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Selmer Varieties for Curves with CM Jacobians

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Selmer Varieties for Curves with CM Jacobians Selmer varieties for curves with CM Jacobians John Coates and Minhyong Kim February 3, 2010 Abstract We study the Selmer variety associated to a canonical quotient of the Qp-pro-unipotent fun- damental group of a smooth projective curve of genus at least two defined over Q whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multi- variable Iwasawa theory is used to prove dimension bounds, which, in turn, lead to a new proof of Diophantine finiteness over Q for such curves. Let X/Q be a smooth proper curve of genus g 2 and b X(Q) a rational point. We assume that X has good reduction outside a finite set S of primes≥ and∈ choose an odd prime p / S. In earlier papers ([14], [15], [16], [17], [18]), a p-adic Selmer variety ∈ 1 Hf (G, U) was defined and studied, with the hope of applying its structure theory to the Diophantine geometry of X. Here, G = Gal(Q¯ /Q), Qp,un ¯ U = π1 (X,b) is the Qp-pro-unipotent étale fundamental group of X¯ = X Spec(Q¯ ), ×Spec(Q) and the subscript f refers to a collection of local ‘Selmer conditions,’ carving out a moduli space of torsors for U on the étale topology of Spec(Z[1/S, 1/p]) that satisfy the condition of being crystalline at p. The Selmer variety is actually a pro-variety consisting of a projective system 1 1 1 1 H (G, Un ) H (G, Un ) H (G, U ) H (G, U ) ···→ f +1 → f +1 →···→ f 2 → f 1 of varieties over Qp associated to the descending central series filtration U = U 1 U 2 U n U n+1 = [U,U n] ⊃ ⊃ ⊃ ⊃··· of U and the corresponding system of quotients n+1 Un = U U \ that starts out with U = V = TpJ Qp, 1 ⊗ the Qp-Tate module of the Jacobian J of X. As a natural extension of the map - - 1 X(Q) J(Q) Hf (G, V ) 1 visible in classical Kummer theory, the Selmer variety is endowed with a system of unipotent Albanese maps emanating from the points of X: . 1 . Hf (G, U4) - ? 1 Hf (G, U3) - ? 1 Hf (G, U2) - ? - 1 X(Q) Hf (G, U1) These maps fit into commutative diagrams - X(Q) X(Qp) ? ? - 1 loc-p 1 -D DR 0 Hf (G, Un) Hf (Gp,Un) Un /F 1 DR 0 0 involving the local Selmer varieties Hf (Gp,Un) and their De Rham realizations Un /F . Here, F refers to the zeroth level of the Hodge filtration U DR F i F i+1 F 0 ⊃···⊃ ⊃ ⊃···⊃ DR on the De Rham fundamental group U of X Spec(Qp). Recall that the De Rham funda- ×Spec(Q) mental group is defined using the Tannakian category of unipotent vector bundles with flat connections −i DR on X Spec(Qp) ([15], section 1), and that the Hodge filtration F on U is the subvariety ×Spec(Q) defined by the ideal F i+1 DR in the coordinate ring DR of U DR (loc. cit. and [4]). F 0U DR turns out to be a subgroup. TheA filtration on DR is definedA over C using the (d, d¯) decomposition on iterated integrals of differential forms, butA descends to any field of characteristic zero [32]. Here and in the following, we will suppress from the notation the object that the Hodge filtration filters when the context provides sufficient clarity. Diophantine motivations oblige us to study the localization map D locp with some care. In fact, one could formulate the dimension hypothesis ◦ 1 DR 0 dimHf (G, Un) < dimUn /F (DHn) for each n, and show that DHn for any fixed n implies the finiteness of X(Q) [15]. Throughout this paper, dimension will refer to that of algebraic varieties over Qp [14], although the dimensions of n n+1 various associated graded objects, e.g., U /U , are just the naive ones of Qp-vector spaces. Given any X, it seems reasonable to believe that DHn should be true for n sufficiently large [15]. 2 An eventual goal is to use the Selmer variety to arrive at a structural understanding of the Diophan- tine set X(Q), or at least some means of effective computation. The hope for effective computation is associated with classical method of Chabauty and Coleman [3], which the study of the unipotent Albanese map generalizes. The related issue of structural understanding, on the other hand, should concern an implication of the form control of L-values control of Selmer varieties ⇒ following a pattern familiar from the theory of elliptic curves ([2], [25]). It should be admitted right away that our current intuition for the nature of such an implication is in a very tentative state. Nevertheless, the cases studied previously of hyperbolic curves of genus zero and one seem to suggest that our expectations are not entirely without ground. The purpose of the present paper is to augment our list of examples where something can be worked out with the case where J is isogenous over Q¯ to a product J Ai ∼ Yi of abelian varieties Ai that have complex multiplication by CM fields Ki of degree 2dimAi. For this discussion we choose the prime p to further satisfy the condition that p is split in the compositum K of the fields Ki, and hence, in each field Ki. Let QT be the maximal extension of Q unramified outside T = S p, and GT = Gal(QT /Q). Now let ∪{ ∞} W = U/[U 2,U 2] be the quotient of U by the third level of its derived series and let W = U/[U 2,U 2]. Of course W itself has a descending central series W = W 1 W 2 W n+1 = [W, W n] ⊃ ⊃···⊃ ⊃··· n+1 and associated quotients Wn = W/W . Theorem 0.1 There is a constant B (depending on X and T ) such that n 2 i i+1 2g−1 dim H (GT , W /W ) Bn . ≤ Xi=1 We will derive this inequality as a rather elementary consequence of multi-variable Iwasawa theory. The key point is to control the distribution of zeros of a reduced algebraic p-adic L-function of sorts, namely an annihilator of a natural ideal class group. In accordance with the ‘motivic nature’ of the construction, W also has a De Rham realization W DR = U DR/[(U DR)2, (U DR)2] over Qp, endowed with a Hodge filtration. The upper bound of theorem 0.1 combines with an easy linear independence argument for sufficiently many elements in (W DR)n/(W DR)n+1, yielding a lower bound for the De Rham realization DR 0 Wn /F of its local Selmer variety. We obtain thereby the easy but important corollary: Corollary 0.2 For n sufficiently large, we have the bound 1 DR 0 dimHf (G, Wn) < dimWn /F . 3 Of course, this implies: Corollary 0.3 (Faltings’ theorem, special case.) X(Q) is finite. For explicit examples where the hypothesis is satisfied, we have of course the Fermat curves xm + ym = zm for m 4 ([26], VI, Satz 1.2, Satz 1.5), but also the twisted Fermat curves ≥ axm + bym = czm for a,b,c, Q 0 , m 4. One might hope (optimistically) that the methods of this paper will eventually∈ lead\{ to} some≥ effective understanding of these twists. Some relatively recent examples of hyperelliptic curves with CM Jacobians can be found in [30]. One from the list there is y2 = 243x6 + 2223x5 1566x4 19012x3 + 903x2 + 19041x 5882 − − − − whose Jacobian has CM by Q( 13+3√13). q− The results here conclude the crude application of Selmer varieties to finiteness over Q in situations where the controlling Galois group of the base is essentially abelian. It remains then to work out the appropriate interaction between non-commutative geometric fundamental groups and the non- commutative Iwasawa theory of number fields. Of course, as far as a refined study of defining ideals for the image of D locp is concerned, work of any serious nature has not yet commenced. In this regard, we note that there◦ is little need in this paper for specific information about the annihilator that occurs in the proof of theorem 0.1. However, it is our belief that structure theorems of the ‘Iwasawa main conjecture’ type will have an important role to play in eventual refinements of the theory. 1 Preliminaries on complex multiplication Let F/Q be a finite extension with the property that the isogeny decomposition J Ai ∼ Yi as well as the complex multiplication on each Ai are defined over F . We assume further that F ∞ r ⊃ Q(J[4p]), so that F := F (J[p ]) has Galois group Γ Z over F . Denote by GF,T , the Galois ∞ ≃ p group Gal(QT F/F ). As a representation of GF,T , we have V := TpJ Qp iVi ⊗ ≃⊕ where Vi := TpAi Qp. ⊗ Let m be a modulus of F that is divisible by the conductor of all the representations Vi. Each factor representation ∗ ρi : GF,T (Ki Qp) Aut(Vi) → ⊗ ⊂ corresponds to an algebraic map Ki fi : Sm Res (Gm), → Q 4 Ki where Sm is the Serre group of F with modulus m ([29], II) and ResQ is the restriction of scalars from Ki to Q. That is, there is a universal representation (op. cit. II.2.3) ǫp : GF,T Sm(Qp) → such that - - Ki ∗ ρi = fi ǫp : GF,T Sm(Qp) Res (Gm)(Qp) = (Ki Qp) . ◦ Q ⊗ Since we have chosen p to split in each Ki, we have Ki ResQ (Gm) Qp [Gm]Qp .
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