The work of Granville{Stark on ABC and Siegel zeros Christian T´afula D´epartement de math´ematiques et de statistique (DMS), Universit´ede Montr´eal(UdeM) Math 726: Topics in Number Theory Modular forms and the theory of complex multiplication T´afula,C. (DMS, UdeM) Granville{Stark October 2020 0 / 22 Contents 1 Introduction 2 Lower bounds for h(D) 3 The Siegel zero T´afula,C. (DMS, UdeM) Granville{Stark October 2020 1 / 22 Contents 1 Introduction 2 Lower bounds for h(D) 3 The Siegel zero T´afula,C. (DMS, UdeM) 1. Intro October 2020 2 / 22 Notation p p D < 0 fundamental discriminant (so D2 = R, D = disc Q( D)); p C`(D) = class group of Q( D); p h(D) = jC`(D)j class number of Q( D); j = j-invariant function: 3 1 + 240 P P d3qn n≥1 djn 2πiτ j(τ) := Q n 24 =(τ) > 0; q = e q n≥1(1 − q ) j(i) = 1728; q-expansion of j: j(τ + 1) = j(τ); 2πi=3 j(e ) = 0; 1 j(−1/τ) = j(τ); j(τ) = + 744 + 196884q + ··· \j(i1) = 11"; q T´afula,C. (DMS, UdeM) 1. Intro October 2020 3 / 22 Singular moduli (1/2) Let τ 2 h (=(τ) > 0) A; B; C 2 ; A > 0; CM-point: τ j Aτ 2 + Bτ + C = 0 Z gcd(A; B; C) = 1; unique Singular modulus: j(τ)(j = j-invariant, τ a CM-point) CM-points Quadratic forms F ! τ 2 h Ax2 + Bxy + Cy2 2πi πi e 3 e 3 disc(τ) := B2 − 4AC • ◦ τ ∼ τ 0 () (A; B; C) ∼ (A0;B0;C0) τ 2 F () (A; B; C) reduced -1 -0.5 0 0.5 1 T´afula,C. (DMS, UdeM) 1. Intro October 2020 4 / 22 Singular moduli (2/2) Heegner points ΛD Reduced prim. quad. forms of disc. D 8 (a; b; c) := ax2 + bxy + cy2 9 CM-points τ 2 F ; < = ! s.t. b2 − 4ac = D; and disc(τ) = D :−a < b ≤ a < c or 0 ≤ b ≤ a = c; p 2 p 2 D −1 + D τD := or $ Principal form p 2 2 Z[τD] = O ( D) |{z} | {z } | {z } Q D≡0(4) D≡1(4) D 1−D (1;0;− 4 ) or (1;1; 4 ) p Write HD := Hilbert class field of Q( D). p p HD = Q( D; j(τD)) [HD : Q( D)] = [Q(j(τD)) : Q] = h(D) p fj(τ) j τ 2 ΛDg = Gal(Q=Q( D))-conjugates of j(τD) j(τD) is an algebraic integer! T´afula,C. (DMS, UdeM) 1. Intro October 2020 5 / 22 The equation x3 − Dy2 = 1728 In class, we saw the equation x3 − Dy2 = 1728 (= j(i)). Factorization of differences of singular moduli 3 2 Solutions of the type (j(τD); j(τD) − 1728) = (x ; Dy ) Idea: ABC =) few such solns =) lower bounds for h(D) ABC : a; b; c 2 Z coprime: 8" > 0; 9C" > 0 s.t. Q 1+" (a + b = c) maxfjaj; jbj; jcjg < C" pjabc p Example (Class number 1) p We've seen that h(D) = 1 =) x; y jDj 2 Z. However, in this case, p abc p 3 2 1+" jj(τD)j ≤ maxfjxj ; jy Dj ; 1728g ≤ C" · jx · y Dj 5 (1+") ≤ C" · jj(τD)j 6 ; so there can only be finitely many D < 0 with h(D) = 1. 4 T´afula,C. (DMS, UdeM) 1. Intro October 2020 6 / 22 How does h(D) grow? What do we know? h(D) ! 1 as D ! −∞ (Heilbronn, 1934) List h(D) (small values of jDj) Estimate h(D) (large values of jDj) −D 2 f3; 4; 7; 8; 11; 19; 43; 67; 163g are the only D s.t. h(D) = 1 Solved by Heegner (∼1952) [based on Weber's work] Reworked and clarified by Stark, Birch (1967∼9) Baker's solution (∼1970) [based on Baker's theorem] h(D) grows roughly like jDj1=2 (GRH =) err ≈ (log log jDj)±1) h(D) Hecke (1917): If ζ p (s) 6= 0 near 1, then (log jDj)−1 Q( D) jDj1=2 h(D ) h(D ) Landau (1918): max 1 ; 2 (log jD D j)−1 1=2 1=2 1 2 jD1j jD2j h(D) Siegel (1935): jDj−" (unconditional but ineffective) jDj1=2 " T´afula,C. (DMS, UdeM) 1. Intro October 2020 7 / 22 Contents 1 Introduction 2 Lower bounds for h(D) 3 The Siegel zero T´afula,C. (DMS, UdeM) 2. LB for h(D) October 2020 8 / 22 The Granville{Stark Theorem Theorem (Granville{Stark, 2000) \No Siegel zeros" for Uniform abc-conj. =) ζ p (s); D < 0 Q( D) Study x3 − y2 = 1728 A. Granville Uniform ABC for number fields (U-abc) U-abc =) lower bounds for h(D) 1 Lower bounds h(D) () 2 \no Siegel zeros" H. Stark T´afula,C. (DMS, UdeM) 2. LB for h(D) October 2020 9 / 22 The equation x3 − y2 = 1728 3 2 Solutions of the type (j(τD); j(τD) − 1728) = (x ; y ) X ABC (for Q) : log maxfjaj; jbj; jcjg < (1 + ") log p + O"(1) pjabc | {z } log -conductor Applying the same logic for x3 − y2 − 1728 = 0, we would get: log maxfjxj3; jyj2g ≤ log maxfjxj3; jyj2; 1728g abc ≤ (1 + ") log -cond(12jxyj) + tax 5 ≤ (1 + ") log maxfjxj3; jyj2g + tax 6 How to make −! this precise? log maxfjxj3; jyj2g < (6 + "0) · tax T´afula,C. (DMS, UdeM) 2. LB for h(D) October 2020 10 / 22 abc-conjecture for number fields (1/2) non Let K=Q be a NF, MK its places, MK ⊆ MK non-arch. places n For a point P = [x0 : ··· : xn] 2 PK , define: (na¨ıve, abs, log) height ht(P ) (log) conductor NK (P ) 1 1 X X fv log(pv) log maxfkxikvg [K : ] [K : Q] i Q v2Mnon v2MK K 9i;j≤n s:t: v(xi)6=v(xj ) For a; b; c 2 coprime, Z $ 8 v ∼ p = p ht([a : b : c]) = log maxfjaj; jbj; jcjg <> v p ∼ p \ Q v v Q NQ([a : b : c]) = log pjabc p :> fv := [Kv : Qpv ] 1 X α integral ) ht(α) = log+ jα∗j jAj For α 2 Q, ht(α) := ht([α : 1]). α∗2A A = fconjugates of αg T´afula,C. (DMS, UdeM) 2. LB for h(D) October 2020 11 / 22 abc-conjecture for number fields (2/2) abc-conjecture for number fields Fix K=Q a number field. Then, for every " > 0, there is C(K; ") 2 R+ such that, 8a; b; c 2 K j a + b + c = 0, we have ht([a : b : c]) < (1 + ") NK ([a : b : c]) + log(rdK ) + C(K; "); 1=[K: ] where rdK := j∆K j Q is the root-discriminant of K. Example (a; b; c integral, coprime) For a; b; c 2 OK coprime (meaning v(a) 6= v(b) 6= v(c) for v < 1), 1 1 1 [K:Q] [K:Q] [K:Q] maxfjNK=Q(a)j ; jNK=Q(b)j ; jNK=Q(c)j g !1+" Y OK < CK;" j∆K j pOK ; p j (abc) with CK;" = exp([K : Q] ·C(K; ")). T´afula,C. (DMS, UdeM) 2. LB for h(D) October 2020 12 / 22 What do we get? 3 2 3 2 (j(τD); j(τD) − 1728) = (x ; y ) x − y = 1728 j(τD) 2 HD He D := HD(x; y) [He D :HD] ≤ 6 3 2 • \ log maxfjxj ; jyj g" < (6 + ") · tax(He D;") More precisely: ht(j(τD)) < 6 (1 + ") log(rd ) + C(HD;") He D e p p p \Factorization": rd ≤ 6 jDj rdH = rd = jDj He D D Q( D) 1 Uniform ABC-conj.: C(K; ") = C(") =) tax(") log jDj . 2 T´afula,C. (DMS, UdeM) 2. LB for h(D) October 2020 13 / 22 Lower bounds for h(D) Lemma [G{S] Unif. ABC-conj. =) ht(j(τD)) ≤ (3 + o(1)) log jDj However, 1 X ax2 + bxy + cy2 ht(j(τ )) = log+ jj(τ )j D h(D) (a;b;c) reduced; (a;b;c) 2 p b − 4ac = D 1 X π jDj = + O(1); h(D) a p (a;b;c) −b + D τ(a;b;c) = and thus: 2a Theorem [G{S] q-expansion! p U-abc π jDj X 1 1 h(D) ≥ + o(1) j(τ) = + O(1) 3 log jDj a q (a;b;c) T´afula,C. (DMS, UdeM) 2. LB for h(D) October 2020 14 / 22 Contents 1 Introduction 2 Lower bounds for h(D) 3 The Siegel zero T´afula,C. (DMS, UdeM) 3. Siegel zero October 2020 15 / 22 p Dedekind zeta function of Q( D) X 1 c−1 ζ (s) = = + c + O(s − 1) (s ! 1) K N(a)s s − 1 0 a⊆OK Conjecture (\no Siegel zeros" for D < 0) δ There is δ > 0 such that ζ p (β) 6= 0 for 1 − ≤ β < 1. Q( D) log jDj π h(D) • Dirichlet's class number formula: c−1(D) = (D < −4) pjDj p 1 π h(D) log jDj −1 ζ (β) ≤ − + c0(D) + O (log jDj) Q( D) δ pjDj U-abc 1 X 1 ≤ − + o(1) + c (D) + o(1) δ a 0 (a;b;c) T´afula,C. (DMS, UdeM) 3. Siegel zero October 2020 16 / 22 Kronecker's limit formula (KLF) X 1 We just need to show that c (D) (as D ! −∞). 0 a (a;b;c) +1 ! 1 X X 1 ζ p (s) = Q( D) 2 (am2 + bmn + cn2)s (a;b;c) m;n=−∞ m;n 6= 0 | {z } Epstein zeta function: Z(a;b;c)(s) KLF [Kronecker 1863, Weber 1881, :::, Chowla{Selberg 1967, :::] The constant term (c0) of Z(a;b;c)(s) as s ! 1 is given by: p p ! π 1 jDj −b+ D 4 2γ − log(2) − log jDj − log η pjDj 2 2a 2a T´afula,C.