ABC...L: The uniform abc -conjecture and zeros of Dirichlet L - PowerPoint PPT Presentation

ABC...L: The uniform abc -conjecture and zeros of Dirichlet L -functions Christian T´ afula 京 都 大 学 数 理 解 析 研究 所 (RIMS, Kyoto University) 2nd Kyoto–Hefei Workshop, August 2020 T´ afula, C. (RIMS, Kyoto U) ABC...L Kyoto–Hefei 2020 0 / 45

Contents 1 Review: Zeros of L -functions 2 Statement of the main theorems 3 Isolating the Siegel zero 4 The bridge: KLF and Duke’s Theorem 1 5 Uniform abc = ⇒ 2 “no Siegel zeros” T´ afula, C. (RIMS, Kyoto U) ABC...L Kyoto–Hefei 2020 1 / 45

Contents 1 Review: Zeros of L -functions 2 Statement of the main theorems 3 Isolating the Siegel zero 4 The bridge: KLF and Duke’s Theorem 1 5 Uniform abc = ⇒ 2 “no Siegel zeros” T´ afula, C. (RIMS, Kyoto U) 1. Review of L -functions Kyoto–Hefei 2020 2 / 45

Characters Let q ≥ 1 be an integer. A Dirichlet character χ (mod q ) is a function χ : Z → C ∗ s.t.: χ ( nm ) = χ ( n ) χ ( m ) for every n, m ; χ ( n + q ) = χ ( n ) for every n ; χ ( n ) = 0 if gcd( n, q ) > 1. Alternatively, χ is the lifting of a homeomorphism χ : ( Z /q Z ) × → C ∗ . χ (mod q ) ( Z /q Z ) × C ∗ Primitive: ∄ d | q ( d � = q ) s.t. χ ′ (mod d ) ( Z /d Z ) × � Principal: ( Z /q Z ) × → C ∗ is trivial (i.e., χ 0 ( n ) = 1 , if ( n, q ) = 1 0 , if ( n, q ) > 1 ) ⇒ Quadratic: χ 2 = χ 0 ) ( ⇐ Real: χ = χ Even: χ ( − 1) = 1, Odd: χ ( − 1) = − 1. T´ afula, C. (RIMS, Kyoto U) 1. Review of L -functions Kyoto–Hefei 2020 3 / 45

Real characters � � �� D � � Real primitive , D fundamental ← → Dirichlet characters · discriminant A fundamental discriminant is an integer D ∈ Z s.t.: ∃ K/ Q quadratic | ∆ K = D ; or, equivalently , � D ≡ 1 (mod 4) , D square-free; or D ≡ 0 (mod 4) , s.t. D/ 4 ≡ 2 or 3 (mod 4) and D/ 4 square-free . � D � : Z → {− 1 , 0 , 1 } is The Kronecker symbol · � D �� D � � D � , ∀ m, n ∈ Z ); Completely multiplicative (i.e., = m n mn  √ 1 , ( p ) splits in Q (  D ) � D � � D �  = = sgn( D ). − 1 , ( p ) is inert · · · p  − 1  0 , ( p ) ramifies · · · (i.e., p | D ) � D � , we have χ D (mod | D | ) real, primitive Writing χ D := · T´ afula, C. (RIMS, Kyoto U) 1. Review of L -functions Kyoto–Hefei 2020 4 / 45

L -functions The Dirichlet L -function associated to non-principal χ (mod q ): � � � � χ ( n ) 1 L ( s, χ ) := = , ( ℜ ( s ) > 1) n s 1 − χ ( p ) p − s p n ≥ 1 (E.g.: 1 + z + z 2 + · · · = Analytic continuation: 1 1 − z ) L ( s, χ ) is entire; Functional equation: For a χ := 0 (if χ even) or 1 (if χ odd), 2 ( s + a χ ) Γ( 1 L ∗ ( s, χ ) := ( π/q ) − 1 2 ( s + a χ )) L ( s, χ ) is entire; L ∗ ( s, χ ) = W ( χ ) L ∗ (1 − s, χ ), where | W ( χ ) | = 1. Reflection Critical strip: � 0 , − 2 , − 4 . . . , ( χ even) (Trivial zeros) Poles of Γ( 1 2 ( s + a χ )), i.e.: − 1 , − 3 , − 5 . . . , ( χ odd) (Non-trivial zeros) All other zeros are in { s ∈ C | 0 < ℜ ( s ) < 1 } T´ afula, C. (RIMS, Kyoto U) 1. Review of L -functions Kyoto–Hefei 2020 5 / 45

Anatomy of ζ ( s ) = � n ≥ 1 n − s • • 50 i • • • 40 i • critical strip 0 < ℜ ( s ) < 1 • • 30 i non-trivial • zeros • 20 i critical line ℜ ( s ) = 1 / 2 • 10 i − 6 − 5 − 4 − 3 − 2 − 1 1 2 0 • • • • pole trivial zeros − 10 i • − 20 i • • T´ afula, C. (RIMS, Kyoto U) 1. Review of L -functions Kyoto–Hefei 2020 6 / 45

Classical (quasi) zero-free regions t [Gronwall 1913, Landau 1918, Titchmarsh 1933] Write s = σ + it ( σ = ℜ ( s ), t = ℑ ( s )), and let χ (mod q ) be a Dirichlet character. There exists c 0 > 0 such that, in the region � � � σ � c 0 � s ∈ C � σ ≥ 1 − , 1 log q ( | t | + 2) 0 1 2 the function L ( s, χ ) has: ( χ complex) no zeros; ( χ real) at most one zero, which is necessarily real and simple – the so-called Siegel zero . T´ afula, C. (RIMS, Kyoto U) 1. Review of L -functions Kyoto–Hefei 2020 7 / 45

q -aspect vs. t -aspect c 0 t -aspect σ < 1 − log q ( | t | + 2) ⇔ 1 1 − σ ≪ log q ( | t | + 2) q -aspect: q → + ∞ , | t | bounded; t c t -aspect: q bounded , e p s a - | t | → + ∞ . q T´ afula, C. (RIMS, Kyoto U) 1. Review of L -functions Kyoto–Hefei 2020 8 / 45

Siegel zeros Let: D ∈ Z be a fundamental discriminant β D the largest real zero of L ( s, χ D ) Conjecture (“no Siegel zeros”) 1 ≪ log | D | 1 − β D Remark. Imprimitive case follows from primitive case. (Siegel, 1935) For every ε > 0, it holds that Ineffective! � � � 1 1 2 − ε , √ D ) ≫ | D | h Q ( for D < 0 ≪ | D | ε ⇐ ⇒ 1 2 − ε , for D > 0 1 − β D h Q ( √ D ) log η D ≫ D 1 − β D = 1 1 1 (GRH + Chowla) 2 (if D < 0), 1 − β D = 1 (if D > 0). T´ afula, C. (RIMS, Kyoto U) 1. Review of L -functions Kyoto–Hefei 2020 9 / 45

In summary (1/5) Siegel zeros are... t about real primitive characters χ exceptional = ⇒ χ = χ D Exceptional := violates (q-)ZFR by at most one (real, simple) zero a q -aspect problem σ Box of height 1 ( | t | ≤ 1) 1 0 1 2 Zeros “very close” to s = 1 related to quadratic fields √ → Q ( χ D ← D ) √ β D ← → h Q ( D ) T´ afula, C. (RIMS, Kyoto U) 1. Review of L -functions Kyoto–Hefei 2020 10 / 45

Contents 1 Review: Zeros of L -functions 2 Statement of the main theorems 3 Isolating the Siegel zero 4 The bridge: KLF and Duke’s Theorem 1 5 Uniform abc = ⇒ 2 “no Siegel zeros” T´ afula, C. (RIMS, Kyoto U) 2. Theorems Kyoto–Hefei 2020 11 / 45

1 Uniform abc = ⇒ 2 “no Siegel zeros” Theorem (Granville–Stark, 2000) “No Siegel zeros” for Uniform abc -conj. = ⇒ χ D (mod | D | ) , D < 0 ht( j ( τ D )) Uniform abc -conj. = ⇒ lim sup ≤ 3 log | D | D →−∞ ⇒ A. Granville L ′ L (1 , χ D ) lim sup < + ∞ log | D | D →−∞ ⇔ δ ( 1 ∃ δ > 0 | β D < 1 − 2 “No Siegel zeros”) log | D | L ′ 1 ← → ← → L (1 , χ D ) ht( j ( τ D )) 1 − β D H. Stark Key to the bridge! T´ afula, C. (RIMS, Kyoto U) 2. Theorems Kyoto–Hefei 2020 12 / 45

Uniform abc -conjecture (1/2) Let K/ Q be a NF, M K its places, M non ⊆ M K non-arch. places K For a point P = [ x 0 : · · · : x n ] ∈ P n K , define: (log) conductor N K ( P ) (na¨ ıve, abs, log) height ht( P ) � � � � 1 1 f v log( p v ) log max i {� x i � v } [ K : Q ] [ K : Q ] v ∈M non v ∈M K K ∃ i,j ≤ n s . t . v ( x i ) � = v ( x j ) For a, b, c ∈ Z coprime, ↔  v ∼ p = p v   ht([ a : b : c ]) = log max {| a | , | b | , | c |} p v ∼ p v ∩ Q � � �  N Q ([ a : b : c ]) = log p | abc p  f v := [ K v : Q p v ] � 1 log + | α ∗ | α integral ⇒ ht( α ) = For α ∈ Q , ht ( α ) := ht ([ α : 1]). |A| α ∗ ∈A A = { conjugates of α } T´ afula, C. (RIMS, Kyoto U) 2. Theorems Kyoto–Hefei 2020 13 / 45

Uniform abc -conjecture (2/2) abc for number fields Fix K/ Q a number field. Then, for every ε > 0 , there is C ( K, ε ) ∈ R + such that, ∀ a, b, c ∈ K | a + b + c = 0 , we have � � ht([ a : b : c ]) < (1 + ε ) N K ([ a : b : c ]) + log(rd K ) + C ( K, ε ) , where rd K := | ∆ K | 1 / [ K : Q ] is the root-discriminant of K . d ( α ) Hermite: ∆ K bdd. ⇒ # { K } < ∞ Northcott: d ( α ), ht( α ) bdd. ⇒ # { α } < ∞ Uniform abc -conjecture (U- abc ) C ( K, ε ) = C ( ε ) ht( α ) d -aspect Vojta’s general conjecture = ⇒ U- abc vs. height-aspect T´ afula, C. (RIMS, Kyoto U) 2. Theorems Kyoto–Hefei 2020 14 / 45

Singular moduli (1/2) Let τ ∈ h ( ℑ ( τ ) > 0) � � A, B, C ∈ Z , A > 0 , CM-point: τ | Aτ 2 + Bτ + C = 0 gcd( A, B, C ) = 1 , unique Singular modulus: j ( τ ) ( j = j -invariant, τ a CM-point) j : h → C is the unique function s.t.: F is holomorphic; j ( i ) = 1728, j ( e 2 πi/ 3 ) = 0, j ( i ∞ ) = ∞ ; � aτ + b � � a b � 2 πi πi e e ∈ SL 2 ( Z ) . 3 3 j = j ( τ ) , ∀ ◦ • c d cτ + d j ( τ ) = 1 q + 744 + 196884 q + · · · -1 -0.5 0 0.5 1 q -expansion of the j -invariant ( q = e 2 πiτ ) T´ afula, C. (RIMS, Kyoto U) 2. Theorems Kyoto–Hefei 2020 15 / 45

Singular moduli (2/2) Heegner points Λ D Reduced bin. quad. forms of disc. D   ( a, b, c ) := ax 2 + bxy + cy 2 � CM-points τ ∈ F , �   s.t. b 2 − 4 ac = D, and ← → disc( τ ) = D   − a < b ≤ a < c or 0 ≤ b ≤ a = c ∈ ∈ √ √ or − 1 + D D τ D := ↔ Principal form Z [ τ D ] = O Q ( √ 2 2 � �� � D ) ���� � �� � (1 , 0 , − D 4 ) or (1 , 1 , 1 − D ) D ≡ 0(4) D ≡ 1(4) 4 √ Write H D := Hilbert class field of Q ( D ). � � √ √ H D = Q ( D, j ( τ D )) [H D : Q ( D )] = [ Q ( j ( τ D )) : Q ] = h Q ( √ D ) √ { j ( τ ) | τ ∈ Λ D } = Gal( Q / Q ( D ))-conjugates of j ( τ D ) � 1 log + | j ( τ ) | j ( τ D ) is an algebraic integer! ⇒ ht( j ( τ D )) = h Q ( √ D ) τ ∈ Λ D T´ afula, C. (RIMS, Kyoto U) 2. Theorems Kyoto–Hefei 2020 16 / 45