Selmer groups and Euler systems

Transcription

1 Semer groups and Euer systems S. M.-C. 21 February Introduction Semer groups are a construction in Gaois cohomoogy that are cosey reated to many objects of arithmetic importance, such as cass groups of number fieds, and Morde-Wei groups and Tate-Shafarevich groups of eiptic curves. We make these connections precise in a moment, but you may aready be abe to imagine the importance measuring the size of Semer groups, because the size of cass groups, Morde-Wei groups, and Tate-Shafarevich groups are of fundamenta importance in number theory. Of course, this is quite a difficut probem. An important too for it is the notion of an Euer system. Essentiay, an Euer system is a system of Gaois cohomoogy casses that restrict the size of a Semer group. The genera definition of an Euer system is sighty compicated (and possiby not even we-estabished, because there are so few exampes to take inspiration from but I may not be up to date on the theory). However, there is a canonica exampe, the Euer system of cycotomic units, which gives a good first idea of the nature and appications of Euer systems in genera. So our pan is to first discuss Semer groups and where they arise in number theory, and then to expain the idea of Euer systems as a way to bound the size of Semer groups, and finay to ook sighty more in-depth at the exampe of cycotomic units. 1.1 References These notes mainy foow the notes of Barry Mazur and Tom Weston from the 2001 Arizona Winter Schoo. I aso consuted the notes of Loeffer and Zerbes from the 2018 AWS as we as Rubin s book Euer Systems. The Mazur and Weston notes are a good first introduction to Euer systems; the Loeffer-Zerbes notes give more recent exampes of Euer systems, and genera toos for constructing them; and Rubin s book is ike a whoe book on Euer systems. 2 Semer Groups Let K be a number fied with absoute Gaois group G K, and T a free Z p -modue (or possiby a finite abeian group, or a Q p -vector space the situation is essentiay simiar) with continuous G K -action. We can then form Gaois cohomoogy groups H i (K, T). If K v is the competion of K at a pace v, then T is aso a G Kv -modue via the natura map G Kv G K, so we get a oca Gaois cohomoogy H 1 (K v, T) and a goba-to-oca restriction map H 1 (K, T) H 1 (K v, T). 1

2 A Semer group is a subgroup of a goba Gaois cohomoogy group consisting of eements satisfying certain oca conditions. A oca condition at v is simpy a subgroup F v H 1 (K v, T). For exampe: the strict oca condition at v is F v = 0; the reaxed oca condition at v is F v = H 1 (K v, T); and the unramified oca condition at v is F v = image(h 1 (G Fv, T I v ) H 1 (K v, T)), the map of cohomoogy groups being induced by the map G Kv G Fv which makes T I v a G Fv -modue (here I v G Kv is the inertia subgroup and F v is the residue fied of K v ). This is a Gaois-cohomoogica interpretation of inertia is kied. Definition 1. A Semer structure F for T is a choice of oca condition F v at each pace v of K, which is required to be the unramified oca condition at amonst a paces. The associated Semer group is the subgroup of H 1 (K, T) consisting of eements satisfying a oca conditions; that is, the Semer group is the kerne 0 S F (K, T) H 1 (K, T) v H 1 (K v, T)/F v. A of this has a natura duaity. Let T := Hom Zp (T, µ p ) be the Cartier dua of T, which is again a G K -modue (with the adjoint action, which I assume to mean τ G K sends f : T µ p to τ F τ 1 or something, using the actions of G K on T and µ p ). Then the pairing T T µ p, together with the cup product on cohomoogy, induces pairings H 1 (K, T) H 1 (K, T ) H 2 (K, µ p ) H 1 (K v, T) H 1 (K v, T ) H 2 (K v, µ p ) Q p /Z p. If F is a Semer structure for T, we can define a dua Semer structure F on T by taking F v to be the annihiator of F v under the oca duaity pairing. We obtain a corresponding Semer group S F (K, T ). As mentioned above, the main reason to care about Semer groups is that they are cosey reated to important arithmetic objects. Let s see some exampes of this. Exampe 2. Let T = Z p (1) := im µ p n be the cycotomic character. Reca the oca Kummer map K v H 1 (K v, T) ( ) α σ σ(α1/n ) α 1/n This induces an isomorphism K v H 1 (K v, Z p (1)) after taking the p-adic competion K v of K v. Using this identification, we can define a Semer structure by taking F v = Ô K v. (This is the unramified oca condition for v p). As aways we take the dua Semer structure on T. 2

3 In this case we find S F (K, T ) = A, where A is the p-part of the cass group of K and A = Hom(A, Q p /Z p ). Exampe 3. Let T = E[p] be the p-torsion subgroup of an eiptic curve. The the Semer group with respect to an appropriate Semer structure fits into an exact sequence 0 E(K)/pE(K) S F (K, T) X(K, E)[p] 0. Thus the Semer group gives us information on the Morde-Wei group E(K) and the Tate- Shafarevich group X(K, E) simutaneousy. In particuar, a bound on the order of S F (K, T) woud give a bound on the order of X(K, E)[p] as we as a bound on the rank of E(K). Exampe 4. Much of the importance of Semer groups can be summed up in the Boch-Kato conjecture. This says that if V is a G K -representation over Q p coming from geometry, then dim Qp S BK (K, V) dim Qp H 0 (K, V) = ord s=0 L(V (1), s) where S BK is the Semer group produced by a certain Boch-Kato Semer structure. This is a generaization of other important conjectures in number theory. For exampe, say V = V p (E) is the rationa Tate modue of an eiptic curve. Then H 0 (K, V) = 0; the Kummer map E(K) Q p H 1 (K, V) ands in S BK (K, V), so rank E(K) dim Qp S BK (K, V), and in fact it s an equaity if the p-part of X is finite (which is conjectured to be true); L(E/K, s) = L(V, s) = L(V (1), s 1) so in particuar ord s=0 L(V (1), s) = ord s=1 L(E/K, s). Combining these, we see that in this case the Boch-Kato conjecture predicts rank E(K) = ord s=1 L(E/K, s), which is the Birch and Swinnerton-Dyer conjecture. 3 Motivation for Euer Systems Now, how can we get a hande on the size of Semer groups? Very vaguey, the idea is to produce eements of Gaois cohomoogy groups that impose restrictions on our Semer group. Reca the duaity pairings H 1 (K, T) H 1 (K, T ) H 2 (K, µ p ), H 1 (K v, T) H 1 (K v, T ) H 2 (K v, µ p ) Q p /Z p. In this way we can interpret eements of Gaois cohomoogy groups as functionas on the Gaois cohomoogy groups (or Semer groups) of the dua representation. Moreover, it s a fact that the oca pairing is a perfect pairing, whie the goba pairing, at east after summing over oca factors H 1 (K, T) H 1 (K, T ) H 2 (K, µ p ) Q p /Z p 3

4 is identicay zero. This impies that if h H 1 (K, T) and σ S F (K, T ) have oca restrictions h v and σ v, then we can regard h v as a functiona on S F (K v, T ) (we coud even use the image of h v in H 1 (K, T)/F v, because F v is the annihiator of F v ), and we have the reation h v (σ v ) = 0. v The idea is that by producing enough reations, we can bound the size of the Semer group. To be more precise, we introduce a way to modify Semer structures. If F is a Semer structure for T and a K is a fractiona idea, we define a modified Semer structure F a by { Fv ord F a,v = v (a) = 0 H 1 (K v, T) ord v (a) = 0. That is, we define F a to be F with the oca conditions at a reaxed. Now there is the foowing exact sequence of Semer groups. 0 S F (K, T) (1) S Fa (K, T) (2) v : ord v (a) =0 H 1 (K v, T)/F v (3) S F (K, T ) (4) S F a (K, T ) 0. The map (3) is regarding eements of H 1 (K v, T) as functionas on H 1 (K v, T ), thus on Fv and then S F (K, T ), by the oca duaity pairing (and we can quotient by F v because eements in F v annihiate Fv ). We put this exact sequence to work in the foowing way. In many cases we can find an a so that S F a (K, T ) vanishes (note that F a is got by reaxing conditions, so Fa is got by imposing extra conditions, which cuts down S F a (K, T )). Then S F (K, T ) = H 1 (K v, T)/F v /S Fa (K, T). v : ord v (a) =0 If we can find eements of S Fa (K, T) that generate v : ord v (a) =0 H1 (K v, T)/F v up to index N, then we get a bound #S F (K, T ) N on the order of a Semer group. In practice, choosing a so that S F a (K, T ) vanishes is easy, whie finding eements of S Fa (K, T) that generate v : ord v (a) =0 H1 (K v, T)/F v up to finite index is hard. One way to produce such eements is using an Euer system. 4 The Euer System of Cycotomic Units We try to give some idea of the favor this method through the exampe of cycotomic units. Fix K = Q(ζ p ) + for some odd p, and T = Z p (1) with the Semer structure defined above. Instead of working ony with the Gaois cohomoogy groups of K, we work with Gaois cohomoogy groups of a tower of extensions of K. For m > 1, define c m := ζ m 1 Q(ζ m ). For any prime, we have the fundamenta norm reation { N Q(ζm )/Q(ζ m ) c cm m m = (1 Frob 1 )c m m. Here Frob is the Frobenius at in Ga(Q(ζ m )/Q), and the action is mutipicative, so (1 Frob 1 )c m = 4 c m Frob 1 c m.

5 Note that we can use the Kummer map Q(ζ m ) H 1 (Q(ζ m ), Z p (1)) to transport the c m to Gaois cohomoogy casses over Q(ζ m ). This is essentiay what an Euer system is: a system of Gaois cohomoogy casses ascending a tower of fied extensions, which satisfy a certain norm reation invoving Euer factors (which seems to be where the name Euer system comes from). Vaguey, the strategy for bounding the Semer group in this case is to produce certain operators in the group agebra of a Gaois group which convert our c m Q(ζ m ) to some eements d m K, which we then transport to Gaois cohomoogy using the Kummer map. It turns out that the index of the submodue these eements generate can be computed, and the resut is the foowing theorem. (See Weston s notes, and the references therein, for more detais). Theorem 5. #S F (K, T ) [O K : C K] where C K is the group of cycotomic units in K. (The group C K of cycotomic units in K = Q(ζ p ) + is defined to be the intersection of O K with the mutipicative group generated by {±ζ p, 1 ζ a p 1 a > p}). Reca that S F (K, T ) is dua to the p-part of the cass group of K. The above theorem therefore gives a bound on the p-part of the cass group of K. Of course, Euer systems are not the first nor the simpest way to reate the cass group of K to cycotomic units. But this argument is of the sort that can be generaized to much more difficut situations. For exampe, the same argument can be used to prove the divisibiity on the eve of χ- components for any character χ : Ga(K/Q) Z p, which in particuar impies Ribet s converse to Herbrand s theorem. As another exampe, these ideas can be extended to prove the main conjecture of Iwasawa theory: essentiay the Euer system aows us to work with a cass groups in the tower simutaneousy. 5