Equidistribution in arithmetic geometry The probabilistic viewpoint and dynamics in arithmetic geometry Juan Rivera-Letelier Roots of Littlewood and Einsenstein polynomials; U. of Rochester Discrepancy; Equidistribution. Parameter Problems in Analytic Dynamics Imperial College, June Roots of Littlewood polynomials Roots of Einsenstein polynomials Figure : Roots of polynomials with coe ffi cients + and − , by Tiozzo . Figure : Roots of monic polynomials of degree with coe ffi cients or . Christensen , Derbyshire , Baez , ...
Roots of single Einsenstein polynomial Discrepancy Theorem (Radial discrepancy) P ( z ) = a d z d + a d − z d − + ··· + a ∈ C [ z ] , a d a � . For every ε in ( , ) , we have � � d # α root of P : | α | < − ε or | α | > − ε � d j = | a j | ≤ d log � . ε | a a d | Hughes – Nikeghbali , ; � � Applications: Roots of Littlewood and Einsenstein polynomials ε ∼ √ . d Discrepancy Discrepancy ⇒ Equidistribution Corollary (Equidistribution) λ : Uniform probability on S ( Haar measure); ( P n ) + ∞ n = : Littlewood polynomials such that d n := deg ( P n ) − n → + ∞ + ∞ . − − − − − → For every continuous function ϕ : C → R with compact support, � � ϕ ( α ) − − − − − − → ϕ ( z ) d λ ( z ) . d n n → + ∞ α root of P n Equivalently: � δ α − − n → + ∞ λ. − − − − → d n Radial discrepancy: Hughes – Nikeghbali , ; Angular discrepancy: Erdös – Turán , α root of P n ; Higher dimension: D’Andrea – Galligo – Sombra , .
Discrepancy Theorem (Radial discrepancy) Heights and equidistribution P ( z ) = a d z d + a d − z d − + ··· + a ∈ C [ z ] , a d a � . For every ε in ( , ) , we have Mahler measure and naive height; � � Arithmetic equidistribution; d # α root of P : | α | < − ε or | α | > − ε Dynamical heights; � d j = | a j | ≤ Proving equidistribution; d log � . ε | a a d | Adelic heights. Hughes – Nikeghbali , ; � � Applications: Roots of Littlewood and Einsenstein polynomials ε ∼ √ . d Mahler measure Naive height � � d � j = | a j | √ can by replaced by: d log α : Algebraic number; | a a d | P α := Minimal polynomial of α (with integer sup z ∈ S | P ( z ) | T ET ( P ) := d log � ; coe ffi cients); | a a d | d α := deg ( P α ) ; Ganelius , ; h W ( α ) := d α log M ( P α ) . Erdös – Turán size of P . Naive or Weil height of α ; � � �� � p Comparison with T ET = sup z ∈ S | P α ( z ) | | P ( z ) | p d λ ( z ) d α log √ . p > , M p ( P ) := ; | a a d | L p measure of P . • h W ( α ) measures the arithmetic complexity of α , e.g. , �� � � p � M ( P ) := exp log | P ( z ) | d λ ( z ) ; = logmax {| p | , | q |} ; h W q = lim p → + M p ( P ) . Exercise! • h W ( α ) ≥ with equality if and only if α is a root of unity. Mahler measure of P (“geometric mean”).
Naive height and equidistribution Equidistribution of roots of unity Theorem ( Bilu , ) ( α n ) + ∞ n = : Algebraic numbers such that h W ( α n ) − − n → + ∞ − − − − → and d α n − n → + ∞ + ∞ . − − − − − → We have, � δ α − n → + ∞ λ. − − − − − → d α n α root of P α n Similar to the Néron – Tate height of an Abelian variety Szpiro – Ullmo – Zhang , . Application: α n primitive root of unity of order n ; Equidistribution of roots of unity Equidistribution of roots of unity
Equidistribution of roots of unity Dynamical heights R ( z ) ∈ Q ( z ) , rational function of degree D ≥ , R : � C → � C ; Discrete time dynamical system on the Riemann sphere � C . ρ R : Maximal entropy measure of R . D n h W ◦ R n . h R := lim n → + ∞ Canonical height of R . • Unique “adelic” height such that h R ◦ R = D · h R ; • h R ( α ) ≥ with equality if and only if α is a periodic point of R (a solution of R n ( z ) − z = , for some n ≥ ). Comparison: Naive height / Néron – Tate height; Roots of unity / torsion points. Dynamical heights Dynamical heights Theorem ( α n ) + ∞ n = : Algebraic numbers such that h R ( α n ) − − n → + ∞ − − − − → and d α n − − n → + ∞ + ∞ . − − − − → We have, � δ α − n → + ∞ ρ R . − − − − − → d α n α root of P α n Baker – Rumely , Chabert-Loir , Favre – RL , ; Application: Equidistribution of periodic points ( Lyubich , ). Figure : Density invariant by a rational map, by Chéritat .
Mandelbrot height Mandelbrot height c ∈ C , Definition P c : C → C , P c ( z ) := z + c ; The Mandelbrot height h M : Q → R is, K c := { z ∈ C : ( P n c ( z )) n ≥ is bounded } . Filled Julia set of P c . h M ( c ) := h P c ( c ) . M := { c ∈ C : K c is connected } . Comparison: Uniformization of � C \ M . The Mandelbrot set. • h M ( c ) ≥ with equality if and only if P c is post-critically finite. ⇔ The orbit of the critical point of P c is finite. Theorem The asymptotic distribution of small points for the Mandelbrot set is given by the harmonic measure of the Mandelbrot set. Application: Equidistribution of post-critically finite parameters, Levin s. Proving equidistribution Proving equidistribution ρ , ρ ′ : (Signed) measures on the Riemann sphere � C . α : Algebraic number; � P α = Minimal polynomial of α (with integer ( ρ,ρ ′ ) := − log | z − z ′ | d ρ ( z ) d ρ ′ ( z ′ ) . coe ffi cients); C × C \ Diag Potential energy. d α = deg ( P α ) ; � � � h W ( α ) = − d α log M ( P α ) = λ,δ � α � h W ( α ) = log | P α ( z ) | d λ ( z ) . d α � � = + λ − δ � α � ,λ − δ � α � log | ∆ ( α ) | . When α is an algebraic integer ( ⇔ P α is monic): d α ∆ ( α ) := discriminant of P α (a nonzero integer). � � h W ( α ) = Morally, Bilu ’s theorem follows from: log | z − α ′ | d λ ( z ) d α Cauchy – Schwarz inequality : α ′ ∈O ( α ) �� � � ρ regular, and ρ C = ⇒ log | z − z ′ | d λ ( z ) d δ � α � ( z ′ ) . = ( ρ,ρ ) ≥ , with equality if and only if ρ = . O ( α ) := Set of roots of P α ; � Details: Case α is not an integer (adelic formula); δ � α � := α ′ ∈O ( α ) δ α ′ . d α δ � α � is not regular (convolution, and error estimate).
Adelic heights Adelic heights ρ : Regular probability measure on � C . Zhang ’s inequality ( ) h ρ := height such that for every algebraic integer α , The essential minimum of an adelic height is nonnegative. � � h ρ ( α ) = + ρ − δ � α � ,ρ − δ � α � log | ∆ ( α ) | . d Definition α An adelic height h ρ is quasi-canonical if its essential Adelic height associated to ρ . minimum is equal to . λ : uniform measure on S , Theorem ( Yuan , ) h λ = h W , the Weil height ; If h ρ is quasi-canonical, then the asymptotic distribution of Small points for h ρ is given by ρ . µ M : harmonic measure of the Mandelbrot set, h µ M = h M , the Mandelbrot height . In dimension : Baker – Rumely , Chabert-Loir , Favre – RL , ; The previous equidistribution results follow by observing: ρ R : measure of maximal entropy of (some) R ∈ Q ( z ) , The Weil height, the dynamical heights, and the h ρ R = h R , canonical height associated to R ; Mandelbrot height are all quasi-canonical. For R with “good reduction at every prime”. Beyond quasi-canonical heights Toric heights ω : Spherical measure on the Riemann sphere � C ; Theorem ( Burgos –Philippon– RL – Sombra , arXiv ) h ω : Adelic height associated to ω . ρ : Regular probability on � C with radial symmetry. Spherical height. Centered case: supp ( ρ ) ⊃ S . Equidsitribution to λ . Theorem (Sombra, ) Bipolar case: supp ( ρ ) disjoint from S , but intersecting both The spherical height is not quasi-canonical. In fact hemispheres. essential minimul of h ω = log . Non-radial, but centered limit measures. Totally unbalanced case: supp ( ρ ) disjoint from S , contained in a hemisphere. Theorem ( Burgos – Phillipon – Sombra , ) Non-radial and non-centered limit measures. Among “toric” heights (= heights with radial symmetry), the only quasi-canonical height is the Weil height (!!).
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