Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue The first non-trivial example A Z 2 -manifold in dimension 2 The Klein bottle: K 2 = � [ − 1 2 , L e 1 , L e 2 �\ R 2 1 ] L e 2 where Λ = Z 2 , F ≃ � [ − 1 1 ] � ≃ Z 2 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Holonomy representation The action by conjugation on Λ by F ≃ Λ \ Γ B L λ B − 1 = L B λ defines the integral holonomy representation ρ : F → GL n ( Z ) This ρ is far from determining a flat manifold uniquely There are (already in dim 4) non-homeomorphic orientable flat manifolds M Γ , M Γ ′ with the same integral holonomy representation, i.e. ρ Γ = ρ Γ ′ but M Γ �≃ M Γ ′ Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Geometric properties Bieberbach theorems M Γ = T Λ / F = ( R n / Λ) / (Γ / Λ) T Λ → M Γ , diffeomorphic ⇔ homeomorphic ⇔ homotopically equivalent Γ ≃ Γ ′ π n ( M Γ ) = π n ( M ′ M Γ ≃ M Γ ′ ⇔ ⇔ Γ ) since π n ( M Γ ) = 0 for n ≥ 2 In each dimension, there is a finite number of affine equivalent classes of compact flat manifolds Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Geometric properties Every finite group can be realized as the holonomy group of a compact flat manifold [Auslander-Kuranishi ‘57] Every compact flat manifold bounds, i.e., if M n is a compact flat manifold, then there is a N n +1 such that ∂ N = M [Hamrick-Royster ‘82] Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Z p -manifolds We will now describe the Z p -manifolds M Γ M Γ satisfies 0 → Λ ≃ Z n → Γ → Z p → 1 M Γ can be thought to be constructed by integral representations of Z p = Z [ Z p ]-modules Z p -modules were classified by Reiner [Proc AMS ‘57] Z p -manifolds were classified by Charlap [Annals Math ‘65] We won’t need Charlap’s classification, just Reiner’s Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Reiner Z p -modules Any Z p -module is of the form Λ( a , b , c , a ) := a ⊕ ( a − 1) O ⊕ b Z [ Z p ] ⊕ c Id where a , b , c ∈ N 0 , a + b > 0 ξ = primitive p th -root of unity O = Z [ ξ ] = ring of algebraic integers in Q ( ξ ) a = ideal in O Z [ Z p ] = group ring over Z Id = trivial Z p -module Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Z p -actions The actions on the modules are given by multiplication by ξ In matrix form, the action of ξ on O and Z [ Z p ] are given by 0 − 1 0 1 1 0 − 1 1 0 0 1 − 1 1 0 C p = . ∈ GL p − 1 ( Z ) , J p = . ∈ GL p ( Z ) ... ... . . . . 0 − 1 0 0 1 0 1 − 1 The action on a is given by C p , a ∈ GL p − 1 ( Z ) with C p , a ∼ C p n J p = 1, n C p = n C p , a = 0 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Properties of Z p -manifolds Proposition Let M Γ = Γ \ R n be a Z p -manifold with Γ = � γ, Λ � , γ = BL b . Then ( BL b ) p = L b p where b p = � p − 1 j =0 B j b ∈ L Λ � ( � p − 1 j =0 B j )Λ As a Z p -module, Λ ≃ Λ( a , b , c , a ) , with c ≥ 1 and n = a ( p − 1) + bp + c a , b , c are uniquely determined by the ≃ class of Γ Γ is conjugate in I ( R n ) to a Bieberbach group ˜ Γ = � ˜ γ, Λ � b where B ˜ b = ˜ b and ˜ b ∈ 1 with ˜ γ = BL ˜ p Λ � Λ Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Properties of Z p -manifolds Proposition (continued) n B = 1 ⇔ ( b , c ) = (0 , 1) and in this case γ = BL b can be chosen so that b = 1 p e n One has H 1 ( M Γ , Z ) ≃ Z b + c ⊕ Z a p H 1 ( M Γ , Z ) ≃ Z b + c and hence n B = b + c = β 1 M Γ is orientable Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue The models For our purposes, it will suffice to work with the “models” M b , c p , Λ b , c p , a ( a ) � \ R n p , a ( a ) = � BL e n where = X a Z n − c ⊥ Λ b , c p , a ( a ) = X a L Z n X − 1 ⊕ Z c a for some X a ∈ GL n ( R ) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue The models and B = diag( B p , . . . , B p , 1 , . . . , 1 ) � �� � � �� � b + c a + b with B ( 2 π p ) B ( 2 · 2 π ) p q = [ p − 1 B p = 2 ] ... B ( 2 q π p ) � cos t − sin t � B ( t ) = t ∈ R sin t cos t Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Exceptional Z p -manifolds In Charlap’s classification there is a distinction between exceptional and non-exceptional Z p -manifolds A Z p -manifold is called exceptional if Λ ≃ Λ( a , 0 , 1 , a ) We will use exceptional Z p -manifolds M 0 , 1 p , a ( a ) of dim n = a ( p − 1) + 1 ( ∴ odd) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Example: the “tricosm” It is the only 3-dimensional Z 3 -manifold It is exceptional: M 3 , 1 = M 0 , 1 3 , 1 ( O ), with O = Z [ 2 π i 3 ] 2 π i 3 ] ⊕ Z As a Z 3 -module, Λ ≃ Z [ e � 0 − 1 � with Z 3 -(integral) action given by C = 1 − 1 1 Thus 3 , L f 1 , L f 2 , L e 3 �\ R 3 M 3 , 1 = � BL e 3 with � � √ − 1 / 2 − 3 / 2 √ B = ∈ SO(3) 3 / 2 − 1 / 2 1 where f 1 , f 2 , e 3 is a Z -basis of Λ 3 , 1 = X Z 2 ⊕ Z and X ∈ GL 3 ( R ) is such that X C X − 1 = B Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Spin group and maximal torus The spin group Spin( n ) is the universal covering of SO( n ) π : Spin( n ) 2 → SO( n ) n ≥ 3 A maximal torus of Spin( n ) is given by � � x ( t 1 , . . . , t m ) : t 1 , . . . , t m ∈ R , m = [ n T = 2 ] m � x ( t 1 , . . . , t m ) := (cos t j + sin t j e 2 j − 1 e 2 j ) j =1 where { e i } is the canonical basis of R n Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Spin group and maximal torus Notation: x a ( t 1 , t 2 , . . . , t q ) := x ( t 1 , t 2 , . . . , t q , . . . , t 1 , t 2 , . . . , t q ) a ∈ N � �� � � �� � a 1 A maximal torus in SO( n ) is given by T 0 = { x 0 ( t 1 , . . . , t m ) : t 1 , . . . , t m ∈ R } � � x 0 ( t 1 , . . . , t m ) := diag B ( t 1 ) , . . . , B ( t m ) , “1” The restriction map π : T → T 0 duplicates angles x ( t 1 , . . . , t m ) �→ x 0 (2 t 1 , . . . , 2 t m ) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Spin representations The spin representation of Spin( n ) is the restriction ( L n , S n ) of any irreducible representation of Cliff ( C n ) dim C S n = 2 [ n / 2] ( L n , S n ) is irreducible if n is odd ( L n , S n ) is reducible if n is even, S n = S + n ⊕ S − n L ± n := L n | S ± n are the half-spin representations Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Characters of spin representations Characters of L n , L ± n are known on the maximal torus Lemma (Miatello-P, TAMS ‘06) m � χ Ln ( x ( t 1 , . . . , t m )) = 2 m cos t j j =1 n ( x ( t 1 , . . . , t m )) = 2 m − 1 � m � m � � cos t j ± i m χ sin t j L ± j =1 j =1 where m = [ n / 2] Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Spin structures Let M = orientable Riemannian manifold B ( M ) = SO( n )-principal bundle of oriented frames on M A spin structure on M is an equivariant double covering p : ˜ B ( M ) → B ( M ) ˜ B ( M ) is a Spin( n )-principal bundle of M , i.e. ✲ · ˜ ˜ B ( M ) B ( M ) ❅ ❅ π ˜ p p ❅ ❅ ❄ ❄ ❘ ❅ ✲ ✲ B ( M ) B ( M ) M π · Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Spin structures on compact flat manifolds The spin structures on M Γ are in a 1–1 correspondence with group homomorphisms ε commuting the diagram Spin( n ) ✒ � � ε π � ❄ � ✲ SO( n ) Γ r Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Spin structures on compact flat manifolds Let M Γ be a Z p -manifold, Γ = � γ, Λ = Z f 1 ⊕ · · · ⊕ Z f n � . Then ε is determined by ε ( γ ) and δ j := ε ( L f j ) ∈ {± 1 } 1 ≤ j ≤ n ∃ necessary and sufficient conditions on ε : Γ → Spin( n ) for defining a spin structure on M Γ when F ≃ Z k 2 or F ≃ Z n [Miatello-P, MZ ‘04] Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Spin structures on flat manifolds Not every flat manifold is spin [Vasquez ‘70] Flat tori are spin [Friedrich ‘84] Z k 2 -manifolds are not spin (in general) but Z 2 -manifolds are always spin [Miatello-P ‘04] Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Spin structures on Z p -manifolds Existence every F -manifold with | F | odd is spin (Vasquez, JDG ‘70) thus every Z p -manifold is spin Number if M is spin, the spin structures are classified by H 1 ( M , Z 2 ) If M is a Z p -manifold, since H 1 ( M , Z 2 ) ≃ Z b + c , 2 # { spin structures of M } = 2 b + c = 2 β 1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Spin structures on the models M b , c p , a ( a ) Proposition A Z p -manifold M admits exactly 2 β 1 spin structures, only one of which is of trivial type. p , a ( a ) , its 2 b + c spin structures are explicitly given by If M = M b , c � , δ b +1 , . . . , δ b + c − 1 , ( − 1) h +1 � ε | Λ = 1 , . . . , 1 , δ 1 , . . . , δ 1 , . . . , δ b , . . . , δ b � �� � � �� � � �� � p p a ( p − 1) � π � ε ( γ ) = ( − 1) ( a + b )[ q +1 2 ]+ h +1 x a + b p , . . . , q π p , 2 π p � � ∈ {± 1 } n Note: here ε | Λ = ε ( L f 1 ) , . . . , ε ( L f n ) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Compact flat manifolds Spectral asymmetry of Dirac operators Z p -manifolds Appendix: Number theoretical tools Spin structures Epilogue Spin structures on exceptional Z p -manifolds Remark If M is an exceptional Z p -manifold, i.e. M ≃ M 0 , 1 p , a ( a ) , then M has only 2 spin structures ε 1 , ε 2 given by � 1 , . . . , 1 , ( − 1) h +1 � ε h | Λ = � π � ε h ( γ ) = ( − 1) a [ q +1 2 ]+ h +1 x a p , 2 π p , . . . , q π p with h = 1 , 2 . In particular, ε 1 is of trivial type Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Twisted Dirac operators on flat manifolds Let ( M Γ , ε ) = compact flat spin n -manifold ρ : Γ → U ( V ) = unitary representation such that ρ | Λ = 1 The spin Dirac operator twisted by ρ is n � L n ( e i ) ∂ D ρ = ∂ x i i =1 where { e 1 , . . . , e n } is an o.n.b. of R n Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Twisted Dirac operators on flat manifolds D ρ acts on smooth sections of the spinor bundle D ρ : Γ ∞ ( S ρ ( M Γ , ε )) → Γ ∞ ( S ρ ( M Γ , ε )) where S ρ ( M Γ , ε ) = Γ \ ( R n × (S n ⊗ V )) → Γ \ R n � � � � γ · ( x , ω ⊗ v ) = γ x , L ε ( γ ) ( ω ) ⊗ ρ ( γ ) v Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Spectrum of D ρ on compact flat manifolds The spectrum of D ρ on ( M Γ , ε ) is �� � � ± 2 πµ, d ± : µ = || v || , v ∈ Λ ∗ Spec D ρ ( M Γ , ε ) = ρ,µ (Γ , ε ) ε where ε = { u ∈ Λ ∗ : ε ( L λ ) = e 2 π i λ · u Λ ∗ ∀ λ ∈ Λ } Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Theorem (Miatello-P, TAMS ‘06) The multiplicities of λ = ± 2 πµ are given by (i) for µ > 0 : � � e − 2 π iu · b χ d ± 1 ρ,µ (Γ , ε ) = χ ρ ( γ ) ( x γ ) | F | L ± σ ( u , x γ ) n − 1 u ∈ (Λ ∗ ε,µ ) B γ = BL b ∈ Λ \ Γ ε,µ ) B = { v ∈ Λ ∗ with (Λ ∗ ε : Bv = v , || v || = µ } (ii) for µ = 0 : � 1 χ ρ ( γ ) χ Ln ( ε ( γ )) ε | Λ = 1 | F | d ρ, 0 (Γ , ε ) = γ ∈ Λ \ Γ 0 ε | Λ � = 1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Eta series of flat manifolds ⋆ For η D ρ ( s ) we have a general expression for arbitrary compact flat manifolds an explicit formula for: Z k 2 -manifolds a family of Z 4 -manifolds Z p -manifolds in the untwisted case ([Miatello-P, TAMS ‘06, PAMQ ‘08], [P, Rev UMA ‘05]) ⋆ We will compute η D ℓ ( s ) for any Z p -manifold Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Notations From now on we consider p = 2 q + 1 an odd prime M = Z p -manifold of dim n ε h = spin structure on M , 1 ≤ h ≤ 2 b + c For 0 ≤ ℓ ≤ p − 1, the characters 2 π ik ℓ ρ ℓ : Z p → C ∗ k �→ e p D ℓ = Dirac operator twisted by ρ ℓ d ± ℓ,µ, h := d ± ρ ℓ ,µ ( M , ε h ) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue The eta series for Z p -manifolds Recall that d + ℓ,µ, h − d − � ℓ,µ, h η ℓ, h ( s ) = (2 πµ ) s ± 2 πµ ∈A Although the expressions for d ± ℓ,µ, h are not explicit, the differences d + ℓ,µ, h − d − ℓ,µ, h can be computed Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue An important reduction For flat manifolds, by a result in [Miatello-P, TAMS ‘06], n B > 1 ∀ BL b ∈ Γ ⇒ Spec D ( M ) is symmetric thus d + ℓ,µ, h = d − ⇒ η D ( s ) ≡ 0 ℓ,µ, h For Z p -manifolds, since n B = 1 ⇔ ( b , c ) = (0 , 1) then η ( s ) ≡ 0 for non-exceptional Z p -manifolds Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue An important reduction We can focus on exceptional Z p -manifolds Thus, it suffices to compute d + ℓ,µ, h − d − ℓ,µ, h , η ℓ, h ( s ) , η ℓ, h for the exceptional Z p -manifolds only In particular, we can assume that M = M 0 , 1 p , a ( a ) (i.e. b = 1 p e n ) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue The differences d + ℓ,µ, h − d − ℓ,µ, h Key lemma For an exceptional Z p -manifold ( M , ε h ) we have p − 1 � ( − 1) k ( h +1) � k � a e 2 π ik ℓ d + ℓ,µ, h − d − sin( 2 πµ k ℓ,µ, h = κ p , a ) p p p k =1 where κ p , a = ( − 1) ( p 2 − 1 ) a +1 i m +1 2 p a 2 − 1 8 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Sketch of proof I Apply the general multiplicity formula to this case p − 1 � � 2 π i ℓ k e − 2 π iu · b k χ d ± ℓ,µ, h = 1 ( ε h ( γ k )) e p ± σ ( u , x γ k ) p L n − 1 k =0 ε h ,µ ) Bk u ∈ (Λ ∗ ε h ) B k = R e n and hence note that (Λ ∗ ε h ,µ ) B k = {± µ e n } (Λ ∗ Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Sketch of proof II Thus, we get � p − 1 � � 2 π ik ℓ d ± ℓ,µ, h = 1 2 m − 1 | Λ ∗ p S ± ε h ,µ | + e µ, h ( k ) p k =1 where − 2 π i µ k 2 π i µ k S ± n − 1 ( ε h ( γ k )) + e n − 1 ( ε h ( γ k )) µ, h ( k ) := e χ L ± χ L ∓ p p (only 2-terms sums) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Sketch of proof III Note that � k π � ε h ( γ k ) = ( − 1) s h , k x a p , . . . , qk π p , 2 k π p for 1 ≤ k ≤ p , where s h , k := k ([ q +1 2 ] a + h + 1) Compute n − 1 ( ε h ( γ k )) = ( − 1) s h , k 2 m − 1 �� q � a ± i m � q � a � � � cos( jk π sin( jk π χ p ) p ) L ± j =1 j =1 compute the blue trigonometric products Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue The differences d + ℓ,µ, h − d − ℓ,µ, h Proposition Let ( M , ε h ) be an exceptional Z p -manifold. Put r = [ n 4 ] . (i) If a is even then d + 0 ,µ, h − d − 0 ,µ, h = 0 � a ± ( − 1) r p p | h ( ℓ ∓ µ ) 2 d + ℓ,µ, h − d − ℓ,µ, h = 0 otherwise Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue The differences d + ℓ,µ, h − d − ℓ,µ, h Proposition (continued) (ii) If a is odd then ℓ,µ, h = ( − 1) q + r �� 2( ℓ − µ ) �� � � 2( ℓ + µ ) a − 1 d + ℓ,µ, h − d − − p 2 p p In particular, � 0 p ≡ 1 (4) d + 0 ,µ, h − d − 0 ,µ, h = � 2 µ � ( − 1) r 2 a − 1 p p ≡ 3 (4) 2 p Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue The differences d + ℓ,µ, h − d − ℓ,µ, h Sketch of proof Rewrite d + 0 ,µ, h − d − 0 ,µ, h in terms of “character Gauß sums” − i m +1 2 p a χ 0 2 − 1 F h ( ℓ, c µ ) a even d + 0 ,µ, h − d − 0 ,µ, h = 2 − 1 ( − 1) ( p 2 − 1 ) F χ p − i m +1 2 p a h ( ℓ, c µ ) a odd 8 where χ 0 = trivial character mod p χ p = quadratic character mod p Compute the blue Gauß sums Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue The eta series η ℓ, h ( s ) η ℓ, h ( s ) can be computed in terms of Hurwitz zeta functions ∞ � 1 ζ ( s , α ) = ( n + α ) s n =0 where α ∈ (0 , 1] Re ( s ) > 1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue The eta series η ℓ, h ( s ) Theorem Let ( M , ε h ) be an exceptional Z p -manifold. Put r = [ n 4 ] , t = [ p 4 ] . (i) If a is even then η 0 , 1 ( s ) = η 0 , 2 ( s ) = 0 and for ℓ � = 0 2 � � η ℓ, 1 ( s ) = ( − 1) r a p ) − ζ ( s , p − ℓ ζ ( s , ℓ (2 π p ) s p p ) � � ( − 1) r a ζ ( s , 1 2 + ℓ p ) − ζ ( s , 1 2 − ℓ (2 π p ) s p p ) 1 ≤ ℓ ≤ q 2 η ℓ, 2 ( s ) = � � ( − 1) r a ζ ( s , 1 2 − p − ℓ p ) − ζ ( s , 1 2 + p − ℓ (2 π p ) s p p ) q < ℓ < p 2 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue The eta series η ℓ, h ( s ) Theorem (continued) (ii) If a is odd then p − 1 � � � η ℓ, 1 ( s ) = ( − 1) t + r a − 1 ( ℓ − j p ) − ( ℓ + j ζ ( s , j (2 π p ) s p p ) p ) 2 j =1 p − 1 � � � η ℓ, 2 ( s ) = ( − 1) q + r a − 1 ( 2 ℓ − (2 j +1) ) − ( 2 ℓ +(2 j +1) ζ ( s , 2 j +1 ( π p ) s p ) 2 p ) 2 p p j =0 In particular, η 0 , h ( s ) = 0 for p ≡ 1 (4) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Papers on eta-invariants Incomplete list of authors M. Atiyah, V. Patodi, I. Singer S. Goette P. Gilkey J. Park W. M¨ uller R. Mazzeo, R. Melrose, P. Piazza N. Hitchin X. Dai, D. Freed H. Donelly J. Br¨ uning, M. Lesch U. Bunke W. Zhang and others Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Computation of eta invariants We will now compute , for 0 ≤ ℓ ≤ p − 1, the eta invariants η ℓ = η ℓ (0) the reduced eta invariants η ℓ = η ℓ + dim ker D ℓ ¯ mod Z 2 the relative eta invariants η ℓ − ¯ ¯ η 0 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Eta invariants η ℓ, h Theorem Let ( M , ε h ) be an exceptional Z p -manifold. Put r = [ n 4 ] , t = [ p 4 ] . (i) If a is even then η 0 , h = 0 and for ℓ � = 0 η ℓ, 1 = ( − 1) r p 2 − 1 ( p − 2 ℓ ) a � � η ℓ, 2 = ( − 1) r p a 2 − 1 2 [ 2 ℓ p ] p − ℓ Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Eta invariants η ℓ, h Theorem (continued) (ii) If a is odd then a − 1 2 S − ( − 1) t + r +1 p 1 ( ℓ, p ) p ≡ 1 (4) η ℓ, 1 = p − 1 2 � � � j � � a − 1 ( − 1) t + r p S + 1 ( ℓ, p ) + 2 j p ≡ 3 (4) p p j =1 2 � � 2 � � a − 1 S − S − ( − 1) q + r +1 p 2 ( ℓ, p ) − 1 ( ℓ, p ) p ≡ 1 (4) p 2 � � 2 � a − 1 S + S + ( − 1) q + r p 2 ( ℓ, p ) + 1 ( ℓ, p ) + η ℓ, 2 = p p − 1 � � 2 � � j � � 1 − ( 2 + p ) j p ≡ 3 (4) p p j =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Eta invariants η ℓ, h where Notation p − ℓ − 1 ℓ − 1 � � � j � � j � S ± 1 ( ℓ, p ) := ± p p j =1 j =1 � � � � 2 ℓ 2 ℓ p + p − 2 ℓ − 1 2 ℓ − p − 1 p � � p � j � � j � S ± 2 ( ℓ, p ) := ± p p j =1 j =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Eta invariants η ℓ, h Sketch of proof Evaluate η ℓ, h ( s ) in s = 0, using that ζ (0 , α ) = 1 2 − α a even trivial, a odd: p − 1 � � � η ℓ, 1 (0) = ( − 1) t + r p a − 1 ( ℓ − j p ) − ( ℓ + j ( 1 2 − j p ) p ) 2 j =1 p − 1 � � � η ℓ, 2 (0) = ( − 1) q + r p a − 1 ( 2 ℓ − (2 j +1) ) − ( 2 ℓ +(2 j +1) ( p − 1 2 p − j ) p ) 2 p p j =0 Study the violet sums! Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Eta invariants η ℓ, h : integrality, parity Corollary (i) If ( p , a ) � = (3 , 1) then η ℓ, h ∈ Z Furthermore, η 0 , h is even, η ℓ, 1 is odd and η ℓ, 2 is even ( ℓ � = 0 ) (ii) If ( p , a ) = (3 , 1) then � − 2 / 3 ℓ = 0 η ℓ, 1 = η ℓ, 2 = 4 / 3 ℓ = 0 , 1 , 2 1 / 3 ℓ = 1 , 2 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue dim ker D ℓ It is known that dim ker D = multiplicity of the 0-eigenvalue = # independent harmonic spinors So, we will compute d ℓ, 0 , h = dim ker D ℓ, h Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue dim ker D ℓ Proposition Let ( M , ε h ) be any Z p -manifold, 1 ≤ h ≤ 2 b + c . Then d ℓ, 0 ( ε h ) = 0 for h � = 1 and � �� )( a + b ) � b + c − 1 2 ( a + b ) q + ( − 1) ( p 2 − 1 d ℓ, 0 ( ε 1 ) = 2 2 p δ ℓ, 0 − 1 8 p In particular, if b + c > 1 then d ℓ, 0 , 1 is even for any 0 ≤ ℓ ≤ p − 1 while if b + c = 1 then d 0 , 0 , 1 is even and d ℓ, 0 , 1 is odd for ℓ � = 0 . Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue dim ker D ℓ sketch of proof: We have p − 1 � 2 π ik ℓ d ℓ, 0 ( ε 1 ) = 1 χ Ln ( ε 1 ( γ k )) e p p k =0 and � k π � ε 1 ( γ k ) = ( − 1) k [ q +1 2 ]( a + b ) x a + b p , 2 k π p , . . . , qk π p Thus p − 1 2 ]( a + b ) � q �� a + b � � � jk π 2 π ik ℓ ( − 1) k [ q +1 d ℓ, 0 , 1 = 2 m cos e p p p k =0 j =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue The reduced eta invariant of Z p -manifolds η ℓ, h = 1 Recall that ¯ 2 ( η ℓ, h + d ℓ, 0 , h ) mod Z Studying the parities of η ℓ, h and d 0 ,ℓ, h we get our main result Theorem Let p be an odd prime and 0 ≤ ℓ ≤ p − 1 . Let M be a Z p -manifold with spin structure ε h , 1 ≤ h ≤ 2 b + c . Then � 2 mod Z p = n = 3 3 η ℓ, h = ¯ 0 mod Z otherwise Moreover, the relative eta invariants are ¯ η ℓ, h − ¯ η 0 , h = 0 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue The exception: the tricosm There is only one Z p -manifold with non-trivial reduced eta invariant The tricosm: the only 3-dimensional Z 3 -manifold M = M 3 , 1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Case ℓ = 0 • In the untwisted case ℓ = 0 we have a better insight • and there is a close relation with number theory We can put η ( s ) is in terms of the L -function ∞ ( n � p ) L ( s , χ p ) = n s n =1 η is in terms of class numbers h − p of imaginary quadratic fields Q ( √− p ) = Q ( i √ p ) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Case ℓ = 0, eta series Theorem ([Miatello-P, PAMQ ‘08]) Let ( M , ε h ) be a Z p -manifold of dimension n. If M is exceptional and n ≡ p ≡ 3 (4) , a ≡ 1 (4) then a − 1 − 2 2 L ( s , χ p ) η 0 , 1 ( s ) = (2 π p ) s p 2 � p ) 2 s � a − 1 2 1 − ( 2 η 0 , 2 ( s ) = (2 π p ) s p L ( s , χ p ) In particular, � � p ) 2 s − 1 ( 2 η 0 , 2 ( s ) = η 0 , 1 ( s ) Otherwise we have η 0 , 1 ( s ) = η 0 , 2 ( s ) ≡ 0 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Case ℓ = 0, eta invariants Theorem ([Miatello-P, PAMQ ‘08]) In the non-trivial case before, we have a − 3 a − 3 (i) If p = 3 then η 0 , 1 = − 2 · 3 and η ε 2 = 4 · 3 2 2 (ii) If p ≥ 7 then a − 1 2 h − p η 0 , 1 = − 2 p � � � 0 p ≡ 7 (8) ( 2 η 0 , 2 = p ) − 1 η ε 1 = a − 1 2 h − p 4 p p ≡ 3 (8) where h − p = the class number of Q ( √− p ) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Spectrum of twisted Dirac operators Z p -manifolds Eta series of Z p -manifolds Spectral asymmetry of Dirac operators Eta invariants of Z p -manifolds Appendix: Number theoretical tools The untwisted case ℓ = 0 Epilogue Case ℓ = 0, trigonometric expressions Proposition ([Miatello-P, PAMQ ‘08]) The eta invariants of an exceptional Z p -manifold ( M , ε h ) can be expressed in the following ways p − 1 p − 1 � � � k � a − 2 a − 2 cot( π k 2 cot( π k η 0 , 1 = − p p ) = − p p ) 2 2 p k =1 k =1 p − 1 � ( − 1) k � k � a − 1 csc( π k η 0 , 2 = p p ) 2 p k =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Legendre symbol Definition For p an odd prime, the Legendre symbol of k mod p is � if x 2 ≡ k ( p ) has a solution � k � 1 := if x 2 ≡ k ( p ) does not have a solution p − 1 if ( k , p ) = 1 and ( k p ) = 0 otherwise We have p 2 − 1 p − 1 ( 2 ( − 1 p ) = ( − 1) p ) = ( − 1) 8 2 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Trigonometric products Lemma Let p = 2 q + 1 be an odd prime, k ∈ N with ( k , p ) = 1 . Then q ) � � � p ) = ( − 1) ( k − 1)( p 2 − 1 2 − q √ p sin( jk π k ( i ) 8 p j =1 q � p ) = ( − 1) ( k − 1)( p 2 − 1 ) 2 − q cos( jk π ( ii ) 8 j =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Sketch of proof (i) use identities of Γ( z ) π sin( π z ) = Γ( z )Γ(1 − z ) d − 1 2 Γ( z ) = d z − 1 d ) · · · Γ( z +( d − 1) 2 Γ( z d )Γ( z +1 (2 π ) ) d Gauß Lemma ( p − 1) / 2 ) � � [ jk � p ] = ( − 1) ( k − 1)( p 2 − 1 k ( − 1) j =1 8 p (ii) follows from (i) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Classical character Gauß sums Definition For ℓ ∈ N 0 the character Gauß sum is p − 1 � � � 2 π i ℓ k k G ( ℓ, p ) := G ( χ p , ℓ ) = e p p k =0 We have � � √ p ℓ p ≡ 1 (4) p � � √ p G ( ℓ, p ) = ℓ i p ≡ 3 (4) p Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Modified character Gauß sums Definition For p ∈ P , ℓ ∈ N 0 , c ∈ N , 1 ≤ h ≤ 2, χ a character mod p we define p − 1 π ik (2 ℓ + δ h , 2 ) � ( − 1) k ( h +1) χ ( k ) e G χ p h ( ℓ ) := k =1 p − 1 � 2 π i ℓ k � π k (2 c + δ h , 2 ) � ( − 1) k ( h +1) χ ( k ) e F χ p h ( ℓ, c ) := sin p k =1 We want to compute G χ h ( ℓ ) and F χ h ( ℓ, c ) for χ = χ 0 = trivial character mod p χ = χ p = quadratic character mod p given by ( · p ) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue The sums G χ h ( ℓ ) p − 1 � 2 ℓπ ik G χ 0 1 ( ℓ ) = e p k =1 p − 1 � (2 ℓ +1) π ik ( − 1) k e G χ 0 2 ( ℓ ) = p k =1 p − 1 � 2 ℓπ ik G χ p ( k 1 ( ℓ ) = p ) e p k =1 p − 1 � (2 ℓ +1) π ik G χ p ( − 1) k ( k 2 ( ℓ ) = p ) e p k =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue The sums G χ 0 h ( ℓ ) Proposition We have � p − 1 p | ℓ χ 0 G 1 ( ℓ ) = − 1 p ∤ ℓ � p − 1 p | 2 ℓ + 1 χ 0 G 2 ( ℓ ) = − 1 p ∤ 2 ℓ + 1 In particular, χ 0 χ 0 G 1 ( ℓ ) ≡ G 2 ( ℓ ) ≡ p − 1 mod p Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue The sums G χ p h ( ℓ ) Proposition We have � √ p � G χ p ℓ 1 ( ℓ ) = δ ( p ) p � � � � √ p G χ p 2 2 ℓ +1 2 ( ℓ ) = δ ( p ) p p where � 1 p ≡ 1 (4) δ ( p ) := i p ≡ 3 (4) In particular, G χ p 1 ( ℓ ) = 0 if p | ℓ and G χ p 2 ( ℓ ) = 0 if p | 2 ℓ + 1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue The sums F χ h ( ℓ, c ) p − 1 � 2 ℓπ ik � 2 c π k � F χ 0 1 ( ℓ, c ) = e p sin p k =1 p − 1 � 2 ℓπ ik � (2 c +1) π k � ( − 1) k e F χ 0 2 ( ℓ, c ) = p sin p k =1 p − 1 � 2 ℓπ ik � 2 c π k � F χ p ( k 1 ( ℓ, c ) = p ) e p sin p k =1 p − 1 � 2 ℓπ ik � (2 c +1) π k � F χ p ( − 1) k ( k 2 ( ℓ, c ) = p ) e p sin p k =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue The sums F χ 0 h ( ℓ, c ) Proposition We have χ 0 1 If p | ℓ then F h ( ℓ, c ) = 0 for h = 1 , 2 2 If p ∤ ℓ then � ± i p if p | ℓ ∓ c χ 0 2 F 1 ( ℓ, c ) = 0 otherwise � ± i p if p | 2( ℓ ∓ c ) ∓ 1 χ 0 2 F 2 ( ℓ, c ) = 0 otherwise Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue The sums F χ p h ( ℓ, c ) Proposition We have � √ p � F χ p ( ℓ − c p ) − ( ℓ + c 1 ( ℓ, c ) = i δ ( p ) p ) 2 � √ p � 2 �� F χ p ( 2( ℓ − c ) − 1 ) − ( 2( ℓ + c )+1 2 ( ℓ, c ) = i δ ( p ) ) p p p 2 In particular, if p | ℓ then � 0 p ≡ 1 (4) F χ p � c � √ p 1 ( ℓ, c ) = p ≡ 3 (4) p � 0 p ≡ 1 (4) F χ p � 2 �� 2 c +1 � √ p 2 ( ℓ, c ) = p ≡ 3 (4) p p Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Sums involving Legendre symbols For 0 ≤ ℓ ≤ p − 1, we want to compute the sums Definition p − 1 �� ℓ − j �� � � � ℓ + j S 1 ( ℓ, p ) := − j p p j =1 p − 1 �� 2 ℓ − (2 j +1) �� � � � 2 ℓ +(2 j +1) S 2 ( ℓ, p ) := − j p p j =0 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Sums involving Legendre symbols Lemma p − 1 � � k ℓ ± j � � k ℓ � = − k ∈ Z p p j =1 p − 1 � � 2 ℓ ± (2 j +1) � = 0 p j =0 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Sums involving Legendre symbols Lemma p − 1 � ℓ + j � � j � p − 1 � j � � ℓ − 1 � � j = p + j p p p j =1 j =1 j =1 �� � p − 1 � ℓ − j � � − 1 p − ℓ − 1 � j � p − 1 � j � � � � j = p + j p p p p j =1 j =1 j =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Sums involving Legendre symbols Lemma � � 2 ℓ 2 ℓ − p − 1 p − 1 � 2 ℓ + j � � j � p − 1 � j � � � � p j = p + j p p p j =1 j =1 j =1 � � 2 ℓ �� p + p − 2 ℓ − 1 � p − 1 � 2 ℓ − j � � − 1 � j � p − 1 � j � � � � p j = p + j p p p p j =1 j =1 j =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Sums involving Legendre symbols Lemma p − 1 p − 1 p − 1 � � � � 2 ℓ ± (2 j +1) � � 2 ℓ ± j � � ℓ ± j � j − ( 2 j = p ) j p p p j =0 j =1 j =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Sums involving Legendre symbols Proposition p S − 1 ( ℓ, p ) p ≡ 1 (4) p − 1 S 1 ( ℓ, p ) = � − p S + ( j 1 ( ℓ, p ) − 2 p ) j p ≡ 3 (4) j =1 � � S − 2 ( ℓ, p ) − ( 2 p ) S − p 1 ( ℓ, p ) p ≡ 1 (4) � � S + p ) S + 2 ( ℓ, p ) − ( 2 − p 1 ( ℓ, p ) + S 2 ( ℓ, p ) = � p − 1 � � � j � ( 2 +2 p ) − 1 j p ≡ 3 (4) p j =1 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Sums involving Legendre symbols where we have used the notations p − ℓ − 1 ℓ − 1 � � � j � � j � S ± 1 ( ℓ, p ) := ± p p j =1 j =1 � � � � 2 ℓ 2 ℓ p + p − 2 ℓ − 1 2 ℓ − p − 1 � � p p � j � � j � S ± 2 ( ℓ, p ) := ± p p j =1 j =1 Note that S ± 1 (0 , p ) = S ± 1 (0 , p ) = 0 since � � j � = 0 1 ≤ j ≤ p − 1 p Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Sums involving Legendre symbols Dirichlet’s class number formula We recall � p − 1 � � j � − h − p p ≥ 5 , j = − 2 h − p 1 ω − p = p p − 2 / 3 p = 3 , j =0 where h − p = class number of Q ( √− p ) ⊂ Q ( ξ p ), ω − p = the number of p th -roots of unity of Q ( √− p ). In fact, and h − 3 = 1, ω − 3 = 6 and ω − p = 2 for p ≥ 5. Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Trigonometric products Spectral asymmetry of Dirac operators Gauß sums Appendix: Number theoretical tools Sums of Legendre symbols Epilogue Sums involving Legendre symbols Corollary For p ≥ 5 , � 0 p ≡ 1 (4) S 1 (0 , p ) = − 2 h − p p ≡ 3 (4) � 0 p ≡ 1 (4) � � S 2 ( ℓ, p ) = ( 2 2 p ) − 1 h − p p ≡ 3 (4) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Bordism Spectral asymmetry of Dirac operators Final remarks Appendix: Number theoretical tools References Epilogue Bordism groups The integrality of η ℓ − η 0 implies Theorem Let ( M , ε, σ p ) and ( M , ε, σ 0 ) denote a Z p -manifold M equipped with a spin structure ε and with the natural and the trivial Z p -structures σ p : Z p → T Λ → M σ 0 : Z p → M × Z p → M Then [( M , ε, σ p )] − [( M , ε, σ 0 )] = 0 in the reduced equivariant spin bordism group � M Spin n ( B Z p ) Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Bordism Spectral asymmetry of Dirac operators Final remarks Appendix: Number theoretical tools References Epilogue Summary of results We have 1 considered the “models” M b , c p , a ( a ) of Z p -manifolds 2 given an explicit description of the spin strucures of M b , c p , a ( a ) 3 explicitly computed, for twisted Dirac operators D ℓ acting on an arbitrary Z p -manifold ( M Γ , ε h ), the following the eta series η ℓ, h ( s ) the eta invariants η ℓ, h the number of independent harmonic spinors d ℓ, 0 , h the reduced eta invariants ¯ η ℓ, h = 0 (except for M 3 , 1 ) the relative eta invariants ¯ η ℓ, h − ¯ η 0 , h = 0 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Bordism Spectral asymmetry of Dirac operators Final remarks Appendix: Number theoretical tools References Epilogue Note on methodology ⋆ There are indirect methods to compute η -invariants (representation techniques, computing Ind ( D ) geo − Ind ( D ) top ) ⋆ However, we have performed the direct approach , that is, we have explicitly computed 1 the spectrum λ = ± 2 πµ , d λ = d ± ℓ,µ, h � d + ℓ,µ, h − d − 2 the eta series η ℓ ( s ) = 1 ℓ,µ, h (2 π ) s | µ | s µ � =0 3 the different eta invariants η ℓ = 1 η ℓ , ¯ 2 ( η ℓ + dim ker D ℓ ) mod Z , η ℓ − ¯ ¯ η 0 Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
Introduction Z p -manifolds Bordism Spectral asymmetry of Dirac operators Final remarks Appendix: Number theoretical tools References Epilogue References R. Miatello - R. Podest´ a Spin structures and spectra of Z k 2 -manifolds , Mathematische Zeitschrift (MZ) 247 (319–335), 2004. The spectrum of twisted Dirac operators on compact flat manifolds , Trans. Amer. Math. Soc. (TAMS) 358 , 10 (4569–4603), 2006. Eta invariants and class numbers , Pure and Applied Mathematics Quarterly (PAMQ), 5 , 2 (1–26), 2009. Ricardo Podest´ a (Universidad Nacional de C´ ordoba, Argentina) Eta invariants of Z p -manifolds
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