Rational Points on Curves

Transcription

1 Rational Points on Curves 1

2 Q algebraic closure of Q. k a number field. Ambient variety: Elliptic curve E an elliptic curve defined over Q. E N = E E. In general A an abelian variety defined over Q. For A defined over k, we denote by A(k) the k-rational points of A, (all coordinates in k). We identify A = A(Q). 2

3 Tori G m = Q the multiplicative group of Q G n m = Q Q, Torsion subgroup of G n m Tor Gm = Roots of unity = {ζ G m : N N,ζ N = 1} Tor G n m = {(ζ 1,,ζ n ) G n m : ζ i Tor Gm } Finitely generated subgroup Γ of G n m 3

4 Algebraic subgroups of G n m An algebraic subgroup B of dimension n s is the kernel of a matrix φ B Mat s,n (Z) of rank s b b 1n φ B =... : G n m G s m b s1... b sn φ B : (x 1,...,x n ) (x b x b 1n n,...,x b s x b sn n ). Up to constants, degb is the maximal Minor of φ B. Choosing φ B appropriately, degb is up to constant b 1... b s. Remark There are only finitely many algebraic subgroups of bounded degree. Tor G n m = dimb=0 B. dimb=0 B dimb 1 B dimb n 1 B. 4

5 Cuves in Tori Let C be an algebraic curve embedded in G n m. Questions When is C Tor G n m finite? When is C Γ finite? When is C dimb r B finite? Note that if C = G m ζ 2 ζ n then none of the above sets is finite!! These are special cases of the toric Manin-Mumford Conjecture, Mordell-Lang Conjecture, Torsion Anomalous Conjecture 5

6 Elliptic Case Let E be an elliptic curve. Consider E N for some integer N. Torsion subgroups Tor E = {P E : m N + with mp = 0} Tor E N = (Tor E ) N Finitely generated subgroup Γ of E N Theorem (Mordell-Weil Theorem) Let k be a number field and let E be defined over k. Then E(k) is finitely generated. Let A be an abelian variety defined over k. Then A(k) is finitely generated. 6

7 Algebraic subgroups of E N An algebraic subgroup B of dimension n s is the kernel of a matrix φ B Mat s,n (End(E)) of rank s b b 1,n φ B =... : E N E s b s,1... b s,n φ B : (x 1,...,x N ) (b 11 x b 1N x N,...,b s1 x b sn x N ). Up to constants, degb is the square of the maximal Minor of φ B. Choosing φ B appropriately, degb is up to constants b b s 2. Remark There are only finitely many algebraic subgroups of bounded degree. Tor E N = dimb=0 B. dimb=0 B dimb 1 B dimb n 1 B. 7

8 Cuves in E N Let C be an algebraic curve embedded in E N. Questions When is C Tor E N finite? When is C(k) finite? When is C dimb r B finite? Note that if C = E + t with t a torsion point, then none of the above sets is finite!! These are special cases of the Manin-Mumford Conjecture, Mordell-Lang Conjecture, Torsion Anomalous Conjecture 8

9 Some Classical results on Curves Let C P 2 be a curve defined by a homogeneous polynomial with coefficients in k. P(x,y,z) = 0 Geometry Arithmetic C(k) Quadrics C(k) = /0 or g(c) = 0 degp 2 C(k) infinite Elliptic C(k) = curves degp = 3 Z r Z/n 1 Z Z/n 2 Z g(c) = 1 non-singular (Mordell-Weil 1922) C(k) g(c) 2 degp 4 finite non-singular (Faltings 1983) 9

10 Effectivity The result of Faltings is not effective, in the sense that it does not give any method for finding the points in C(k). This is due to the non existence of an effective bound for the height of the points in C(k). 10

11 Height Function is a measure of the complexity of the coordinates of a point. h : E N (Q) R + On P N (Q) we consider the logarithmic Weil height. On an abelian variety we consider the canonical Néron-Tate height. This is the square of a norm. Theorem (Northcott Theorem) A set of bounded (Néron-Tate) height and bounded degree is finite. 11

12 Effective Methods The method of Chabauty-Coleman provides a bound on the number of rational points on curves with Jacobian of Q-rank strictly smaller than the genus. Example Flynn gives explicit examples: he finds the rational points for a selection of curves of genus 2 with Jacobian of Q-rank 1. 12

13 The Manin-Dem janenko method applies to curves that admit many Q-independent morphisms towards an abelian variety. Example Kulesz, Girard, Matera, Schost and others find all rational points on some families of curves: these curves have genus 2 (resp. 3) and elliptic Jacobian of Q-rank 1 (resp. 2) with some special properties. For instance with factors given by a Weierstrass equation y 2 = x 3 + a 2 x, with a square-free integer and such that the Mordell-Weil group has rank 1. No explicit height s bound The bounds for the height must be worked out with ad hoc methods case by case and for the technique to be successful the equations of the curve must be of a special shape and small genus. 13

14 Joint work with S. Checcoli and F. Veneziano Let E be an elliptic curve given in the form y 2 = x 3 + Ax + B. with A, B algebraic integers. Let ĥ be the Néron-Tate height on E N. Let h(c) be the normalised height of C. 14

15 Theorem Let E be an elliptic curve of Q-rank 1. Let C E N be a curve of genus at least 2. Then P C(Q) has height bounded as ĥ(p) 4 3 N 2 N!degC ( C 1 h(c)(degc) + 4C 1 c 1 (degc) 2 + 2c 1 ). Moreover if N = 2 ĥ(p) C 1 h(c)degc + 4C 1 c 1 (degc) 2 + 4c 1 C 1 = 145 c 1 = c 1 (E) = 2h W (A) + 2h W (B) + 4, 15

16 Compared with previous bounds In a previous work, for C not contained in any translate of a subgroup of E N with N 3 then ĥ(p) B 1 (N)4(N 1)C 1 h(c)(degc) N 1 + B 2 (N)(N 1)C 2 (degc) N + + N 2 C 3 where B 1 (N) B 2 (N) N N2 (N!) N. While here ĥ(p) 4(N 1)C 1 h(c)degc + (N 1)C 2 (degc) 2 + N 2 C 3, 16

17 Explicit Bounds Assume that E is without CM, defined over a number field k and that E(k) has rank 1. Let (x 1,y 1 ) (x 2,y 2 ) be the affine coordinates of E 2 Corollary Let C be the curve given in E 2 cut by the additional equation p(x 1 ) = y 2, with p(x) k[x] a non-constant polynomial of degree n. Then for P C(k) we have ĥ(p) 2595(h W (p) + logn + 4c 1 (E))(2n + 3) 2 + 4c 1 (E) where h W (p) = h W (1 : p 0 :... : p n ) is the height of the coefficients of p(x) and c 1 = 2log(3 + A + B )

18 C is transverse. Moreover degc = 6n + 9 and h(c) 6(2n + 3)(h W (p) + logn + 2c 1 (E)) For almost all E, the C n have genus 4n + 2 and the D n have increasing genus. 18

19 Explicit Examples Definition Let {C n } n be the family of curves C n E 2 defined for n 1 via the additional equation x n 1 = y 2. Let {D n } n be the family of curves D n E 2 defined for n 1 via the additional equation Φ n (x 1 ) = y 2, where Φ n (x) is the n-th cyclotomic polynomial. 19

20 Choice of the ambient variety: E 1 : y 2 = x 3 + x 1, E 2 : y 2 = x x , E 3 : y 2 = x x , E 4 : y 2 = x x , E 5 : y 2 = x x These are five elliptic curves without CM and of rank 1 over Q. 20

21 y1 2 = x1 3 + x 1 1 E 1 C n = y2 2 = x2 3 + x 2 1 E 1 n x 1 = y 2 y1 2 = x1 3 + x 1 1 E 1 D n = y2 2 = x2 3 + x 2 1 E 1 Φ n (x 1 ) = y 2 The C n have genus 4n + 2 and the D n have increasing genus. 21

22 degc n = 6n + 9, h(c n ) 6(2n + 3)log(3 + A + B ). degd n = 6ϕ(n) + 9, h(d n ) 6(2ϕ(n) + 3) (2 ω 2(n) log2 + 2log(3 + A + B ) ), where ϕ(n) is the Euler function, ω 2 (n) is the number of distinct odd prime factors of n. 22

23 For every n 1 and every point P C n (Q) we have ĥ(p) 1301(4c 6 (E))(2n + 3) 2 + 4c 6 (E). For every n 2 and every point P D n (Q) we have ( ) ĥ(p) ω 2(n) log2 + 4c6 (E) (2ϕ(n) + 3) 2 + 4c 6 (E) were c 6 (E) = log(3 + A + B ). 23

24 Explit Examples Example For the curves C n E 1 E 1 we have C n (Q) = {(1,±1) (1,1)}. For the curves C n E i E i with i = 2,3,4,5, we have C n (Q) =. 24

25 Example For i = 2,3,4,5, the curves D n E i E i have For the curves D n E 1 E 1 we have D n (Q) =. D 1 (Q) = (2,±3) (1,1) D 2 (Q) = (2,±3) (2,3) D 3 k (Q) = (1,±1) (2,3) D 47 k (Q) = (1,±1) (13,47) D p k (Q) = if p 3,47 or p = 2 and k > 1 D 6 (Q) = (1,±1) (1,1) and (2,±3) (2,3) D n (Q) = (1,±1) (1,1) if n 6 has at least two distinct prime factors. 25

26 Theorem (Manin-Mumford Conjecture) Raynaud 1983 Let A be an abelian variety and Tor A its torsion. Let C A be a curve of genus 2. Then, C Tor A is finite. Theorem ( Mordell-Lang Conjecture) Faltings 1983, Vojta 1996, Hindry 1988 Let C A of genus at least 2. Let Γ be a subgroup of A of finite rank. Then C Γ is finite. Mordell-Lang Conjecture implies Manin-Mumford Conjecture Mordell-Lang Conjecture + Mordell-Weil Theorem imply C(k) is non dense. 26

27 Theorem (Torsion Anomalous Conjecture) Let C be weak-transverse in A. Then the set C dimb N 2 B is finite. Here B ranges over all algebraic subgroups of dimension N 2. Show that C dimb N 2 B has bounded height. C dimb N 2 B has bounded degree. Remark This theorem implies the Mordell-Lang conjecture for curves. 27

28 Effective/Explicit Methods For Curves The proof of the theorem is effective only if C is transverse in E N (i.e. not contained in any translate of an algebraic subgroup of E N ) (Use the geometry of numbers and properties of the height) In principle effective For N 3 and C weak-transverse in E N, E with CM and C dimb=1 B. (Use a Lehmer type bound) explicit For N 2 and C in E N of genus at least 2, E without CM and C dimb=1 B. IMPLEMENTABLE FOR N = 2 explicit (preprint 2016) For N 2 and C transverse in E N and C dimb N 1 B. This explicit result implies new cases of the Effective Mordell Conjecture. Let C be transverse in E N and rank of E(k) N 1. Then C(k) has height bounded by an explicit constant. 28

29 Normalized Height The height of a subvariety V of A is its normalized height. The normalized height of an algebraic subgroup is 0. Theorem (Arithmetic Bézout Theorem) Let V and W be irreducible subvarieties of P m. Let Z 1,...,Z n be the irreducible components of V W. Then n i=1 h(z i ) degv h(w) + degw h(v) + c(m)degv degw. 29

30 Essential Minimum Definition: h : V R + 0 } {{ } µ(v) } {{ } Essential Minimum h 1 non dense h 1 dense µ(v) = sup{ε : h 1 [0,ε) non dense in V} Theorem (Zhang Inequality) Let X be an irreducible subvariety of P m, then 1 h(x) (1 + dimx) degx µ(x) h(x) degx. 30