Colmez Conjecture in Average Shou-Wu Zhang Princeton University May 28, 2015 Shou-Wu Zhang Colmez Conjecture in Average
Faltings Heights A / K : abelian variety defined over a number field of dim g . Shou-Wu Zhang Colmez Conjecture in Average
Faltings Heights A / K : abelian variety defined over a number field of dim g . A / O K : unit connected component of the N´ eron model of A . Shou-Wu Zhang Colmez Conjecture in Average
Faltings Heights A / K : abelian variety defined over a number field of dim g . A / O K : unit connected component of the N´ eron model of A . Ω( A ) := Lie ( A ) ∗ , invariant differetnial 1-forms on A / O K . Shou-Wu Zhang Colmez Conjecture in Average
Faltings Heights A / K : abelian variety defined over a number field of dim g . A / O K : unit connected component of the N´ eron model of A . Ω( A ) := Lie ( A ) ∗ , invariant differetnial 1-forms on A / O K . ω ( A ) := det Ω( A ) with metric for each archimedean place v of K : � α ∈ ω ( A v ) = Γ( A v , Ω g � α � 2 v := (2 π ) − g | α ∧ ¯ α | , A v ) . A v ( C ) Shou-Wu Zhang Colmez Conjecture in Average
Faltings Heights A / K : abelian variety defined over a number field of dim g . A / O K : unit connected component of the N´ eron model of A . Ω( A ) := Lie ( A ) ∗ , invariant differetnial 1-forms on A / O K . ω ( A ) := det Ω( A ) with metric for each archimedean place v of K : � α ∈ ω ( A v ) = Γ( A v , Ω g � α � 2 v := (2 π ) − g | α ∧ ¯ α | , A v ) . A v ( C ) ω ( A ) := ( ω ( A ) , � · � ) ¯ 1 Faltings height of A = h ( A ) := [ K : Q ] deg ω ( A ) . Shou-Wu Zhang Colmez Conjecture in Average
Faltings Heights A / K : abelian variety defined over a number field of dim g . A / O K : unit connected component of the N´ eron model of A . Ω( A ) := Lie ( A ) ∗ , invariant differetnial 1-forms on A / O K . ω ( A ) := det Ω( A ) with metric for each archimedean place v of K : � α ∈ ω ( A v ) = Γ( A v , Ω g � α � 2 v := (2 π ) − g | α ∧ ¯ α | , A v ) . A v ( C ) ω ( A ) := ( ω ( A ) , � · � ) ¯ 1 Faltings height of A = h ( A ) := [ K : Q ] deg ω ( A ) . Assume A is semiabelian, then height is invariant under base change. Shou-Wu Zhang Colmez Conjecture in Average
Colmez conjecture E : CM field with totally real subfield F , [ F : Q ] = g . Shou-Wu Zhang Colmez Conjecture in Average
Colmez conjecture E : CM field with totally real subfield F , [ F : Q ] = g . Φ : E ⊗ R ≃ C g a CM-type. Shou-Wu Zhang Colmez Conjecture in Average
Colmez conjecture E : CM field with totally real subfield F , [ F : Q ] = g . Φ : E ⊗ R ≃ C g a CM-type. I ⊂ O E : an ideal. Shou-Wu Zhang Colmez Conjecture in Average
Colmez conjecture E : CM field with totally real subfield F , [ F : Q ] = g . Φ : E ⊗ R ≃ C g a CM-type. I ⊂ O E : an ideal. A Φ , I = C g / Φ( I ), CM abelian variety by O E . Shou-Wu Zhang Colmez Conjecture in Average
Colmez conjecture E : CM field with totally real subfield F , [ F : Q ] = g . Φ : E ⊗ R ≃ C g a CM-type. I ⊂ O E : an ideal. A Φ , I = C g / Φ( I ), CM abelian variety by O E . CM theory: A Φ , I defined over a # field K with a smooth A / O K Shou-Wu Zhang Colmez Conjecture in Average
Colmez conjecture E : CM field with totally real subfield F , [ F : Q ] = g . Φ : E ⊗ R ≃ C g a CM-type. I ⊂ O E : an ideal. A Φ , I = C g / Φ( I ), CM abelian variety by O E . CM theory: A Φ , I defined over a # field K with a smooth A / O K Colmez: h ( A Φ ) is independent of I ; denote h ( A Φ ) = h (Φ) Shou-Wu Zhang Colmez Conjecture in Average
Colmez conjecture E : CM field with totally real subfield F , [ F : Q ] = g . Φ : E ⊗ R ≃ C g a CM-type. I ⊂ O E : an ideal. A Φ , I = C g / Φ( I ), CM abelian variety by O E . CM theory: A Φ , I defined over a # field K with a smooth A / O K Colmez: h ( A Φ ) is independent of I ; denote h ( A Φ ) = h (Φ) Comez conjecture: h (Φ) is a precise linear combination of logarithmic derivatives of Artin L-functions at 0. Shou-Wu Zhang Colmez Conjecture in Average
Colmez conjecture E : CM field with totally real subfield F , [ F : Q ] = g . Φ : E ⊗ R ≃ C g a CM-type. I ⊂ O E : an ideal. A Φ , I = C g / Φ( I ), CM abelian variety by O E . CM theory: A Φ , I defined over a # field K with a smooth A / O K Colmez: h ( A Φ ) is independent of I ; denote h ( A Φ ) = h (Φ) Comez conjecture: h (Φ) is a precise linear combination of logarithmic derivatives of Artin L-functions at 0. Known cases: Shou-Wu Zhang Colmez Conjecture in Average
Colmez conjecture E : CM field with totally real subfield F , [ F : Q ] = g . Φ : E ⊗ R ≃ C g a CM-type. I ⊂ O E : an ideal. A Φ , I = C g / Φ( I ), CM abelian variety by O E . CM theory: A Φ , I defined over a # field K with a smooth A / O K Colmez: h ( A Φ ) is independent of I ; denote h ( A Φ ) = h (Φ) Comez conjecture: h (Φ) is a precise linear combination of logarithmic derivatives of Artin L-functions at 0. Known cases: (1) E / Q is abelian by Colmez and Shou-Wu Zhang Colmez Conjecture in Average
Colmez conjecture E : CM field with totally real subfield F , [ F : Q ] = g . Φ : E ⊗ R ≃ C g a CM-type. I ⊂ O E : an ideal. A Φ , I = C g / Φ( I ), CM abelian variety by O E . CM theory: A Φ , I defined over a # field K with a smooth A / O K Colmez: h ( A Φ ) is independent of I ; denote h ( A Φ ) = h (Φ) Comez conjecture: h (Φ) is a precise linear combination of logarithmic derivatives of Artin L-functions at 0. Known cases: (1) E / Q is abelian by Colmez and (2) [ E : Q ] = 4 by Tonghai Yang. Shou-Wu Zhang Colmez Conjecture in Average
Main theorem d F : the absolute discriminant of F Shou-Wu Zhang Colmez Conjecture in Average
Main theorem d F : the absolute discriminant of F d E / F := d E / d 2 F the norm of the relative discriminant of E / F . Shou-Wu Zhang Colmez Conjecture in Average
Main theorem d F : the absolute discriminant of F d E / F := d E / d 2 F the norm of the relative discriminant of E / F . η E / F : the corresponding quadratic character of A × F . Shou-Wu Zhang Colmez Conjecture in Average
Main theorem d F : the absolute discriminant of F d E / F := d E / d 2 F the norm of the relative discriminant of E / F . η E / F : the corresponding quadratic character of A × F . L f ( s , η ): the finite part of the completed L-function L ( s , η ). Theorem (Xinyi Yuan –) � L ′ f ( η E / F , 0) 1 h (Φ) = − 1 L f ( η E / F , 0) − 1 4 log( d E / F d F ) . 2 g 2 Φ where Φ runs through the set of CM types E. Shou-Wu Zhang Colmez Conjecture in Average
Two Remarks Remark When combined with a recent work of Jacob Tsimerman, The above Theorem implies the AO for Siegel moduli A g Shou-Wu Zhang Colmez Conjecture in Average
Two Remarks Remark When combined with a recent work of Jacob Tsimerman, The above Theorem implies the AO for Siegel moduli A g Remark Recently, a proof of the following weaker form of the averaged formula has been announced by Andreatta, Howard, Goren, and Madapusi Pera: � L ′ � f ( η E / F , 0) 1 h (Φ) ≡ − 1 mod Q log p . 2 g 2 L f ( η E / F , 0) Φ p | d E Shou-Wu Zhang Colmez Conjecture in Average
Two Remarks Remark When combined with a recent work of Jacob Tsimerman, The above Theorem implies the AO for Siegel moduli A g Remark Recently, a proof of the following weaker form of the averaged formula has been announced by Andreatta, Howard, Goren, and Madapusi Pera: � L ′ � f ( η E / F , 0) 1 h (Φ) ≡ − 1 mod Q log p . 2 g 2 L f ( η E / F , 0) Φ p | d E Our proof is different than theirs: we use neither high dimensional Shimura varieties nor Borcherds’ liftings. Shou-Wu Zhang Colmez Conjecture in Average
Ideal of proof: g = 1 If g = 1, then ω ( A ) has a ( Q -section) given by modular form ℓ of weight 1 with q -expansion at the Tate curve G m / q Z : Shou-Wu Zhang Colmez Conjecture in Average
Ideal of proof: g = 1 If g = 1, then ω ( A ) has a ( Q -section) given by modular form ℓ of weight 1 with q -expansion at the Tate curve G m / q Z : η ( q ) = q 1 / 24 � ℓ = η ( q ) 2 du (1 − q n ) . u , n Shou-Wu Zhang Colmez Conjecture in Average
Ideal of proof: g = 1 If g = 1, then ω ( A ) has a ( Q -section) given by modular form ℓ of weight 1 with q -expansion at the Tate curve G m / q Z : η ( q ) = q 1 / 24 � ℓ = η ( q ) 2 du (1 − q n ) . u , n � � � � � η ( q σ ) 24 (4 π Im τ σ ) 6 � 1 � h ( A ) = log | disc ( A ) | − log . 12[ K : Q ] σ : K → C Shou-Wu Zhang Colmez Conjecture in Average
Ideal of proof: g = 1 If g = 1, then ω ( A ) has a ( Q -section) given by modular form ℓ of weight 1 with q -expansion at the Tate curve G m / q Z : η ( q ) = q 1 / 24 � ℓ = η ( q ) 2 du (1 − q n ) . u , n � � � � � η ( q σ ) 24 (4 π Im τ σ ) 6 � 1 � h ( A ) = log | disc ( A ) | − log . 12[ K : Q ] σ : K → C When A has CM, apply either Kronecker–Limit or Chowla–Selberg formula. Shou-Wu Zhang Colmez Conjecture in Average
Ideal of proof: g > 1 If g > 1, there is no natural Q -sections for ω ( A ). Shou-Wu Zhang Colmez Conjecture in Average
Ideal of proof: g > 1 If g > 1, there is no natural Q -sections for ω ( A ). We will use generating series T ( q ) of arithmetic Hecke divisors on the product X × X of Shimura curves X over O F . Shou-Wu Zhang Colmez Conjecture in Average
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