Quillen metrics on modular curves Mathieu Dutour Institut de Mathématiques de Jussieu - Paris Rive Gauche French - Korean LIA : Inaugural Conference November 2019 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 1 / 21
Contents Quillen metrics in the compact case 1 First attempt with modular curves 2 The Riemann-Roch isometry of Deligne 3 The case of modular curves 4 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 2 / 21
Determinant line bundle Let X be a compact Riemann surface, and E be a holomorphic vector bundle over X , both endowed with smooth metrics. Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 3 / 21
Determinant line bundle Let X be a compact Riemann surface, and E be a holomorphic vector bundle over X , both endowed with smooth metrics. Definition The determinant line bundle λ ( E ) is defined as det H 0 ( X , E ) ⊗ det H 1 ( X , E ) ∨ . λ ( E ) = Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 3 / 21
Determinant line bundle Let X be a compact Riemann surface, and E be a holomorphic vector bundle over X , both endowed with smooth metrics. Definition The determinant line bundle λ ( E ) is defined as det H 0 ( X , E ) ⊗ det H 1 ( X , E ) ∨ . λ ( E ) = Using Hodge theory, we can put the L 2 -metric on λ ( E ) . Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 3 / 21
Quillen metric The Quillen metric on λ ( E ) is a renormalization of the L 2 -metric to account for all the eigenvalues of the Dolbeault Laplacian ∆ ∂, E . Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 4 / 21
Quillen metric The Quillen metric on λ ( E ) is a renormalization of the L 2 -metric to account for all the eigenvalues of the Dolbeault Laplacian ∆ ∂, E . Definition The Quillen metric �·� Q on λ ( E ) is defined as � − 1 / 2 � �·� L 2 . �·� Q = det ∆ ∂, E Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 4 / 21
Required conditions The Quillen metric, contrary to its L 2 -counterpart, satisfies three conditions : Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 5 / 21
Required conditions The Quillen metric, contrary to its L 2 -counterpart, satisfies three conditions : Smoothness in family 1 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 5 / 21
Required conditions The Quillen metric, contrary to its L 2 -counterpart, satisfies three conditions : Smoothness in family 1 Spectral interpretation 2 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 5 / 21
Required conditions The Quillen metric, contrary to its L 2 -counterpart, satisfies three conditions : Smoothness in family 1 Spectral interpretation 2 Riemann-Roch type theorem 3 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 5 / 21
The problem with modular curves Let X = Γ \ H be a compactified modular curve, where Γ is a fuchsian group of the first kind, without torsion. Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 6 / 21
The problem with modular curves Let X = Γ \ H be a compactified modular curve, where Γ is a fuchsian group of the first kind, without torsion. Let E be a flat, unitary, holomorphic vector bundle of rank r over X , coming from a representation : Γ − → U r ( C ) . ρ Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 6 / 21
The problem with modular curves Let X = Γ \ H be a compactified modular curve, where Γ is a fuchsian group of the first kind, without torsion. Let E be a flat, unitary, holomorphic vector bundle of rank r over X , coming from a representation ρ : Γ − → U r ( C ) . The Poincaré metric on X and the metric on E inherited from the hermitian metric on C r are then singular at the cusps, and the previous definition does not make sense. Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 6 / 21
Contents Quillen metrics in the compact case 1 First attempt with modular curves 2 The Riemann-Roch isometry of Deligne 3 The case of modular curves 4 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 7 / 21
The Selberg zeta function Let X = Γ \ H be a compactified modular curve without elliptic points, Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 8 / 21
The Selberg zeta function Let X = Γ \ H be a compactified modular curve without elliptic points, and E be a flat, unitary vector bundle over X . Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 8 / 21
The Selberg zeta function Let X = Γ \ H be a compactified modular curve without elliptic points, and E be a flat, unitary vector bundle over X . Definition The Selberg zeta function associated to X and E is defined by + ∞ � I − ρ ( γ ) N ( γ ) − s − k � � � Z ( s , Γ , ρ ) = det . k = 0 { γ } hyp Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 8 / 21
The Selberg zeta function Let X = Γ \ H be a compactified modular curve without elliptic points, and E be a flat, unitary vector bundle over X . Definition The Selberg zeta function associated to X and E is defined by + ∞ � I − ρ ( γ ) N ( γ ) − s − k � Z ( s , Γ , ρ ) = � � det . { γ } hyp k = 0 This function exists on the half-plane Re s > 1, and can be meromorphically continued. Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 8 / 21
First attempt at a Quillen metric Assuming E is stable, Takhtajan and Zograf defined in 2007 a Quillen metric on λ ( End ( E )) . Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 9 / 21
First attempt at a Quillen metric Assuming E is stable, Takhtajan and Zograf defined in 2007 a Quillen metric on λ ( End ( E )) . Definition (Takhtajan-Zograf, 2007) The regularized determinant is defined as ∂ det ∆ = ∂ s | s = 1 Z ( s , Γ , Ad ρ ) where Ad ρ is the adjoint representation, and the Quillen metric by (det ∆) − 1 / 2 �·� L 2 . �·� Q = Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 9 / 21
First attempt at a Quillen metric Assuming E is stable, Takhtajan and Zograf defined in 2007 a Quillen metric on λ ( End ( E )) . Definition (Takhtajan-Zograf, 2007) The regularized determinant is defined as ∂ det ∆ = ∂ s | s = 1 Z ( s , Γ , Ad ρ ) where Ad ρ is the adjoint representation, and the Quillen metric by (det ∆) − 1 / 2 �·� L 2 . �·� Q = Their aim was to get a curvature formula. Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 9 / 21
Conditions on the Quillen metric As inspired by the compact case, this Quillen metric should satisfy : Smoothness in family 1 Spectral interpretation 2 Riemann-Roch type theorem 3 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 10 / 21
Conditions on the Quillen metric As inspired by the compact case, this Quillen metric should satisfy : Smoothness in family 1 Spectral interpretation 2 Riemann-Roch type theorem 3 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 10 / 21
Conditions on the Quillen metric As inspired by the compact case, this Quillen metric should satisfy : Smoothness in family 1 Spectral interpretation 2 Riemann-Roch type theorem 3 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 10 / 21
Conditions on the Quillen metric As inspired by the compact case, this Quillen metric should satisfy : Smoothness in family 1 Spectral interpretation 2 Riemann-Roch type theorem 3 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 10 / 21
Conditions on the Quillen metric As inspired by the compact case, this Quillen metric should satisfy : Smoothness in family 1 Spectral interpretation 2 Riemann-Roch type theorem 3 We will work to get a functorial Riemann-Roch theorem on modular curves, similar to the one proved by Deligne in 1987. Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 10 / 21
Contents Quillen metrics in the compact case 1 First attempt with modular curves 2 The Riemann-Roch isometry of Deligne 3 The case of modular curves 4 Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 11 / 21
Functorial isomorphism Let f : X − → S be a family of compact Riemann surfaces of genus g , and E be a holomorphic vector bundle over X of rank r . Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 12 / 21
Functorial isomorphism Let f : X − → S be a family of compact Riemann surfaces of genus g , and E be a holomorphic vector bundle over X of rank r . Theorem (Deligne, 1987) We have an isomorphism of line bundles over S � r � � 6 IC 2 X / S ( E ) − 12 λ ( E ) 12 det E , det E ⊗ ω − 1 � ≃ ω X / S , ω X / S X / S X / S which is compatible with base change. Mathieu Dutour (IMJ-PRG) Quillen metrics on modular curves November 2019 12 / 21
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