1 + + + + ··· = 24 34 44 90 1 1 1 π3 1 − + − + ··· = 33 53 73 32

∞ X 1 = π2k · rational number n2k n=1 ∞ X 1 =? n2k+1 n=1 Riemann Zeta function

∞ X 1 ζ(s) = ns n=1 ζ(s) =?, s = 2, 3, 4, 5, ....

Dirichlet L-function

∞ X η(n) L(η, s) = ns n=1 × × η :(Z/mZ) → C character example: m = 4 × ∼  :(Z/4Z) =∼ Z/2Z −→{±1} Generalisation to number fields F

X 1 Y 1 ζ (s) = = (Dedekind Zeta) F Nas 1 − Np−s a⊆OF p 0 =ζ(Spec(OF ), s) = L(h (Spec(F )), s)

Analytic class number formula

2r1 (2π)r2 · h · R Res ζF (s) = s=1 w · p|D|

× log | |v M
Σ=0 × OF −−−−→ R , R = covol(OF ) v|∞

∼ r1 r2 F ⊗Q R = R × C × h = #Pic(OF ), w = #OF ,tors, D = disc(OF /Z) ∼ × F = Q(ζm), G := Gal(F /Q) = (Z/mZ) Y ζF (s) = L(η, s) η∈Gˆ

m = 4, ζQ(i)(s) = ζ(s)L(, s) 2π · 1 · 1 π Res ζ (i)(s) = L(, 1) = = s=1 Q 4p| − 4| 4

Examples

F = Q, r2 = 0, r1 = 1 h = R = D = 1, w = 2 21 · 1 · 1 Res ζ(s) = = 1 s=1 2 · 1 Examples

F = Q, r2 = 0, r1 = 1 h = R = D = 1, w = 2 21 · 1 · 1 Res ζ(s) = = 1 s=1 2 · 1

∼ × F = Q(ζm), G := Gal(F /Q) = (Z/mZ) Y ζF (s) = L(η, s) η∈Gˆ

m = 4, ζQ(i)(s) = ζ(s)L(, s) 2π · 1 · 1 π Res ζ (i)(s) = L(, 1) = = s=1 Q 4p| − 4| 4 Generalisation to s=2,3,4,...

∼ ∼ × Res ζF (s) involves K0(OF ) = ⊕ Pic(OF ) and K1(OF ) = O s=1 Z F ζF (m) involves K2m−2(OF ) and K2m−1(OF ) Theorem (Lichtenbaum,Quillen,Borel, 1973-77) a) Kn(OF ) is a finitely generated abelian group b)  0 n ≡ 0 mod 2  r +r −1 Q 1 2 n = 1 Kn(OF ) ⊗ Q = r +r Q 1 2 n ≡ 1 mod 4  Qr2 n ≡ 3 mod 4 c) dm = r1 + r2 (resp. r2) for m ≡ 1 (resp. 0) mod 2

md−dm π R2m−1 ζF (m) ∼ × Q p|D| Computation of Kn(Z) Theorem (Suslin,Voevodsky,Rost et al 1997-2008)

n 0 1 2 3 4 5 6 7 Kn(Z) Z Z/2 Z/2 Z/48 0 Z 0 Z/240

8 9 10 11 12 13 14 15 0? Z ⊕ Z/2 Z/2 Z/1008 0? Z 0 Z/480 For n < 20000 one knows n 8a 8a + 1 8a + 2 8a + 3 Kn(Z) 0? Z ⊕ Z/2 Z/2c2a+1 Z/2w4a+2

8a + 4 8a + 5 8a + 6 8a + 7 0? Z Z/c2a+2 Z/w4a+4 B c = Numerator of k k 4k × wk = max{m|k · (Z/mZ) = 0}, Example: w2 = 24 K-Theory vs. Cohomology

2 #K ( ) ζ(2) = (2π)2 = (2π)2 2 Z 48 #K3(Z) #K ( ) R ? dm−dm 2m−2 Z 2m−1 ? ζ(m) = (2π) · p · 2 #K2m−1(Z)tor |D| This is proven using ´etalecohomology. For primes l and i = 1, 2 one has 1 Chern class maps: K ( ) → Hi ( [ ], (m)) 2m−i Z et Z l Zl inducing isomorphisms 1 K ( ) ⊗ =∼ Hi ( [ ], (m)) 2m−i Z Z Zl et Z l Zl for m ≥ 2 and all l if i = 2, odd l if i = 1. But 1 0 → /2 → K ( ) ⊗ → H1 ( [ ], (2)) → 0 Z Z 3 Z Z Z2 et Z 2 Z2

1 1 Note : H1 ( [ ], (2)) =∼ H0 ( [ ], / (2)) =∼ /8 et Z 2 Z2 et Z 2 Q2 Z2 Z Z Tamagawa number conjecture for M = Q(2)F and F totally real.

Fundamental line   M Ξ(M) = det M /M0 ⊗ det−1 M+ = det F ⊗ det−1 (2πi)2 Q dR dR Q B Q Q  Q v|∞ There is a period isomorphism ∼ ϑ∞ : R = Ξ(M) ⊗Q R and an isomorphism 1 ϑ : Ξ(M) ⊗ =∼ det RΓ ( [ ], (2)) l Q Ql Ql c Z l Ql for all primes l. The conjecture says 1 · ϑ ϑ−1(ζ (2)) = det RΓ ( [ ], (2)) Zl l ∞ F Zl c Z l Zl

1 1 RΓ ( [ ], (2)) → RΓ( [ ], (2)) → RΓ( , (2)) ⊕ RΓ( , (2)) c Z l Zl Z l Zl Ql Zl R Zl M = Q(2), F = Q ctd

ϑ ((2π)2) = det O ⊗ det−1 H0( , (2πi)2 ) ∞ Z F Z R Z

∼ 1 exp : FQl = H (Ql , Ql (2)) −2 1 2 −1 exp(OF ) = (1 − l )H (Ql , Zl (2)) · #(H (Ql , Zl (2))) −2 ϑl also involves (1 − l )

1 1 0 → H0( , (2)) → H1( [ ], (2)) → H1( [ ], (2)) → R Zl c Z l Zl Z l Zl 1 1 H1( , (2)) → H2( [ ], (2)) → H2( [ ], (2)) → Ql Zl c Z l Zl Z l Zl 1 H2( , (2)) ⊕ H2( , (2)) → H3( [ ], (2)) → 0 Ql Zl R Zl c Z l Zl

2 ∼ H (R, Zl (2)) = Z/2Z for l = 2 Insertion: Determinants of perfect complexes What is a perfect complex? R ring. A complex C • of R-Modules is called perfect if C • is quasi-isomorphic to a complex · · · → 0 → Pn → Pn+1 → · · · → Pn+k → 0 → · · · where each Pi is a finitely generated projective R-module. Example: R regular Noetherian, e.g. R = Z, C • perfect ⇔ Hi (C •) finitely generated and zero for almost all i R commutative, C • perfect. Define rank

• X i i 0 rankR C := (−1) rankZ P ∈ H (Spec(R), Z) i∈Z and the determinant

i  i (−1) rankR P • O ^ i detR C =  P  i∈Z an invertible R-module. Determinants of perfect complexes, ctd

Main properties: A) A short exact sequence of complexes

• • • 0 → C1 → C2 → C3 → 0 induces canonical isomorphism

• ∼ • • detR C2 = detR C1 ⊗R detR C3 B) For acyclic C • there is a canonical isomorphism

• ∼ detR C = R

• ∼ • C) C1 −→ C2 quasi-isomorphism induces

• ∼ • detR C1 −→ detR C2 D) For R regular Noetherian there are canonical isomorphisms

• ∼ O (−1)i i • detR C = detR H (C ) i∈Z Determinants of perfect complexes, ctd

The isomorphism in A) depends on the ordering. Take free, rank one R-module V . The diagram

detR (V ⊕ V ) −−−−→ detR V ⊗ detR V V ⊗ V       yflip yflip yflip

detR (V ⊕ V ) −−−−→ detR V ⊗ detR V V ⊗ V does not commute since v1 ∧ v2 = −v2 ∧ v1 but v1 ⊗ v2 = v2 ⊗ v1. Hence define graded line bundles (L, α) where α ∈ H0(Spec(R), Z) with (L, α) ⊗ (M, β) := (L ⊗ M, α + β) and commutativity constraint

l ⊗ m = (−1)αβm ⊗ l.

DetR : Perfect complexes → Graded line bundles • • • DetR C = (detR C , rankR C ) Determinants of perfect complexes, ctd

If R = Zl and M is a finite R-module then

detQl (M ⊗Zl Ql ) = detQl (0) = Ql and −1 detZl (M) = Zl · |M| ⊂ Ql

I R a finite free Zl -algebra I T a finitely generated projective R-module with continuous R-linear  1  action of π1,et Spec(Z S )  1  I RΓc (Z S , T ), RΓ(Ql , T ) are perfect complexes of R-modules,  1  RΓ(Z S , T ) is not in general. The Class number formula from the point of view of TNC 2r1 (2π)r2 · h · R Res ζF (s) = s=1 w · p|D|

Motive: M = Q(1)F L Betti realization: MB = τ∈Σ(2πi)Qτ, Σ = Hom(F , C) Fundamental Line: Ξ(M) := Det−1(H1(M)) ⊗ Det−1(H0(M∗(1))∗) ⊗ Det (M /M0 ) ⊗ Det−1(M+) Q f Q f Q dR dR Q B M =∼ Det−1((O×) ) ⊗ Det−1( ) ⊗ Det F ⊗ Det−1 (2πi)(τ − τ¯ )
Q F Q Q Q Q Q v v Q v|∞ v complex

+ αM M ∼ 0 → M −−→ Fv = F → coker(αM ) → 0 B,R R v|∞

× log | |v 0 → (OF )R −−−−→ coker(αM ) → R → 0 induces ∼ ϑ∞ : R = Ξ(M) ⊗Q R ! (2π)r2 R ^ ϑ = (η ∧· · ·∧η )−1⊗(1∗)−1⊗(ω ∧· · ·∧ω )⊗ (τ −τ¯ )−1 ∞ p 1 r1+r2−1 1 d v v |D| v The Class number formula from the point of view of TNC, ctd

× η1, ··· , ηr1+r2−1 Z-basis of OF /tor ω1, ··· , ωd Z-basis of OF 0 1 Z-basis of H (Spec F , Z) 1 RΓ (O [ ], (1)) → RΓ (F , (1)) → RΓ (F , (1)) ⊕ RΓ (F , (1)) c F l Zl f Zl f Ql Zl f R Zl

+ ∼ M 0 1 1 × 0 → M ⊗ l = H (Fv , l (1)) →H (OF [ ], l (1)) → O ⊗ l B,Z Z Z c l Z F Z v|∞ M ˆ× M 1 2 1 → O ⊕ H (Fv , Zl (1)) →Hc (OF [ ], Zl (1)) → Pic(OF ) ⊗ Zl → 0 Fv l v|l v|∞ 1 H2(O [ ], (1)) =∼ c F l Zl Zl

⊂ H1(F , (1)) := Oˆ× −−−−→ F × ⊗ =∼ H1(F , (1)) f v Zl Fv v Zl v Zl The Class number formula from the point of view of TNC, ctd

x2 The exponential series exp(x) = 1 + x + 2 + ··· gives an isomorphism

O ⊃ (l) = me =∼ 1 + me ⊂ O× Fv v v Fv hence Nv − 1 × det OF = det Oˆ Zl v Nv Zl Fv inside det Oˆ× ⊗ = det H1(F , (1)). Ql Fv Zl Ql Ql f v Ql But 1 ϑ : Ξ(M) ⊗ =∼ det RΓ (O [ ], (1)) l Q Ql Ql c F l Ql also involves a factor 1 − Nv −1 by definition. Hence 1 · ϑ ϑ−1(ζ (1)∗) = Det RΓ (O [ ], (1)) Zl l ∞ F Ql c F l Zl Advantages of the Fontaine/Perrin-Riou formulation

∗ I Generalizes to a description of the leading Taylor-coefficients ζF (s) at s = 0, −1, −2, ... I Generalizes to Hasse-Weil L-functions L(M, s) of motives M over number fields at any s ∈ Z I Generalizes to motives with coefficients in a semisimple Q-algebra A. Time line

Year Mathematician Theorem or Conjecture π 1673 Leibniz formula for 4 1734 Euler formula for ζ(2k) 1838 Dirichlet Analytic Class number formula for quadratic fields 1850 Kummer Analytic Class number formula for cyclotomic fields 1859 Riemann Analytic continuation and functional equation of ζ(s), Riemann hypothesis 1896 Dedekind Analytic Class number formula for number fields 1917 Hecke Analytic continuation and functional equation of ζF (s) 1921 Artin Zeta function for global fields of finite characteristic 1934 Hasse Riemann hypothesis in char. p and g = 1 1949 Weil Weil-Conjectures 1960 Dwork Rationality of Zeta-Functions in char. p 1963 Ono Tamagawa numbers of Tori 1965 Birch, Sw.-Dyer Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves Year Mathematician Theorem or Conjecture 1963 Grothendieck et al Cohomological formula for Zeta-Functions in char. p 1966 Tate BSD-Conjecture for abelian varieties, Tate Conjec- ture 1969 Serre Definition of motivic L-Functions 1972 Quillen Definition of algebraic K-groups 1974 Borel Relation between ζF (m) and K2m−1(OF ) 1974 Deligne Riemann hypothesis in char. p 1976 Coates, Wiles rank-part of BSD for CM elliptic curves of rank 0 1978 Bloch K2(E) ∼ L(E, 2) for certain elliptic curves 1979 Deligne rationality conjecture for L(M, 0) for critical motives M 1983 Gross, Zagier Gross-Zagier formula for L0(E, 1) 1984 Mazur, Wiles Iwasawa main conjecture for cyclotomic fields 1985 Beilinson rationality conjecture for L(M, s) 1986 Rubin first examples of finite Tate-Shafarevich groups 1988 Bloch, Kato integrality conjecture for L(M, s) for w(M) − 2s < 0 1988 Kolyvagin Euler systems, rank-part of BSD for ords=1L(E, s) ≤ 1 Year Mathematician Theorem or Conjecture 1991 Fontaine, Perrin-Riou integrality conjecture for L(M, s) for all s, determi- nants 1999 Burns, Flach equivariant conjecture with non-commutative coeffi- cients, Galois module theory 2000 Kings TNC for L(E, s), E CM, s = 2, 3, 4... 2003 Huber, Kings TNC for Dirichlet L-functions 2003 Burns, Greither ETNC for Dirichlet L-functions 2005 Fukaya, Kato General Iwasawa main conjecture 2014 Skinner, Urban, Kato Iwasawa main conjecture for elliptic curves over Q

Partly based on: Tejaswi Navilarekallu, On the Equivariant Tamagawa Number Conjecture, Thesis, Caltech 2006.