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The Isogeny Invariance of the BSD Conjecture Over Number Fields

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The Isogeny Invariance of the BSD Conjecture Over Number Fields ISOGENY INVARIANCE OF THE BSD CONJECTURE OVER NUMBER FIELDS JEREMY BOOHER 1. Introduction Let A be an abelian variety over a number field K. The Birch and Swinnerton-Dyer conjecture states that L(A=K; s) has a zero of order rank(A) at s = 1, and that L(A=K; s) #X R Ω (1.1) lim = A A A s!1 rank(A) (s − 1) #A(K)tors#Ab(K)tors Implicit in this are the assertions that L(A=K; s) has an analytic continuation and that the Tate- Shafarevich group XA is finite. Suppose f : A ! B is an isogeny of abelian varieties over K. As a sanity check for the BSD conjecture, the right side of (1.1) should be the same for A and for B. We have seen examples where the individual pieces of this expression change under an isogeny (see [L5]). It requires a delicate analysis to show that the entire expression is invariant under isogeny. In Section 2 we analyze the effect of an isogeny on the terms in the conjecture. In Section 3 we review important results in Galois cohomology and illustrate them, and in Section 4 we use them to deduce an isogeny has no overall effect on the constant: Theorem 1.1 (Tate). If the Birch and Swinnerton-Dyer conjecture is true for A, it is true for all K-isogenous abelian varieties. Proof. Combine Lemma 2.2, the calculations (2.2), (2.3), and (2.4), and Proposition 4.1. Remark 1.2. The arguments we give come from [Mil06, xI.7] (itself modeled on Tate's Bourbaki talk on BSD), with minor adaptations to fit in with the rest of the learning seminar. Remark 1.3. Everything we do can be easily adapted to case of a global function field K when char(K) does not divide the degree of the isogeny. One has to use flat cohomology and infinitesimal group schemes to push through a generalization of the technique to prove Theorem 1.1 without restriction on the degree of the isogeny. Notational Conventions. Let S denote a finite set of places of K that contains the archimedean places, and define KS to be the maximal extension of K unramified outside of S. Let GS = Gal(KS=K) and OK;S be the ring of S-integers of K. When we talk about cohomology groups, we always use continuous cohomology. If g : X ! Y is a homomorphism of abelian groups with finite kernel and cokernel, we define # ker(g) h(g) = ; #coker(g) this is a measure of how g effects the size of the groups: if X and Y are finite, h(g) = #X=#Y . If f : A ! B is a K-homomorphism of abelian varieties over K, denote the induced maps on K points and Kv points by f(K) and f(Kv) respectively. We also need to consider several types of duality. Date: October 28 and November 4, 2015. 1 2 JEREMY BOOHER • For a locally compact Hausdorff topological abelian group X, we define the dual X∗ = 1 Homcont(M; S ). (If M is torsion and discrete then this is the same as Hom(M; Q=Z). • For a finite flat (locally free) commutative group scheme G over a base T , the Cartier dual D 0 G represents the functor on T -schemes given by T HomT 0 (GT 0 ; Gm;T 0 ). D × • If M is a finite discrete Gk-module for some field k, we define M = HomAb(M; ks ) (with its evident structure of finite discrete Gk-module). 2. The Effect of Isogenies Let f : A ! B be a degree-n isogeny of abelian varieties over K. Let S be a finite set of places of K containing: the archimedean places, the places where A (and B) have bad reduction, and the places dividing n. (Later we will add a few more places into S.) The significance of such a choice of S is that f extends to a map of abelian schemes fe : A!B over OK;S (namely, the N´eronmodels over OK;S are abelian schemes). The kernel of this map is finite ´etaleby using properness, fibral flatness, and degree considerations, so passing to OKS -points gives an exact sequence (exercise!) and hence by properness this is an exact sequence (2.1) 0 ! ker(f)(KS) ! A(KS) ! B(KS) ! 0; for details on this see [Mil06, Lemma I.6.1]. 1 An important feature at places v away from S is that triviality in H (Kv;A) is equivalent to unramifiedness: Proposition 2.1. If F is a non-archimedean local field and A is an abelian variety over F with good reduction then the unramified classes in H1(F; A) are trivial. Proof. We have to show that if X is an A-torsor admitting an F un-point then X(F ) is non-empty. un Let A be the N´eronmodel of A over OF ; this is an abelian scheme by hypothesis. Since X(F ) is non-empty, there is a finite unramified extension F 0=F such that X(F 0) is non-empty, which is 0 to say XF 0 ' AF 0 as AF 0 -torsors. Thus, X is described by Galois descent for AF 0 through F =F compatibly with the A-torsor structure, and by the N´eronianproperty the Galois descent datum extends to a descent datum for the Galois finite ´etalecovering Spec(OF 0 ) ! Spec(OF ). That provides an A -torsor X with generic fiber X. The special fiber X0 is a torsor for the abelian variety A0 over a finite field κ, so it is a trivial torsor by Lang's theorem. By smoothness of X over the henselian local ring OF , the natural map X (OF ) ! X (κ) is surjective. Hence, X (OF ) is non-empty, so passing to the generic fiber gives that X(F ) is non-empty. (A more subtle version of the argument is implicit in [L3, x5], where it is necessary to distinguish the henselization and completion of a localization of a ring of S-integers at a maximal ideal; no such distinction is relevant here.) Before analyzing the effect of an isogeny on the terms in the BSD conjecture, we first check some basic compatibilities: Lemma 2.2. We have: (1) The L-functions L(A=K; s) and L(B=K; s) are equal. (2) The ranks of A and B are the same. (3) XA is finite if and only if XB is finite. Proof. The key is that since ker f is killed by [n]A, there exists a K-isogeny g : B = A= ker(f) ! A=A[n] = A such that f ◦ g = [n]B and g ◦ f = [n]A. In particular, V`(g) is an inverse to V`(f) up to in- vertible n-multiplication on the rationalized `-adic Tate module. Since the L-function was defined solely in terms of the rationalized Tate module V`(A) (see [L1, x1.3]), we get the equality of L- functions. Likewise, the induced map A(K)Q ! B(K)Q via f is an isomorphism, or more concretely ISOGENY INVARIANCE OF THE BSD CONJECTURE OVER NUMBER FIELDS 3 A(K)=A(K)tors ! B(K)=B(K)tors is injective since ker f is finite, giving that rank(A) ≤ rank(B), and g provides the reverse inequality. The third assertion follows from (2.1): taking the long exact sequence of GS-cohomology gives an exact sequence 1 1 1 ::: ! H (GS; ker(f)) ! H (GS;A) ! H (GS;B) ! ::: 1 Finiteness results for Galois cohomology (see Fact 3.11 below) gives that H (GS; ker(f)) is finite. Thus, using Proposition 2.1, 1 1 1 1 ker(XA ! XB) ⊂ ker(H (GS;A) ! H (GS;B)) = Im(H (GS; ker(f)) ! H (GS;A)) is finite. Hence, if XB is finite, we deduce that XA is finite. For the converse, apply the same argument with g in place of f. The main work is analyze the effect of an isogeny on each of the pieces of (1.1), and then to assemble them all together to show that the product of the discrepancies is equal to 1; this product calculation will use essentially all of Tate's global duality theorems. 2.1. The Tate-Shafarevich Group. Recall we defined the Tate-Shafarevich group as 1 Y 1 XA = ker(H (K; A) ! H (Kv;A)) v in [L3, x5]. As A is K-isogenous to B, Ab, and Bb, Lemma 2.2 shows that if XA is finite then so are , , and . XB XAb XBb The Cassels{Tate pairing will be defined and analyzed in an upcoming lecture. We need the following result, which is also proven in [Mil06, Theorem I.6.13a]. Fact 2.3. Assuming is finite, the Cassels{Tate pairing × ! Q=Z is non-degenerate. XA XA XAb Furthermore, it is functorial in the sense that XA × X / Q=Z O Ab X(f) X(fb) × / Q=Z XB XBb is commutative. We now assume finiteness of XA, hence the Tate{Shafarevich groups of Ab, B, and Bb are all finite. The perfectness of the Cassels{Tate pairing and its functoriality give that #coker(X(f)) = # ker(X(fb)). Now let us consider the effect of an isogeny on #XA. Looking at the tautological exact sequence X(f) 0 ! ker X(f) ! XA ! XB ! coker(X(f)) ! 0; we see that #X # ker(X(f)) # ker(X(f)) (2.2) A = = : #XB #coker(X(f)) # ker(X(fb)) Put this in the fridge; we'll take it out later.
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