The Manin-Mumford Conjecture: A Brief Survey of Results and Related Conjectures, Study notes of Pre-Calculus

Download The Manin-Mumford Conjecture: A Brief Survey of Results and Related Conjectures and more Study notes Pre-Calculus in PDF only on Docsity! THE MANIN-MUMFORD CONJECTURE: A BRIEF SURVEY PAVLOS TZERMIAS Introduction. This is a survey paper on the Manin-Mumford conjecture for number fields with some emphasis on effectivity. It is based on the author’s lecture at the Arizona Winter School on Arithmetical Algebraic Geometry (March 1999). We discuss some of the history of this conjecture (and of related conjectures) and some recent explicit results. 1. finiteness results The Manin-Mumford conjecture for number fields is a deep and important finite- ness question (raised independently by Manin and Mumford) regarding the inter- section of a curve with the torsion subgroup of its Jacobian: Conjecture 1.1. Let K be a number field. Let C be a curve of genus g ≥ 2 defined over K. We will denote by J the Jacobian of C. Fix an embedding C ↪→ J defined over K. Then the set C(K) ∩ J(K)tors is finite. Conjecture 1.1 was proved by Raynaud in [48]. Various other proofs and general- izations were subsequently given by Raynaud ([49]), Serre ([53]), Coleman ([19]), and Hindry ([30]) (see also the end of this section where recent developments are mentioned). According to Lang ([35]), Manin was led to ask the above question in connection with another famous conjecture, namely the Mordell Conjecture: Conjecture 1.2. Let K be a number field and let C be a curve of genus g ≥ 2 defined over K. Then C(K) is finite. The Mordell conjecture was proved by Faltings in his landmark paper [25] (see also [26]). Different proofs were shortly afterwards given by Vojta ([57]) and Bombieri ([6]). The function field analogue of the Mordell conjecture in characteristic 0 was first proved by Manin in [39], using the theorem of the kernel. Coleman discovered a gap in Manin’s proof of the latter theorem and managed to prove a weaker version of the theorem (sufficient for the proof of Mordell’s conjecture for function fields in characteristic 0; see [21]). Manin’s initial version of the theorem of the kernel was later on proved by Chai ([14]), using work of Deligne and Coleman. Long before either of the Conjectures 1.1 or 1.2 was settled, it was Serge Lang ([34]) who realized that the two statements are special cases of the following more general conjecture, which is usually called the Mordell-Lang conjecture in charac- teristic 0: Conjecture 1.3. Let X be a closed geometrically integral subvariety of a semi- abelian variety A defined over a field K of characteristic 0. Let Γ be a finitely generated subgroup of A(K) and Γ′ a subgroup of the divisible hull of Γ (i.e. for 1991 Mathematics Subject Classification. 11G10, 11G30, 11G35, 14G25, 14H25, 14H40. 1 2 PAVLOS TZERMIAS each x ∈ Γ′ there exists a non-zero integer n such that nx ∈ Γ). If X is not a translate of a semi-abelian subvariety of A, then X(K)∩ Γ′ is not Zariski dense in X. It is not hard to check the following proposition: Proposition 1.4. Conjecture 1.3 implies Conjectures 1.1 and 1.2. Proof. For the Manin-Mumford conjecture, let X = C, A = J , Γ = {0} and Γ′ = J(K)tors. X is a curve of genus at least 2, so it is not a translate of a semi- abelian subvariety of A. By the Mordell-Lang conjecture, X(K)∩ Γ′ is not Zariski dense in X, hence it is finite. For the Mordell conjecture, let X = C, A = J , Γ = Γ′ = J(K) (note that, by the Mordell-Weil theorem, Γ is finitely generated). As before, X(K) ∩ Γ′ = C(K) is finite. The Mordell-Lang conjecture in characteristic 0 was proved in its entirety by Mc- Quillan ([43]), following work of Faltings ([25], [26]), Raynaud ([48], [49]), Hindry ([30]), Vojta ([58]) and Buium ([8]). One should also mention that special cases of Conjecture 1.3 were settled earlier by Tate and Lang (see [34]), Liardet ([37], [38]), Laurent ([36]) and Bogomolov ([5]). A generalization of the Manin-Mumford conjecture was proposed by Bogomolov: Conjecture 1.5. Let X be a curve of genus g ≥ 2 defined over a number field K. Fix an embedding of X into its Jacobian J . Let hNT denote the Néron-Tate height on J(K). Then for sufficiently small ² > 0, the set {P ∈ X(K) : hNT (P ) ≤ ²} is finite. Conjecture 1.5 has been settled by Ullmo ([56]) using work of Szpiro, Ullmo and Zhang ([55]). There is also the generalized Bogomolov conjecture: Conjecture 1.6. Let A be a semi-abelian variety defined over K and let X be a closed geometrically integral subvariety of A which is not a translate of a semi- abelian subvariety of A by a torsion point. Let h be a canonical height on A(K) (for example, if A is an abelian variety, h can be taken to be the Néron-Tate height on A(K)). Then for sufficiently small ² > 0, the set {P ∈ X(K) : h(P ) ≤ ²} is not Zariski dense in X. When A is an abelian variety, Conjecture 1.6 was settled by Zhang ([63]). A quantitative version (and also another proof) of the same result was given by David and Philippon ([24]). Zhang also proved Conjecture 1.6 when A is a torus ([62]). A proof of Conjecture 1.6 for almost split semi-abelian varieties (i.e. semi-abelian varieties which are isogenous to the product of an abelian variety and a torus) was recently announced by Chambert-Loir ([15]). Poonen has recently proposed an even more general conjecture that includes both the Mordell-Lang and the generalized Bogomolov conjecture as special cases (see [46]). Let notation be as in Conjecture 1.6. Let Γ be a finitely generated subgroup of A(K) and let Γ′ be the divisible hull of Γ. Fix a canonical height on A(K). For ² > 0, define Γ′² = {γ + P : γ ∈ Γ′, P ∈ A(K), h(P ) ≤ ²}. THE MANIN-MUMFORD CONJECTURE: A BRIEF SURVEY 5 Theorem 3.3. Suppose that, in addition to the hypotheses of Conjecture 3.1, C has ordinary reduction at v and J has potential complex multiplication. Then #T ≤ pg. The following two examples illustrate that Coleman’s bound is sharp and, more importantly, that the hypotheses on C and J are essential for the bound to hold. Example 2 (Boxall and Grant, [7]). Let C be the genus-two curve with affine model y2 = x5 + x. Embed C into its Jacobian J by sending the point at infinity to 0. The Hasse-Witt matrix of C at 11 is easily seen to be invertible, so C has ordinary reduction at 11. Also J has complex multiplication induced by (x, y) 7→ (ζ2x, ζy), where ζ is a primitive 8-th root of unity in Q. Now the six hyperelliptic branch points of C lie in the torsion packet T (with respect to the given embedding). Moreover, using standard arguments in the arithmetic of genus two curves (see [13]), it is not hard to show that the 16 points (x, y) for which x4 + 4x2 + 1 = 0 or x4 − 4x2 + 1 = 0 have order 6 in J . Therefore, #T ≥ 22. On the other hand, by Coleman’s bound, #T ≤ 22, so we are done. Now an easy computation shows that C is superspecial at 7. By what has been said above, #T ≥ 15. Therefore, the hypothesis of ordinariness of C is essential for Coleman’s bound to hold. Example 3 (Coleman, [16]). The modular curve X1(13) has genus 2, ordinary reduction at 5 and 22 points in its cuspidal torsion packet (see [16] for details). Therefore, X1(13) does not have complex multiplication. This also shows that the CM hypothesis on J is essential for Coleman’s bound to hold. Coleman’s paper ([16]) contains a number of interesting examples, especially for curves of genus 2 and 3. Examples 2 and 3 seem to hold the record for the maximum number of points in a torsion packet on a genus-two curve. Poonen has in fact constructed ([47]) countably many pairwise non-isomorphic genus-two curves over Q, each with at least 22 points in the hyperelliptic torsion packet. A remarkable (and almost unconditional) bound for the cardinality of a torsion packet was given by Buium in [11]: Theorem 3.4. Suppose that, in addition to the hypotheses of Conjecture 3.1, we have p ≥ 2g + 1. Then #T ≤ g! p4g 3g (p(2g − 2) + 6g). Buium uses p-jets to prove an unramified version of the above statement. The result then follows from Theorem 3.2. Buium has in fact given explicit bounds in a number of different contexts ([9], [10], [11] and [12]). It is worth noting at this point that a new bound for the Manin-Mumford con- jecture follows from the work of Hrushovski ([32]): 6 PAVLOS TZERMIAS Theorem 3.5. Fix a projective embedding of J . There exist constants α and β such that #T ≤ α (deg(C))β , where deg(C) is the degree of C with respect to the given projective embedding. The constants α and β do not depend on C or on K, but on the genus of C. They also depend on a prime of good reduction for C (I thank Alexandru Buium for pointing this out to me). 4. recent explicit examples In this section we will focus on Question 2 of the previous section. We will review some recent explicit examples of torsion packets for specific curves. In the next section, we will briefly discuss some of the ideas involved in the proofs hoping that the relevant techniques might prove to be useful in different contexts as well. The examples below are listed roughly in chronological order. Example 4 (Coleman, Kaskel and Ribet, [22]). Consider the modular curve X0(37) and its Jacobian J0(37). Let C0 and C∞ be the two cusps on X0(37) and consider the Albanese embedding X0(37) −→ J0(37) by sending C∞ to 0. By a theorem of Drinfeld and Manin, it follows that C0 lies in the corresponding torsion packet T . In fact, one has: Theorem 4.1. T = {C0, C∞}. Example 5 (Coleman, Tamagawa and Tzermias, [23]). Consider the Fermat curve FN : XN + Y N + ZN = 0, where N is an integer such that N ≥ 4. The set of cusps CN on FN is the set of points (X, Y, Z) (over Q) satisfying XY Z = 0. Embed FN into JN by using a cusp as a base-point and let TN be the corresponding torsion packet. Rohrlich ([51]) has shown that CN ⊆ TN . In fact, one has: Theorem 4.2. TN = CN . It should be noted that Coleman had settled some special cases of Theorem 4.2 using rigid analytic geometry ([18]). These special cases were used in the proof of Theorem 4.2. Also, Theorem 4.2 has an analogue (which is however conditional upon a weak version of Vandiver’s conjecture) for the non-hyperelliptic Fermat quotients Fp,s : yp = xs(1 − x), where p is a prime such that p ≥ 11 and s is an integer such that 1 ≤ s ≤ p − 2 and s 6= 1, (p − 1)/2, p − 2. The latter analogue is obtained by means of the work of Greenberg ([28]) and Kurihara ([33]) and the question whether it remains valid unconditionally (i.e. independently of Vandiver’s conjecture) is open (see [23] for details). Shaulis ([54]) has recently computed the cuspidal torsion packets on the hyperelliptic Fermat quotient curves (s ∈ {1, (p− 1)/2, p− 2}). Example 6 (Boxall and Grant, [7]). In this recent article, a general method is developed that can sometimes explicitly compute the hyperelliptic torsion packet on a genus 2 curve (i.e. the torsion packet corresponding to the embedding of the curve in its Jacobian by taking a hyperelliptic branch point as a base-point). THE MANIN-MUMFORD CONJECTURE: A BRIEF SURVEY 7 Example 7 (Voloch, [61]). Consider the curve C : y2 = x6 + 1. Voloch shows that: Theorem 4.3. The hyperelliptic torsion packet on C consists of the six hyperellip- tic branch points together with the two points at infinity. Example 8 (Baker, [3]). Let p be a prime, with p ≥ 23. As in Example 4, embed the modular curve X0(p) into J0(p) by sending C∞ to 0. When X0(p) is hyperelliptic (i.e. when p =23, 29, 31, 37, 41, 47, 59, 71), the hyperelliptic branch points belong to the cuspidal torsion packet Tp provided that p 6= 37. The last condition has to be imposed since, by a result of Mazur and Swinnerton-Dyer ([42]), the Atkin-Lehner involution on X0(37) does not coincide with the hyperelliptic involution. We define a set Cp as follows: If C is hyperelliptic and p 6= 37, then Cp is the set consisting of C0, C∞ and the hyperelliptic branch points; in all other cases, Cp is the set consisting only of C0 and C∞. Baker proved the Coleman-Kaskel-Ribet conjecture: Theorem 4.4. Tp = Cp. Baker also obtains similar results for the curves X+0 (p). He also studies other torsion packets (besides the cuspidal one). His results also apply to more general modular curves, namely X0(N) and X1(N), for N composite. For details one should consult Baker’s preprint ([3]). A different proof of Theorem 4.4 has been recently announced by Akio Tamagawa. 5. remarks on relevant techniques We will now briefly record some observations regarding the techniques employed in settling the above examples. This is by no means a comprehensive list of such techniques. However, it is our opinion that there exist similarities (broadly con- strued) and therefore we feel it might be useful to record some of them here. 1. Coleman’s conjecture (where known to hold) gives valuable information about the primes dividing the exponent of the torsion packet T . Stated differently, rigid analytic techniques and their consequences seem to be a starting point for tackling such problems. In Examples 4 and 8 it is useful to use the Chinese Remainder Theorem to decompose a potential torsion point P as a sum of its l-primary com- ponents Pl. Coleman, Kaskel and Ribet prove ([22]) that for l 6= 2, 3, the image of Pl in J0(p) is in the cuspidal group, unless either l = p or 5 ≤ l ≤ 2g or X0(p) does not have ordinary reduction at l and l is ramified in the Hecke algebra. In other words, one gets very precise information for “most” primes l. Ribet has shown that, under a mild hypothesis, the situation for the prime l = 2 is completely understood (see [50]). Regarding Example 5, we have the following result of Coleman ([20]) : the exponent of the cuspidal torsion packet on the curve yp = xs(1 − x) is a power of p, unless the curve is hyperelliptic, in which case 2 is the only other prime that can possibly divide the exponent of T . 2. Studying the Galois representation on J(K)tors also gives important informa- tion on the torsion packet T . This idea goes back to Lang ([34]), who showed that the following statement implies the Manin-Mumford conjecture: