Lower bounds for average values of Arakelov-Green functions and - PowerPoint PPT Presentation

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Petsche’s quantitative refinement Following Petsche (2006), we will sketch a proof of the following result. Theorem (Hindry–Silverman, as refined by Petsche) Let K be a number field, let E / K be an elliptic curve, let d = [ K : Q ] , and let σ = σ E / K . There are explicit absolute constants C 1 , C 2 , C 3 > 0 such that h E ( P ) ≤ log N K / Q ( D E / K ) } ≤ C 1 d σ 2 log( C 2 d σ 2 ) . # { P ∈ E ( K ) | ˆ C 3 d σ 2 In particular: 1 # E ( K ) tor ≤ C 1 d σ 2 log( C 2 d σ 2 ) . ˆ 1 h E ( P ) ≥ C 4 d 3 σ 6 log 2 ( C 2 d σ 2 ) log N K / Q D E / K for all non-torsion 2 P ∈ E ( K ) (for an absolute constant C 4 > 0 ). Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Average values of local and global canonical heights Let K be a global field, and let ˆ h = ˆ h E : E ( ¯ K ) → R ≥ 0 be the N´ eron–Tate canonical height on E . For P � = 0, we can write ˆ d v h ( P ) = � d λ v ( P ), where v ∈ M K λ v : E ( C v ) \{ 0 } → R is the normalized N´ eron local height function. Given a set Z = { P 1 , . . . , P N } of distinct points of E ( K ), set Λ( Z ) = 1 � ˆ h ( P i − P j ) N 2 i , j and Λ v ( Z ) = 1 � λ v ( P i − P j ) . N 2 i � = j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Average values of local and global canonical heights Let K be a global field, and let ˆ h = ˆ h E : E ( ¯ K ) → R ≥ 0 be the N´ eron–Tate canonical height on E . For P � = 0, we can write ˆ d v h ( P ) = � d λ v ( P ), where v ∈ M K λ v : E ( C v ) \{ 0 } → R is the normalized N´ eron local height function. Given a set Z = { P 1 , . . . , P N } of distinct points of E ( K ), set Λ( Z ) = 1 � ˆ h ( P i − P j ) N 2 i , j and Λ v ( Z ) = 1 � λ v ( P i − P j ) . N 2 i � = j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Average values of local and global canonical heights Let K be a global field, and let ˆ h = ˆ h E : E ( ¯ K ) → R ≥ 0 be the N´ eron–Tate canonical height on E . For P � = 0, we can write ˆ d v h ( P ) = � d λ v ( P ), where v ∈ M K λ v : E ( C v ) \{ 0 } → R is the normalized N´ eron local height function. Given a set Z = { P 1 , . . . , P N } of distinct points of E ( K ), set Λ( Z ) = 1 � ˆ h ( P i − P j ) N 2 i , j and Λ v ( Z ) = 1 � λ v ( P i − P j ) . N 2 i � = j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Strategy of the proof Key idea: The expression Λ( Z ) can be bounded from above in terms of the ˆ h ( P i )’s using the parallelogram law , and each Λ v ( Z ) can be bounded from below by a negative quantity tending to 0 as N → ∞ using Fourier-style averaging arguments. At a fixed archimedean place v 0 of K , use the pigeonhole principle to pass to a subset Z ′ of Z of positive density such that all points of Z ′ are close to each other in E ( C ). This makes Λ v 0 ( Z ′ ) large , and for N ≫ 0 all other Λ v ( Z ′ ) will be h ( P i ) for P i ∈ Z ′ are sufficiently small, almost non-negative. If the ˆ we get a contradiction. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Strategy of the proof Key idea: The expression Λ( Z ) can be bounded from above in terms of the ˆ h ( P i )’s using the parallelogram law , and each Λ v ( Z ) can be bounded from below by a negative quantity tending to 0 as N → ∞ using Fourier-style averaging arguments. At a fixed archimedean place v 0 of K , use the pigeonhole principle to pass to a subset Z ′ of Z of positive density such that all points of Z ′ are close to each other in E ( C ). This makes Λ v 0 ( Z ′ ) large , and for N ≫ 0 all other Λ v ( Z ′ ) will be h ( P i ) for P i ∈ Z ′ are sufficiently small, almost non-negative. If the ˆ we get a contradiction. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Strategy of the proof Key idea: The expression Λ( Z ) can be bounded from above in terms of the ˆ h ( P i )’s using the parallelogram law , and each Λ v ( Z ) can be bounded from below by a negative quantity tending to 0 as N → ∞ using Fourier-style averaging arguments. At a fixed archimedean place v 0 of K , use the pigeonhole principle to pass to a subset Z ′ of Z of positive density such that all points of Z ′ are close to each other in E ( C ). This makes Λ v 0 ( Z ′ ) large , and for N ≫ 0 all other Λ v ( Z ′ ) will be h ( P i ) for P i ∈ Z ′ are sufficiently small, almost non-negative. If the ˆ we get a contradiction. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Archimedean case Let λ ( z ) = λ v 0 ( z ) for some fixed archimedean place v 0 of K . Lemma (Hindry–Silverman) If z = r 1 + r 2 τ ∈ C \{ 0 } and max {| r 1 | , | r 2 |} ≤ 1 24 , then 1 λ ( z ) ≥ 288 max { 1 , log | j ( τ ) |} . Theorem (Elkies) Λ v 0 ( z ) ≥ − log N 12 N log + | j E | − 16 1 − 5 N . 2 N Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Archimedean case Let λ ( z ) = λ v 0 ( z ) for some fixed archimedean place v 0 of K . Lemma (Hindry–Silverman) If z = r 1 + r 2 τ ∈ C \{ 0 } and max {| r 1 | , | r 2 |} ≤ 1 24 , then 1 λ ( z ) ≥ 288 max { 1 , log | j ( τ ) |} . Theorem (Elkies) Λ v 0 ( z ) ≥ − log N 12 N log + | j E | − 16 1 − 5 N . 2 N Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Archimedean case Let λ ( z ) = λ v 0 ( z ) for some fixed archimedean place v 0 of K . Lemma (Hindry–Silverman) If z = r 1 + r 2 τ ∈ C \{ 0 } and max {| r 1 | , | r 2 |} ≤ 1 24 , then 1 λ ( z ) ≥ 288 max { 1 , log | j ( τ ) |} . Theorem (Elkies) Λ v 0 ( z ) ≥ − log N 12 N log + | j E | − 16 1 − 5 N . 2 N Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Hint of the proof of Elkies’ theorem The key tools in the proof of Elkies’ theorem are the eigenfunction expansion ∞ 1 � λ ( x − y ) = f n ( x ) f n ( y ) λ n n =1 together with smoothing properties of convolution with the heat kernel . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Non-Archimedean case Let v ∈ M ◦ K be a non-Archimedean place of K . Following Rumely, we write λ v ( P − Q ) = i v ( P , Q ) + j v ( P , Q ) where i v is a non-negative term coming from arithmetic intersection theory. If E has good reduction, then j v = 0 and thus λ v ( P − Q ) ≥ 0. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Non-Archimedean case Let v ∈ M ◦ K be a non-Archimedean place of K . Following Rumely, we write λ v ( P − Q ) = i v ( P , Q ) + j v ( P , Q ) where i v is a non-negative term coming from arithmetic intersection theory. If E has good reduction, then j v = 0 and thus λ v ( P − Q ) ≥ 0. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Tate curves Now suppose j E is non-integral, so that E / K has a Tate uniformization E ( C v ) ∼ → C ∗ v / q Z , where q = q E is the Tate − parameter of E . Let r be the composition of the Tate isomorphism v / q Z → R / Z sending z to (log | z | v ) / (log | q | v ). with the map C ∗ Then j v ( P , Q ) = 1 2 B 2 ( r ( P − Q )) log | j E | v , where B 2 ( t ) = ( t − [ t ]) 2 − 1 2( t − [ t ]) + 1 6 is the periodic second Bernoulli polynomial. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Tate curves Now suppose j E is non-integral, so that E / K has a Tate uniformization E ( C v ) ∼ → C ∗ v / q Z , where q = q E is the Tate − parameter of E . Let r be the composition of the Tate isomorphism v / q Z → R / Z sending z to (log | z | v ) / (log | q | v ). with the map C ∗ Then j v ( P , Q ) = 1 2 B 2 ( r ( P − Q )) log | j E | v , where B 2 ( t ) = ( t − [ t ]) 2 − 1 2( t − [ t ]) + 1 6 is the periodic second Bernoulli polynomial. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Fourier analysis of B 2 ( t ) Lemma Let E / C v be a Tate curve, and let δ v = log | j E | > 0 . Then � 1 � 1 − 1 Λ v ( Z ) ≥ 12 δ v . δ 2 N v The proof is based on Parseval’s formula together with the Fourier expansion 1 1 � m 2 e 2 π imt . B 2 ( t ) = 2 π 2 m ∈ Z \{ 0 } Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Fourier analysis of B 2 ( t ) Lemma Let E / C v be a Tate curve, and let δ v = log | j E | > 0 . Then � 1 � 1 − 1 Λ v ( Z ) ≥ 12 δ v . δ 2 N v The proof is based on Parseval’s formula together with the Fourier expansion 1 1 � m 2 e 2 π imt . B 2 ( t ) = 2 π 2 m ∈ Z \{ 0 } Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Interpretation via the Berkovich analytification The map r : E ( C v ) → R / Z defined above can be identified with the retraction map to the skeleton Γ of the Berkovich analytic space E Berk , v , which is isometric to a circle of length log | j E | v : The function j v ( P , Q ) = 1 2 B 2 ( r ( P − Q )) log | j E | v on Γ is the Arakelov-Green function with respect to the normalized Haar measure µ v on Γ. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Interpretation via the Berkovich analytification The map r : E ( C v ) → R / Z defined above can be identified with the retraction map to the skeleton Γ of the Berkovich analytic space E Berk , v , which is isometric to a circle of length log | j E | v : The function j v ( P , Q ) = 1 2 B 2 ( r ( P − Q )) log | j E | v on Γ is the Arakelov-Green function with respect to the normalized Haar measure µ v on Γ. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Local discrepancy: Archimedean case Let K be a number field. Given a sequence Z = { P 1 , . . . , P N } of distinct points in E ( ¯ K ), we want to define a non-negative “smoothing” D v ( Z ) of Λ v ( Z ), which we call the local discrepancy of Z . Over C , Elkies’ theorem is proved by convolving with the heat kernel to get a 1-parameter family { λ t } t > 0 of smooth functions λ t : E ( C ) → R such that lim t → 0 λ t = λ v and 1 � i , j λ t ( P i − P j ) > 0 for all t . N 2 The best choice for our purposes is to take t = 1 / N . We set, for Archimedean v : D v ( Z ) := 1 � λ 1 N ( P i − P j ) . N 2 i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Local discrepancy: Archimedean case Let K be a number field. Given a sequence Z = { P 1 , . . . , P N } of distinct points in E ( ¯ K ), we want to define a non-negative “smoothing” D v ( Z ) of Λ v ( Z ), which we call the local discrepancy of Z . Over C , Elkies’ theorem is proved by convolving with the heat kernel to get a 1-parameter family { λ t } t > 0 of smooth functions λ t : E ( C ) → R such that lim t → 0 λ t = λ v and 1 � i , j λ t ( P i − P j ) > 0 for all t . N 2 The best choice for our purposes is to take t = 1 / N . We set, for Archimedean v : D v ( Z ) := 1 � λ 1 N ( P i − P j ) . N 2 i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Local discrepancy: Archimedean case Let K be a number field. Given a sequence Z = { P 1 , . . . , P N } of distinct points in E ( ¯ K ), we want to define a non-negative “smoothing” D v ( Z ) of Λ v ( Z ), which we call the local discrepancy of Z . Over C , Elkies’ theorem is proved by convolving with the heat kernel to get a 1-parameter family { λ t } t > 0 of smooth functions λ t : E ( C ) → R such that lim t → 0 λ t = λ v and 1 � i , j λ t ( P i − P j ) > 0 for all t . N 2 The best choice for our purposes is to take t = 1 / N . We set, for Archimedean v : D v ( Z ) := 1 � λ 1 N ( P i − P j ) . N 2 i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Local discrepancy: Non-Archimedean case In the non-Archimedean case, the singularity of λ v at O comes from the non-negative intersection term i v ( P , Q ). This makes life easier than in the Archimedean case. Here, a naive construction works just fine: we set λ ∗ v ( O ) = 0 and λ ∗ v ( P ) = λ v ( P ) for P � = O , and define D v ( Z ) := 1 � λ ∗ v ( P i − P j ) . N 2 i , j Although λ ∗ v can be negative, Parseval’s formula shows that D v ( Z ) ≥ 0. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Local discrepancy: Non-Archimedean case In the non-Archimedean case, the singularity of λ v at O comes from the non-negative intersection term i v ( P , Q ). This makes life easier than in the Archimedean case. Here, a naive construction works just fine: we set λ ∗ v ( O ) = 0 and λ ∗ v ( P ) = λ v ( P ) for P � = O , and define D v ( Z ) := 1 � λ ∗ v ( P i − P j ) . N 2 i , j Although λ ∗ v can be negative, Parseval’s formula shows that D v ( Z ) ≥ 0. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Local discrepancy: Non-Archimedean case In the non-Archimedean case, the singularity of λ v at O comes from the non-negative intersection term i v ( P , Q ). This makes life easier than in the Archimedean case. Here, a naive construction works just fine: we set λ ∗ v ( O ) = 0 and λ ∗ v ( P ) = λ v ( P ) for P � = O , and define D v ( Z ) := 1 � λ ∗ v ( P i − P j ) . N 2 i , j Although λ ∗ v can be negative, Parseval’s formula shows that D v ( Z ) ≥ 0. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics The Height-Discrepancy Inequality Define the global discrepancy of Z to be d D v ( Z ), and define ˆ h ( Z ) := 1 P ∈ Z ˆ d v D ( Z ) := � � h ( P ). v ∈ M K N Theorem (B.–Petsche) Let K be a number field, let E / K be an elliptic curve, and let Z = { P 1 , . . . , P N } be a set of N distinct points in E ( ¯ K ) . Then � 1 � h ( Z ) + 1 2 log N + 1 12 h ( j E ) + 16 0 ≤ D ( Z ) ≤ 4ˆ . N 5 Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics The Height-Discrepancy Inequality Define the global discrepancy of Z to be d D v ( Z ), and define ˆ h ( Z ) := 1 P ∈ Z ˆ d v D ( Z ) := � � h ( P ). v ∈ M K N Theorem (B.–Petsche) Let K be a number field, let E / K be an elliptic curve, and let Z = { P 1 , . . . , P N } be a set of N distinct points in E ( ¯ K ) . Then � 1 � h ( Z ) + 1 2 log N + 1 12 h ( j E ) + 16 0 ≤ D ( Z ) ≤ 4ˆ . N 5 Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Local Discrepancy and Equidistribution Intuitively, the local discrepancy measures how far Z = { P 1 , . . . , P N } is from being equidistributed with respect to the canonical measure at a given place v of K . For v ∈ M K Archimedean, set E Berk , v := E ( C ) and let µ v be the normalized Haar measure on E Berk , v . For v ∈ M K non-Archimedean, let µ v be the normalized Haar measure on the circle Γ (pushed forward to E Berk , v ). Theorem Let Z n be a sequence of finite subsets of E ( C v ) ⊂ E Berk , v with # Z n → ∞ . If D v ( Z n ) → 0 , then the sequence Z n is equidistributed with respect to µ v . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Local Discrepancy and Equidistribution Intuitively, the local discrepancy measures how far Z = { P 1 , . . . , P N } is from being equidistributed with respect to the canonical measure at a given place v of K . For v ∈ M K Archimedean, set E Berk , v := E ( C ) and let µ v be the normalized Haar measure on E Berk , v . For v ∈ M K non-Archimedean, let µ v be the normalized Haar measure on the circle Γ (pushed forward to E Berk , v ). Theorem Let Z n be a sequence of finite subsets of E ( C v ) ⊂ E Berk , v with # Z n → ∞ . If D v ( Z n ) → 0 , then the sequence Z n is equidistributed with respect to µ v . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Local Discrepancy and Equidistribution Intuitively, the local discrepancy measures how far Z = { P 1 , . . . , P N } is from being equidistributed with respect to the canonical measure at a given place v of K . For v ∈ M K Archimedean, set E Berk , v := E ( C ) and let µ v be the normalized Haar measure on E Berk , v . For v ∈ M K non-Archimedean, let µ v be the normalized Haar measure on the circle Γ (pushed forward to E Berk , v ). Theorem Let Z n be a sequence of finite subsets of E ( C v ) ⊂ E Berk , v with # Z n → ∞ . If D v ( Z n ) → 0 , then the sequence Z n is equidistributed with respect to µ v . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Local Discrepancy and Equidistribution Intuitively, the local discrepancy measures how far Z = { P 1 , . . . , P N } is from being equidistributed with respect to the canonical measure at a given place v of K . For v ∈ M K Archimedean, set E Berk , v := E ( C ) and let µ v be the normalized Haar measure on E Berk , v . For v ∈ M K non-Archimedean, let µ v be the normalized Haar measure on the circle Γ (pushed forward to E Berk , v ). Theorem Let Z n be a sequence of finite subsets of E ( C v ) ⊂ E Berk , v with # Z n → ∞ . If D v ( Z n ) → 0 , then the sequence Z n is equidistributed with respect to µ v . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Global Discrepancy and Equidistribution of Small Points Using the Height-Discrepancy Inequality, we obtain the following quantitative refinement of the Szpiro-Ullmo-Zhang equidistribution theorem for elliptic curves: Theorem (B.–Petsche) Let Z n be a sequence of Gal ( ¯ K / K ) -invariant finite subsets of E ( ¯ K ) with # Z n → ∞ . If there is a place v ∈ M K such that Z n is not equidistributed in E Berk , v with respect to µ v , then h ( Z n ) ≥ 1 lim inf ˆ 4 lim inf D ( Z n ) > 0 . In particular, if ˆ h ( Z n ) → 0 , then Z n is equidistributed in E Berk , v with respect to µ v for all v ∈ M K . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Global Discrepancy and Equidistribution of Small Points Using the Height-Discrepancy Inequality, we obtain the following quantitative refinement of the Szpiro-Ullmo-Zhang equidistribution theorem for elliptic curves: Theorem (B.–Petsche) Let Z n be a sequence of Gal ( ¯ K / K ) -invariant finite subsets of E ( ¯ K ) with # Z n → ∞ . If there is a place v ∈ M K such that Z n is not equidistributed in E Berk , v with respect to µ v , then h ( Z n ) ≥ 1 lim inf ˆ 4 lim inf D ( Z n ) > 0 . In particular, if ˆ h ( Z n ) → 0 , then Z n is equidistributed in E Berk , v with respect to µ v for all v ∈ M K . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Totally real and totally p -adic points The following theorem is proved by combining the height-discrepancy inequality with Fourier analysis on E ( R ) and a real-analytic version of the Tate uniformization: Theorem (B.–Petsche) Let Q tr be the maximal totally real subfield of ¯ Q , and let E / Q tr be an elliptic curve. There are explicit constants C 1 , C 2 > 0 depending polynomially on h ( j E ) such that # E ( Q tr ) tor ≤ C 1 and ˆ h ( P ) ≥ C 2 for every non-torsion point P ∈ E ( Q tr ) . We prove a similar completely explicit result for totally p-adic points . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Totally real and totally p -adic points The following theorem is proved by combining the height-discrepancy inequality with Fourier analysis on E ( R ) and a real-analytic version of the Tate uniformization: Theorem (B.–Petsche) Let Q tr be the maximal totally real subfield of ¯ Q , and let E / Q tr be an elliptic curve. There are explicit constants C 1 , C 2 > 0 depending polynomially on h ( j E ) such that # E ( Q tr ) tor ≤ C 1 and ˆ h ( P ) ≥ C 2 for every non-torsion point P ∈ E ( Q tr ) . We prove a similar completely explicit result for totally p-adic points . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Totally real and totally p -adic points The following theorem is proved by combining the height-discrepancy inequality with Fourier analysis on E ( R ) and a real-analytic version of the Tate uniformization: Theorem (B.–Petsche) Let Q tr be the maximal totally real subfield of ¯ Q , and let E / Q tr be an elliptic curve. There are explicit constants C 1 , C 2 > 0 depending polynomially on h ( j E ) such that # E ( Q tr ) tor ≤ C 1 and ˆ h ( P ) ≥ C 2 for every non-torsion point P ∈ E ( Q tr ) . We prove a similar completely explicit result for totally p-adic points . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Elkies’ theorem for Riemann surfaces Elkies’ result extends (with more or less the same proof) to compact Riemann surfaces X of arbitrary genus by using the Arakelov–Green function g µ ( x , y ) attached to some volume form µ on X ( C ) in place of λ ( x − y ). Definition: The Arakelov–Green function g µ ( x , y ) is the unique function of two variables on X ( C ) which is continuous away from the diagonal and satisfies: (Differential equation) ∆ y g µ ( x , y ) = δ x − µ . �� (Normalization) g µ ( x , y ) µ ( x ) µ ( y ) = 0. Theorem (Elkies) Λ µ ( Z ) := 1 g µ ( P i , P j ) ≫ − log N � . N 2 N i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Elkies’ theorem for Riemann surfaces Elkies’ result extends (with more or less the same proof) to compact Riemann surfaces X of arbitrary genus by using the Arakelov–Green function g µ ( x , y ) attached to some volume form µ on X ( C ) in place of λ ( x − y ). Definition: The Arakelov–Green function g µ ( x , y ) is the unique function of two variables on X ( C ) which is continuous away from the diagonal and satisfies: (Differential equation) ∆ y g µ ( x , y ) = δ x − µ . �� (Normalization) g µ ( x , y ) µ ( x ) µ ( y ) = 0. Theorem (Elkies) Λ µ ( Z ) := 1 g µ ( P i , P j ) ≫ − log N � . N 2 N i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Elkies’ theorem for Riemann surfaces Elkies’ result extends (with more or less the same proof) to compact Riemann surfaces X of arbitrary genus by using the Arakelov–Green function g µ ( x , y ) attached to some volume form µ on X ( C ) in place of λ ( x − y ). Definition: The Arakelov–Green function g µ ( x , y ) is the unique function of two variables on X ( C ) which is continuous away from the diagonal and satisfies: (Differential equation) ∆ y g µ ( x , y ) = δ x − µ . �� (Normalization) g µ ( x , y ) µ ( x ) µ ( y ) = 0. Theorem (Elkies) Λ µ ( Z ) := 1 g µ ( P i , P j ) ≫ − log N � . N 2 N i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Elkies’ theorem for Riemann surfaces Elkies’ result extends (with more or less the same proof) to compact Riemann surfaces X of arbitrary genus by using the Arakelov–Green function g µ ( x , y ) attached to some volume form µ on X ( C ) in place of λ ( x − y ). Definition: The Arakelov–Green function g µ ( x , y ) is the unique function of two variables on X ( C ) which is continuous away from the diagonal and satisfies: (Differential equation) ∆ y g µ ( x , y ) = δ x − µ . �� (Normalization) g µ ( x , y ) µ ( x ) µ ( y ) = 0. Theorem (Elkies) Λ µ ( Z ) := 1 g µ ( P i , P j ) ≫ − log N � . N 2 N i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Elkies’ theorem for Riemann surfaces Elkies’ result extends (with more or less the same proof) to compact Riemann surfaces X of arbitrary genus by using the Arakelov–Green function g µ ( x , y ) attached to some volume form µ on X ( C ) in place of λ ( x − y ). Definition: The Arakelov–Green function g µ ( x , y ) is the unique function of two variables on X ( C ) which is continuous away from the diagonal and satisfies: (Differential equation) ∆ y g µ ( x , y ) = δ x − µ . �� (Normalization) g µ ( x , y ) µ ( x ) µ ( y ) = 0. Theorem (Elkies) Λ µ ( Z ) := 1 g µ ( P i , P j ) ≫ − log N � . N 2 N i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics A Non-Archimedean Elkies’ inequality? If v is non-Archimedean and X Berk is the Berkovich analytification of an algebraic curve over C v , the space X Berk deformation retracts onto a finite metrized graph Γ. One can define the Arakelov–Green function g µ ( x , y ) attached to any probability measure µ on X Berk exactly as above, using Thuillier’s Laplacian operator. If µ is supported on Γ, it turns out (just as in the case of elliptic curves) that g µ ( x , y ) = i ( x , y ) + j µ ( x , y ) where i ( x , y ) is a non-negative term coming from arithmetic intersection theory and j µ ( x , y ) depends only on the retraction of x and y to Γ. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics A Non-Archimedean Elkies’ inequality? If v is non-Archimedean and X Berk is the Berkovich analytification of an algebraic curve over C v , the space X Berk deformation retracts onto a finite metrized graph Γ. One can define the Arakelov–Green function g µ ( x , y ) attached to any probability measure µ on X Berk exactly as above, using Thuillier’s Laplacian operator. If µ is supported on Γ, it turns out (just as in the case of elliptic curves) that g µ ( x , y ) = i ( x , y ) + j µ ( x , y ) where i ( x , y ) is a non-negative term coming from arithmetic intersection theory and j µ ( x , y ) depends only on the retraction of x and y to Γ. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics A Non-Archimedean Elkies’ inequality? If v is non-Archimedean and X Berk is the Berkovich analytification of an algebraic curve over C v , the space X Berk deformation retracts onto a finite metrized graph Γ. One can define the Arakelov–Green function g µ ( x , y ) attached to any probability measure µ on X Berk exactly as above, using Thuillier’s Laplacian operator. If µ is supported on Γ, it turns out (just as in the case of elliptic curves) that g µ ( x , y ) = i ( x , y ) + j µ ( x , y ) where i ( x , y ) is a non-negative term coming from arithmetic intersection theory and j µ ( x , y ) depends only on the retraction of x and y to Γ. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An example In fact, we have j µ ( x , y ) = g µ, Γ ( r ( x ) , r ( y )) where g µ, Γ ( x , y ) is a continuous function of two variables on Γ defined using the Laplacian operator on metrized graphs. For example, when Γ is a circle, we have g µ, Γ ( x , y ) = 1 2 B 2 ( x − y ) ℓ (Γ) . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An example In fact, we have j µ ( x , y ) = g µ, Γ ( r ( x ) , r ( y )) where g µ, Γ ( x , y ) is a continuous function of two variables on Γ defined using the Laplacian operator on metrized graphs. For example, when Γ is a circle, we have g µ, Γ ( x , y ) = 1 2 B 2 ( x − y ) ℓ (Γ) . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An Elkies-style inequality for metrized graphs Theorem (B.–Rumely) Let Γ be a metrized graph. 1 The function g µ ( x , y ) on Γ has an eigenfunction expansion ∞ 1 � f n ( x ) f n ( y ) λ n n =1 which converges uniformly on Γ × Γ to g µ ( x , y ) . 2 There exists C > 0 (depending on Γ and µ ) such that Λ µ ( Z ) := 1 g µ ( P i , P j ) ≥ − C � N . N 2 i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An Elkies-style inequality for metrized graphs Theorem (B.–Rumely) Let Γ be a metrized graph. 1 The function g µ ( x , y ) on Γ has an eigenfunction expansion ∞ 1 � f n ( x ) f n ( y ) λ n n =1 which converges uniformly on Γ × Γ to g µ ( x , y ) . 2 There exists C > 0 (depending on Γ and µ ) such that Λ µ ( Z ) := 1 g µ ( P i , P j ) ≥ − C � N . N 2 i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An Elkies-style inequality for metrized graphs Theorem (B.–Rumely) Let Γ be a metrized graph. 1 The function g µ ( x , y ) on Γ has an eigenfunction expansion ∞ 1 � f n ( x ) f n ( y ) λ n n =1 which converges uniformly on Γ × Γ to g µ ( x , y ) . 2 There exists C > 0 (depending on Γ and µ ) such that Λ µ ( Z ) := 1 g µ ( P i , P j ) ≥ − C � N . N 2 i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An example Another example of an eigenfunction expansion on a metrized graph is the following: Example: Let Γ = [0 , 1] and let µ = δ 0 . Then 2 ) sin( n π y sin( n π x 2 ) � g µ ( x , y ) = min ( x , y ) = 8 . π 2 n 2 n ≥ 1 odd Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An example Another example of an eigenfunction expansion on a metrized graph is the following: Example: Let Γ = [0 , 1] and let µ = δ 0 . Then 2 ) sin( n π y sin( n π x 2 ) � g µ ( x , y ) = min ( x , y ) = 8 . π 2 n 2 n ≥ 1 odd Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Canonical heights in arithmetic dynamics Following Call and Silverman, one has an analogue of the N´ eron–Tate canonical height in arithmetic dynamics. Let φ : P 1 → P 1 be a morphism of degree d ≥ 2 defined over a global field K . We can lift φ to a map F = ( F 1 ( x , y ) , F 2 ( x , y )) : A 2 → A 2 where the F i are homogeneous of degree d and have no common factor. Write F ( n ) = ( F ( n ) ( x , y ) , F ( n ) ( x , y )) for the n th iterate of F . 1 2 For v ∈ M K , define the local canonical height ˆ H F , v : C 2 v \{ 0 } → R by 1 d n log max {| F ( n ) ( z 1 , z 2 ) | v , | F ( n ) ˆ H F , v ( z 1 , z 2 ) = lim ( z 1 , z 2 ) | v } . 1 2 n →∞ Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Canonical heights in arithmetic dynamics Following Call and Silverman, one has an analogue of the N´ eron–Tate canonical height in arithmetic dynamics. Let φ : P 1 → P 1 be a morphism of degree d ≥ 2 defined over a global field K . We can lift φ to a map F = ( F 1 ( x , y ) , F 2 ( x , y )) : A 2 → A 2 where the F i are homogeneous of degree d and have no common factor. Write F ( n ) = ( F ( n ) ( x , y ) , F ( n ) ( x , y )) for the n th iterate of F . 1 2 For v ∈ M K , define the local canonical height ˆ H F , v : C 2 v \{ 0 } → R by 1 d n log max {| F ( n ) ( z 1 , z 2 ) | v , | F ( n ) ˆ H F , v ( z 1 , z 2 ) = lim ( z 1 , z 2 ) | v } . 1 2 n →∞ Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Canonical heights in arithmetic dynamics Following Call and Silverman, one has an analogue of the N´ eron–Tate canonical height in arithmetic dynamics. Let φ : P 1 → P 1 be a morphism of degree d ≥ 2 defined over a global field K . We can lift φ to a map F = ( F 1 ( x , y ) , F 2 ( x , y )) : A 2 → A 2 where the F i are homogeneous of degree d and have no common factor. Write F ( n ) = ( F ( n ) ( x , y ) , F ( n ) ( x , y )) for the n th iterate of F . 1 2 For v ∈ M K , define the local canonical height ˆ H F , v : C 2 v \{ 0 } → R by 1 d n log max {| F ( n ) ( z 1 , z 2 ) | v , | F ( n ) ˆ H F , v ( z 1 , z 2 ) = lim ( z 1 , z 2 ) | v } . 1 2 n →∞ Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Canonical heights in arithmetic dynamics (continued) The global canonical height of z = ( z 1 : z 2 ) ∈ P 1 ( ¯ K ) is defined to be d v ˆ � ˆ h φ ( z ) = H F , v ( z 1 , z 2 ) . d v ∈ M K By the product formula , this is independent of the lift F and of the coordinate representation ( z 1 : z 2 ). Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Canonical heights in arithmetic dynamics (continued) The global canonical height of z = ( z 1 : z 2 ) ∈ P 1 ( ¯ K ) is defined to be d v ˆ � ˆ h φ ( z ) = H F , v ( z 1 , z 2 ) . d v ∈ M K By the product formula , this is independent of the lift F and of the coordinate representation ( z 1 : z 2 ). Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Canonical measures in arithmetic dynamics For each place v of K there is a canonical measure µ φ, v on P 1 Berk , v which governs equidistribution of periodic points and iterated preimages. In the Archimedean case, the canonical measure is the well-known measure of maximal entropy studied by Brolin, Lyubich, and Freire-Lopes-Ma˜ n´ e. For example, if φ ( z ) = z 2 then for v Archimedean µ φ, v is Haar measure on the complex unit circle in P 1 ( C ). For v non-Archimedean µ φ, v is a point mass at the Gauss point of P 1 Berk , v . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Canonical measures in arithmetic dynamics For each place v of K there is a canonical measure µ φ, v on P 1 Berk , v which governs equidistribution of periodic points and iterated preimages. In the Archimedean case, the canonical measure is the well-known measure of maximal entropy studied by Brolin, Lyubich, and Freire-Lopes-Ma˜ n´ e. For example, if φ ( z ) = z 2 then for v Archimedean µ φ, v is Haar measure on the complex unit circle in P 1 ( C ). For v non-Archimedean µ φ, v is a point mass at the Gauss point of P 1 Berk , v . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Canonical measures in arithmetic dynamics For each place v of K there is a canonical measure µ φ, v on P 1 Berk , v which governs equidistribution of periodic points and iterated preimages. In the Archimedean case, the canonical measure is the well-known measure of maximal entropy studied by Brolin, Lyubich, and Freire-Lopes-Ma˜ n´ e. For example, if φ ( z ) = z 2 then for v Archimedean µ φ, v is Haar measure on the complex unit circle in P 1 ( C ). For v non-Archimedean µ φ, v is a point mass at the Gauss point of P 1 Berk , v . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Arakelov-Green functions in arithmetic dynamics There is a corresponding Arakelov-Green function at each place. The Arakelov-Green function for φ ( z ) = z 2 is given in both the Archimedean and non-Archimedean cases by the formula g φ, v (( x 1 , y 1 ) , ( x 2 , y 2 )) = − log | x 1 y 2 − x 2 y 1 | v + log max {| x 1 | v , | y 1 | v } + log max {| x 2 | v , | y 2 | v } . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Arakelov-Green functions in arithmetic dynamics There is a corresponding Arakelov-Green function at each place. The Arakelov-Green function for φ ( z ) = z 2 is given in both the Archimedean and non-Archimedean cases by the formula g φ, v (( x 1 , y 1 ) , ( x 2 , y 2 )) = − log | x 1 y 2 − x 2 y 1 | v + log max {| x 1 | v , | y 1 | v } + log max {| x 2 | v , | y 2 | v } . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An explicit formula for the dynamical Arakelov-Green function The formula for g φ, v when φ ( z ) = z 2 generalizes nicely to arbitrary rational maps: Theorem (B.–Rumely) Let φ ∈ K ( z ) be a rational map of degree d ≥ 2 . For any place v of K, we have g φ, v (( x 1 , y 1 ) , ( x 2 , y 2 )) = − log | x 1 y 2 − x 2 y 1 | v + ˆ H F , v ( x 1 , y 1 ) + ˆ H F , v ( x 2 , y 2 ) 1 − d ( d − 1) log | Res ( F 1 , F 2 ) | v . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An explicit formula for the dynamical Arakelov-Green function The formula for g φ, v when φ ( z ) = z 2 generalizes nicely to arbitrary rational maps: Theorem (B.–Rumely) Let φ ∈ K ( z ) be a rational map of degree d ≥ 2 . For any place v of K, we have g φ, v (( x 1 , y 1 ) , ( x 2 , y 2 )) = − log | x 1 y 2 − x 2 y 1 | v + ˆ H F , v ( x 1 , y 1 ) + ˆ H F , v ( x 2 , y 2 ) 1 − d ( d − 1) log | Res ( F 1 , F 2 ) | v . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Relation with the dynamical canonical height Corollary: For every x , y ∈ P 1 ( ¯ K ) we have d v h φ ( x ) + ˆ ˆ � h φ ( y ) = d g ϕ, v ( x , y ) . v ∈ M K Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics A Mahler-Elkies style lower bound for average values of the dynamical Arakelov-Green function Theorem (B.) Let φ ∈ C v ( z ) be a rational map of degree d ≥ 2 . There exists a constant C > 0 depending on φ such that if Z = { P 1 , . . . , P N } is a set of N distinct points in P 1 ( ¯ K ) , then Λ φ, v ( Z ) := 1 g φ, v ( P i , P j ) ≥ − C log N � . N 2 N i , j Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Commentary Note that we don’t have Fourier analysis or eigenfunction expansions available to us in this setting. The proof uses the explicit formula for g φ, v ( x , y ) together with a rather elaborate algebraic analysis of certain determinants and resultants. It generalizes an old estimate due to Mahler for the usual Weil height based on van der Monde determinants and Hadamard’s inequality. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Commentary Note that we don’t have Fourier analysis or eigenfunction expansions available to us in this setting. The proof uses the explicit formula for g φ, v ( x , y ) together with a rather elaborate algebraic analysis of certain determinants and resultants. It generalizes an old estimate due to Mahler for the usual Weil height based on van der Monde determinants and Hadamard’s inequality. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Commentary Note that we don’t have Fourier analysis or eigenfunction expansions available to us in this setting. The proof uses the explicit formula for g φ, v ( x , y ) together with a rather elaborate algebraic analysis of certain determinants and resultants. It generalizes an old estimate due to Mahler for the usual Weil height based on van der Monde determinants and Hadamard’s inequality. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics A global application We deduce the following Hindry–Silverman type estimate: Theorem (B.) There are constants A , B > 0 depending on φ and L such that if [ L : K ] = D then h φ ( P ) ≤ A # { P ∈ P 1 ( L ) | ˆ D } ≤ B · D log D . The proof uses a pigeonhole principle argument at a fixed place of K (using the compactness of P 1 ( C ) or P 1 Berk , v ) together with the Mahler-Elkies style lower bound from the previous slide. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics A global application We deduce the following Hindry–Silverman type estimate: Theorem (B.) There are constants A , B > 0 depending on φ and L such that if [ L : K ] = D then h φ ( P ) ≤ A # { P ∈ P 1 ( L ) | ˆ D } ≤ B · D log D . The proof uses a pigeonhole principle argument at a fixed place of K (using the compactness of P 1 ( C ) or P 1 Berk , v ) together with the Mahler-Elkies style lower bound from the previous slide. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics A lower bound for the degree of the field of definition of preperiodic points Corollary: There is a constant C > 0 depending on φ and L such that if P 1 , . . . , P N are distinct preperiodic points of φ defined over N L , then [ L : K ] ≥ C log N . Note that the corollary is nearly sharp in the case φ ( z ) = z 2 , since N [ Q ( ζ N ) : Q ] = ϕ ( N ) ≫ log log N . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics A lower bound for the degree of the field of definition of preperiodic points Corollary: There is a constant C > 0 depending on φ and L such that if P 1 , . . . , P N are distinct preperiodic points of φ defined over N L , then [ L : K ] ≥ C log N . Note that the corollary is nearly sharp in the case φ ( z ) = z 2 , since N [ Q ( ζ N ) : Q ] = ϕ ( N ) ≫ log log N . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An open problem It is an interesting open problem to study the dependence of the constants A , B on the map φ in the estimate: h φ ( P ) ≤ A # { P ∈ P 1 ( L ) | ˆ D } ≤ BD log D . The constants are ineffective as it stands because of the compactness argument invoked. It would be very interesting to have an analogue in arithmetic dynamics of the Hindry-Silverman theorem (that Szpiro’s conjecture implies uniform boundedness of rational torsion points) relating some variant of the ABC Conjecture to the Morton-Silverman conjecture on uniform boundedness of rational preperiodic points. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics An open problem It is an interesting open problem to study the dependence of the constants A , B on the map φ in the estimate: h φ ( P ) ≤ A # { P ∈ P 1 ( L ) | ˆ D } ≤ BD log D . The constants are ineffective as it stands because of the compactness argument invoked. It would be very interesting to have an analogue in arithmetic dynamics of the Hindry-Silverman theorem (that Szpiro’s conjecture implies uniform boundedness of rational torsion points) relating some variant of the ABC Conjecture to the Morton-Silverman conjecture on uniform boundedness of rational preperiodic points. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Potentially good reduction and isotriviality Let v be non-Archimedean. We say that φ ∈ C v ( T ) has potentially good reduction if it has good reduction after a change of coordinates (i.e., after conjugating by a M¨ obius transformation). We say that φ has genuinely bad reduction if it does not have potentially good reduction. If K is a function field (with arbitrary constant field), we say that φ ∈ K ( T ) is isotrivial if, after a change of coordinates and a finite extension of K , it is defined over the field of constants. Theorem (B.) Let K be a function field, and let φ ∈ K ( T ) be a rational map of degree at least 2. Then φ is isotrivial if and only if φ has potentially good reduction over C v for all v ∈ M K . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Potentially good reduction and isotriviality Let v be non-Archimedean. We say that φ ∈ C v ( T ) has potentially good reduction if it has good reduction after a change of coordinates (i.e., after conjugating by a M¨ obius transformation). We say that φ has genuinely bad reduction if it does not have potentially good reduction. If K is a function field (with arbitrary constant field), we say that φ ∈ K ( T ) is isotrivial if, after a change of coordinates and a finite extension of K , it is defined over the field of constants. Theorem (B.) Let K be a function field, and let φ ∈ K ( T ) be a rational map of degree at least 2. Then φ is isotrivial if and only if φ has potentially good reduction over C v for all v ∈ M K . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Potentially good reduction and isotriviality Let v be non-Archimedean. We say that φ ∈ C v ( T ) has potentially good reduction if it has good reduction after a change of coordinates (i.e., after conjugating by a M¨ obius transformation). We say that φ has genuinely bad reduction if it does not have potentially good reduction. If K is a function field (with arbitrary constant field), we say that φ ∈ K ( T ) is isotrivial if, after a change of coordinates and a finite extension of K , it is defined over the field of constants. Theorem (B.) Let K be a function field, and let φ ∈ K ( T ) be a rational map of degree at least 2. Then φ is isotrivial if and only if φ has potentially good reduction over C v for all v ∈ M K . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Potentially good reduction and isotriviality Let v be non-Archimedean. We say that φ ∈ C v ( T ) has potentially good reduction if it has good reduction after a change of coordinates (i.e., after conjugating by a M¨ obius transformation). We say that φ has genuinely bad reduction if it does not have potentially good reduction. If K is a function field (with arbitrary constant field), we say that φ ∈ K ( T ) is isotrivial if, after a change of coordinates and a finite extension of K , it is defined over the field of constants. Theorem (B.) Let K be a function field, and let φ ∈ K ( T ) be a rational map of degree at least 2. Then φ is isotrivial if and only if φ has potentially good reduction over C v for all v ∈ M K . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Potentially good reduction and dynamical Green functions Theorem φ has genuinely bad reduction over C v if and only if g φ, v ( x , x ) > 0 for all x ∈ P 1 Berk , v \ P 1 ( C v ) . Corollary: If φ has genuinely bad reduction, then there exists a constant β > 0 and a covering of P 1 ( C v ) by finitely many analytic open sets V 1 , . . . , V t such that for each 1 ≤ i ≤ t , we have g φ, v ( x , y ) ≥ β for all x , y ∈ V i . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Potentially good reduction and dynamical Green functions Theorem φ has genuinely bad reduction over C v if and only if g φ, v ( x , x ) > 0 for all x ∈ P 1 Berk , v \ P 1 ( C v ) . Corollary: If φ has genuinely bad reduction, then there exists a constant β > 0 and a covering of P 1 ( C v ) by finitely many analytic open sets V 1 , . . . , V t such that for each 1 ≤ i ≤ t , we have g φ, v ( x , y ) ≥ β for all x , y ∈ V i . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Isotriviality and preperiodicity Using the above results and a Hindry-Silverman style pigeonhole argument, we deduce the following global consequence: Theorem Let K be a function field and let φ ∈ K ( T ) be a rational map of degree d ≥ 2 . Assume that φ is not isotrivial. Then there exists ε > 0 (depending on K and φ ) such that the set { P ∈ P 1 ( K ) : ˆ h φ ( P ) ≤ ε } is finite. Corollary: If φ is not isotrivial, then a point P ∈ P 1 ( ¯ K ) satisfies ˆ h φ ( P ) = 0 if and only if P is preperiodic for φ . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Isotriviality and preperiodicity Using the above results and a Hindry-Silverman style pigeonhole argument, we deduce the following global consequence: Theorem Let K be a function field and let φ ∈ K ( T ) be a rational map of degree d ≥ 2 . Assume that φ is not isotrivial. Then there exists ε > 0 (depending on K and φ ) such that the set { P ∈ P 1 ( K ) : ˆ h φ ( P ) ≤ ε } is finite. Corollary: If φ is not isotrivial, then a point P ∈ P 1 ( ¯ K ) satisfies ˆ h φ ( P ) = 0 if and only if P is preperiodic for φ . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Isotriviality and preperiodicity Using the above results and a Hindry-Silverman style pigeonhole argument, we deduce the following global consequence: Theorem Let K be a function field and let φ ∈ K ( T ) be a rational map of degree d ≥ 2 . Assume that φ is not isotrivial. Then there exists ε > 0 (depending on K and φ ) such that the set { P ∈ P 1 ( K ) : ˆ h φ ( P ) ≤ ε } is finite. Corollary: If φ is not isotrivial, then a point P ∈ P 1 ( ¯ K ) satisfies ˆ h φ ( P ) = 0 if and only if P is preperiodic for φ . Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Another open problem Recall the statement of the previous corollary: Corollary: If φ has genuinely bad reduction, then there exists a constant β > 0 and a covering of P 1 ( C v ) by finitely many analytic open sets V 1 , . . . , V t such that for each 1 ≤ i ≤ t , we have g φ, v ( x , y ) ≥ β for all x , y ∈ V i . It would be interesting to find good explicit bounds for t and β in terms of the map φ . This would yield an extension of Benedetto’s “ s log s ” bound for the number of preperiodic points of a polynomial map φ to arbitrary rational maps. Matt Baker Lower bounds for Arakelov-Green functions

Motivation: The work of Hindry–Silverman Equidistribution and Non-Equidistribution on Elliptic Curves Arakelov-Green functions on metrized graphs Arithmetic dynamics Another open problem Recall the statement of the previous corollary: Corollary: If φ has genuinely bad reduction, then there exists a constant β > 0 and a covering of P 1 ( C v ) by finitely many analytic open sets V 1 , . . . , V t such that for each 1 ≤ i ≤ t , we have g φ, v ( x , y ) ≥ β for all x , y ∈ V i . It would be interesting to find good explicit bounds for t and β in terms of the map φ . This would yield an extension of Benedetto’s “ s log s ” bound for the number of preperiodic points of a polynomial map φ to arbitrary rational maps. Matt Baker Lower bounds for Arakelov-Green functions