Some. Manin-Mumford. Problems

Transcription

1 Some Manin-Mumford Problems S. S. Grant 1

2 Key to Stark s proof of his conjectures over imaginary quadratic fields was the construction of elliptic units. A basic approach to elliptic units is as follows. Let E be any elliptic curve given in Weierstrass form y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 over a number field K, with discriminant. Let p be a prime of K not dividing. Suppose that p divides the rational prime p, and m is prime to p. Then the m-torsion E[m] of E injects mod p, so if u ±v E[m] O, x(u) and x(v) are integral at p, and ξ = (x(u) x(v)) 6 / 0 mod p. 2

3 Hence ξ is an S-unit where S consists only of primes dividing m. But ξ is invariant under change of model of E, so in fact S consists only of primes dividing m and the primes of multiplicative reduction for E. Therefore if E has everywhere potentially good reduction (for example if E has CM), then S consists only of primes dividing m. Furthermore, using inclusion-exclusion and a formula of Stark for the norm of ξ, if m is divisible by at least 2 distinct primes, then ξ is a unit. 3

4 We ve just heard about the important advances of Goren and Lauter on generalizing Siegel units to imaginary quartic fields, but there are just a few known instances where torsion points on abelian varieties have been used to build (S)- units. Boxall and Bavencoffe built units from 3- and 4-torsion on the Jacobian of y 2 = x 5 1. I did the same for any curve of genus 2 with everywhere potentially good reduction. 4

5 S. G. S. Arledge generalized this for 3- torsion on the Jacobian of a hyperelliptic curve of genus 3. The same argument holds for certain 4- torsion on the Jacobian of any hyperelliptic curve with everywhere potentially good reduction. Fukuda and Komatsu built units from the 6-torsion on the Jacobian of y 2 y = x 5 and I did the same for 5-torsion. 5

6 What is the difficulty? If A is an abelian variety over a number field K, and f K(A) and u J tors (K), just evaluating f(u) is problematic, since in order to do so, u must not lie in the codimension one subvariety of A which is the support of the divisor of f. The best one generally has to help is: Theorem 1 (Raynaud 83) A proper subvariety V of an abelian variety A over a number field K that does not contain a translate of a proper abelian subvariety, has only finite intersection with A tors (K). The case that V is a curve is the famed Manin- Mumford conjecture. Even in this case it takes considerable effort to find which torsion points in A tors (K) lie on V. 6

7 When V is the image of a curve C in its Jacobian J under an albanese embedding with base point P, the torsion points in V J tors (K) have been found by: Ribet and Baker (and Tamagawa) when C is a modular curve X 0 (N) and P is a cusp. Coleman, Tamagawa, and Tzermias, when C is a Fermat curve or a non-hyperelliptic image of a Fermat curve and P is one of the obvious rational points. S. G. S. Shaulis when C is a hyperelliptic image of a Fermat curve. 7

8 Much less is known explicitly when the dimension of V is greater than 1. There is a general result of Hindry, but there is only one case for which there are any more specific results the case of Jacobians of quotients of Fermat curves. Let l be an odd prime and ζ a primitive l th - root of unity. For 1 a l 2, we let F be the nonsingular projective curve over Q defined by the affine model x l = y(1 y) a. Then F is a quotient of the l th -Fermat curve, and the genus of F is g = (l 1)/2. Let J be the Jacobian of F. 8

9 Let denote the lone point at infinity on this model. Let φ : F J be the albanese embedding with as base point, and let Θ (a theta divisor) be the (g 1)-fold sum of φ(c). Note that J has complex multiplication by the ring of integers Z[ζ] of K = Q(ζ). For any ideal a Z[ζ], let J[a] denote the torsion points of J annihilated by a. If λ = 1 ζ, it is easy to see that J[λ] is generated by φ((0, 0)). 9

10 Using his theory of p-adic solitons, Anderson showed for any first degree prime p in K, that J[pλ] Θ J[λ]. ( ) Using formal groups, I was able to get (*) with p replaced by any power of a first or second degree prime p, but only in the case that F is hyperelliptic. The most recent results are due to S. G. S. Simon. Assume now that J has non-degenerate CM-type (χ(a + 1) χ(a) + 1 for all odd characters χ mod l.) Using CM theory, a generalization of a theorem of Lefschetz, some neat geometric tricks, and the geometry of numbers, he was able to prove the following for primes p of K of any degree: 10

11 Let N(p) denote the absolute norm of p. Theorem 1 (Simon) Suppose that N(p) > l 2l. For every Q J tors (K) Θ, if n p (Q) is the precise power of p that divides the order of Q, then n p (Q) 1. Using the proof of the Theorem, one can use a similar argument to recover Anderson s and my results if N(p) > l 2l and the CM-type of J is non-degenerate. (Hindry s work would also give n p (Q) 1, but only for p with N(p) bigger than a bound which appears to be doubly-exponential in l.) 11

12 There is a recent proof of the Manin-Mumford conjecture, due to Ribet (and Baker). Let G be a commutative algebraic group defined over a perfect field k. Let Γ k be the Galois group of k over k. Ribet defined a point P G(k) to be almost rational over k if whenever σ, τ Γ k are such that σ(p ) + τ(p ) = 2P, then σ(p ) = τ(p ) = P. (So in particular, rational points are almost rational.) Using a result of Serre s on Galois representations on torsion points of abelian varieties, Ribet showed that the set of almost rational torsion points over K on an abelian variety defined over a number field K is finite. 12

13 Let C be a nonsingular projective curve of genus at least 2 over K, and φ Q : C J an Albanese embedding of C into its Jacobian J with a K- rational point Q as base point. Then for any P C(K) which is not a hyperelliptic Weierstrass point, φ Q (P ) is almost rational over k. Hence Ribet s result gives a new proof of the Manin-Mumford conjecture. Where one has better knowledge of the Galois representations, say for abelian varieties with complex multiplication, one can get sharper results. 13

14 Boxall and I showed that for given integers d and g, there exists an integer V d,g such that if A is an abelian variety of dimension at most g with (potential) complex multiplication, and A is defined over a number field K of degree at most d, then the almost rational torsion points of A over K are contained in A[V d,g ]. This implies that for a given g > 1 and d, if C is a curve of genus g defined over a number field K of degree d and Q C(K), and if the Jacobian J of C has (potential) complex multiplication, then there is an integer W d,g such that φ Q (C) J tors (K) J[W d,g ]. Coleman has a sharp bound for the order of this set that depends on the reduction type of C and the ramification in K. 14