Euler Systems
1 Heegner points (and BSD) 1.1 Introduction Let be an elliptic curve over , ∼ r(E,F ). E Q [F : Q] < ∞ ⇒ E(F ) = E(F )tor ⊗ Z Q −s −1 Q −s 1−2s −1 Given E, we have an L-function, L(E, s) = (1 − app ) (1 − app + p ) with Re(s) > 3/2 p|N p-N p + 1 − #E(Fp) p - N 0 E additive at p and ap = , N = cond(E). 1 split multi at p −1 non split multi at p Q Q p Q p Note. L(E, 1) ” = ” () · = where Np = #E( p)ns. p|N p-N p−ap+1 p Np F Conjecture (Birch-Swinnerton-Dyer). Let E/Q be an elliptic curve 1. ords=1L(E, s) = rkZE(Q) = r
2. L(E,s) det(hPi,Pj i) Q , where are generators of . lims→1 r = #X(E, Q) 2 v|N cv {Pi} E(Q) (s−1) #E(Q)tor
Suppose that E is modular (now we know this is always true): there exists f ∈ S2(Γ0(N)) a newform f(q) = P∞ n such that P∞ −s. Or equivalently . This implies that n=1 anq L(E, s) = L(f, s) = n=1 ann ap(E) = ap(f) L(E, s) has analytic continuation to C. It has a functorial equation ∧(E, s) = (2π)−sΓ(s)N s/2L(E, s). We have ∧(E, s) = − ∧ (E, 2 − s) where ∈ {±1} and is determined by wN (f) = · f where w is the Atkin-Lehner function. We shall call − the sign(E, Q). modular representation. non-constant morphism de ned over . f φ : X0(N)/Q / E/Q Q O Hecke−op AJ ( J0(N) = JacX0(N)
1.2 Class eld theory Let be an imaginary quadratic eld. Let be an order, , an integer. K ⊆ C On ⊆ K On = Z + nOK n ≥ 1 ∼ By class eld theory: There is a map rec : Pic(On) = Gal(Kn/K) where Kn/K is an abelian extension unrami ed away from n. Recall that Pic(On) = I(n)/P (n) where I(n) = {fractional ideals coprime to n}, P (n) = . The map is de ned by −1 h(α): α ∈ OK , α ≡ a mod nOK , a ∈ Zi [p] 7→ Frobp Lemma. Let . Then ∼ ∼ ∗ ∗. In particular, if is Gn = Gal(Kn/K1) Gn = Pic(On)/Pic(OK ) = (OK /nOK ) / (Z/nZ) ` ( ` + 1 ` inert in K an odd prime unrami ed in K, then G` is cyclic and [K` : K1] = ` − 1 ` split in K If is square free then ∼ Q . n Gn = `|n G`
1 1.3 Complex Multiplication
0 0 X0(N) classi es (up to isomorphism) cyclic N-isogonies, A → A where ker(A → A ) is cyclic of order N. Let imaginary eld satisfying Heegner Hypothesis: splits in . Can pick an ideal such that K/Q `|N ⇒ ` K N ⊆ OK ∼ (cyclic). Consider , −1 where OK /N = Z/NZ On = Z + nOK χn = [C/On → C/Nn ] ∈ X0(N)(C) Nn = N ∩ On ( ∼ ). ⇒ On/Nn = Z/NZ Theorem (Main Theorem of Complex Multiplication). Let ∼ so σ ∈ Aut(C/K) σ|Kn ∈ Gal(Kn/K) = Pic(On) . σ|Kn [aσ] σ −1 −1 −1 χn = C/aσ → C/Nn aσ Remark. Suppose . Then σ . Hence . σ|Kn = 1 χn = χn χn ∈ X0(N)(Kn) Hecke action (On ): For each , is a correspondence on χ0(N) ` - N T` X0(N) T` : (DivX0(N))(F ) → de ned by 0 P 0 . DivX0(N)(F ) [φ : A 7→ A ] 7→ C⊂A[`],cyclic subgp of order `[A/C → A /φ(C)] Also have the Trace map: Tr` : DivX0(N)(Kn`) → DivX0(N)(Kn). Proposition. Consider , where . Let , then {χn}n (n, ND) = 1 D = disc(K) ` - ND T χ if ` n is inert in K ` n - 1. As elements in , −1 DivX0(N)(Kn) Tr`(χn`) = (T` − Frobλ − Frobλ )χn if ` = λλ - n split in K T`χn − χn/` `|n
2. If is inert in , , is a prime in above . Then Frobλn ` - N K λ = `OK λn Kn λ redλn` (χn`) = redλn (χn ) ∈ . X0(N)(Fλn )
1.4 Heegner points on E
Let φ : X0(N) → E. yn := φ(χn) is the Heegner point of conductor n (in E(Kn)). (in ) is the basis Heegner point yK = TrK1/K (y1) E(K) Proposition. ` - ND : a y if ` is inert in K, ` n ` n - −1 Frobλn Tr`(yn`) = (a` − Frobλ − Frobλ )yn redλn`(yn`) = redλn(yn ) a`yn − yn/` E(Fλn ) Proposition. If is a complex conjugation, then there exists such that τ σ on τ σ ∈ Gal(Kn/K) yn = yn E(K)/E(K)tors .
2 Local Cohomology 2.1 Introduction to cohomology
Let G be a group and M a G-module. Both G and M have topology. 1-cocycles: Are {f : G → M|f continuous and ∀g, h ∈ G, f(gh) = f(g) + gf(h)} 1-coboundary: Are {f : G → M|f continuous and f(g) = g · m − m for some m ∈ M} De nition 2.1. H1(G, M) ={1-cocylce}/{1-coboundary}, H0(G, M) = M G Consider the short exact sequence of G-modules, 0 → M 0 → M → M 00 → 0 and M 00 → M is a continuous section of sets. Then we get a long exact sequence
0 → M 0G → M G → M 00G → H1(G, M 0) → H1(G, M) → H1(G, M 00) → H2 ... We have some useful maps between cohomology groups:
2 • Let H ≤ G, we get a map Res : H1(G, M) → H1(H,M).
1 H 1 • Let H C G be a normal closed subgroup of G, then we have inf : H (G/H, M ) → H (G, M) We have the following relationship between inf and Res called the Hochschild-Serre spectral sequence
0 → H1(G/H, M H ) →inf H1(G, M) Res→ H1(H,M)G/H → H2(G/H, M H ) → ...
2.2 Galois Cohomology Fox prime , and and let un be the maximal unrami ed extension of and let un ` p |K : Q`| < ∞ K K IK = Gal(K/K ) be the Inertia group Denote by and un un ∼ (∼ ) GK = Gal(K/K) GK = Gal(K /K) = GK /IK = Zb Let be a nite dimensional -vector space with discrete action. We say that is unrami ed if acts T Fp GK T IK trivially on T . 1 1 ∗ ∗ There is a perfect pairing of -vector spaces where ∞ . Fp H (GK ,T ) ⊗Fp H (GK ,T ) → Fp T = Hom(T,Mp ) Fact. If and is unrami ed then 1 un and 1 un ∗ are exact orthogonal complements with respect ` 6= p T H (GK ,T ) H (GK ,T ) to h , i. De nition 2.2. A local Selmer structure for is a choice of -subspace of 1 denoted 1 . F T Fp H (GK ,T ) Hf,F (GK ,T ) We call the quotient 1 1 1 the singular quotient. HS,F (GK ,T ) := H (GK ,T )/Hf,F (GK ,T )
1 1 1 0 → Hf,F (GK ,T ) → H (GK ,T ) → HS,F (GK ,T ) → 0 (†)
We say is an unrami ed structure if 1 1 un IK (via inf). In this case identi es with inf-res. F Hf,F (GK ,T ) = H (GK ,T ) (†) Using the Tate pairing we de ne the Dual Selmer structure F ∗ on T ∗ to be the exact orthogonal complement of 1 . Hf,F (GK ,T ) 1 ∗ In particular ∗ we get an induced perfect pairing. HS,F (GK ,T ) ⊗Fp Hf,F (GK ,T ) → Fp 2.3 Local cohomology for elliptic curves Let be an elliptic curve over . We let . We have the Weil pairing E K T = E[p] = E(K)[P ] E[P ] ⊗ FpE[P ] → µp ∗ ∼ (⇒ E[P ] = E[P ]). We have a short exact sequence of GK -module
p 0 → E[P ] → E(K) → E(K) → 0
If we take the associated long exact sequence
GK GK GK 1 1 0 → E[P ] → E(K) → E(K) → H (GK ,E[P ]) → H (GK ,E(K)) → ... (††)
From (††) we get: δ 1 1 0 → E(K)/pE(K) → H (GK ,E[p]) → H (GK ,E(K))[p] → 0 where δ is de ned as follows: for Q ∈ E(K) x L ∈ E(K) such that pL = Q. Then δ(Q) is the 1-cocycle which sends σ ∈ GK to σ(L) − L ∈ E[P ].
1 De nition 2.3. The geometric local Selmer structure F on E[P ] is the image of δ in H (GK ,E[p]). Fact. The dual geometric local Selmer structure is the geometric local Selmer structure (it make sense since E[P ]∗ =∼ E[P ]) If E has good reduction over K and ` 6= p. Then E[p] is an unrami ed GK -module and the geometric structures agrees with the unrami ed structure.
3 3 Global Section
In this talk K will be a number eld, v a place of K, Kv the completion of K at v, Gv = Gal(Kv/Kv) and Iv the inertia subgroup of . As in the last talk we let to be a nite dimensional -vector space with a discrete Gv T Fp GK -action. GK × T → T is continuous when T is given the discrete topology. We choose an embedding K,→ Kv which gives us an embedding Gv ,→ GK . Since T is nite dimensional and GK acts discretely, this implies that T is unrami ed almost everywhere. 1 1 For each place v we have a restriction map Resv : H (K,T ) → H (Kv,T ).
3.1 Selmer Groups De nition 3.1. A global Selmer structure on is a choice of a local Selmer structure 1 for each F T Hf,F (Kv,T ) place such that 1 1 unr Iv almost everywhere. v Hf,F (Kv,T ) = H (Kv ,T ) The Selmer group of is de ned to be 1 1 . SelF (K,T ) F ker H (K,T ) → ⊕vHs,F (Kv,T ) If we take E to be an elliptic curve over K and T = E[p]. De nition 3.2. The geometric global Selmer structure F on E[p] is de ned to be the global Selmer structure obtained by setting 1 to be the local geometric structure at each . Hf,F (Kv,T ) v (p) In this setting SelF (K,E[p]) = Sel (E).
3.2 Global Duality
De nition 3.3. Let F be a global Selmer structure on T . The Cartier dual Selmer structure F ∗ on T ∗ = is de ned to be the global Selmer structure obtained by setting 1 ∗ to be the local Cartier HomFp (T, µp) Hf,F ∗ (Kv,T ) dual Selmer structure. We now x an F and F ∗ and start omitting it from notation. Given an ideal , we de ne 1 1 for all . a C OK Sela(K,T ) = {c ∈ H (K,T ): cv ∈ Hf,F (Kv,T ) v - a} a Sel (K,T ) = {c ∈ SelF (K,T ): cv = 0, ∀v|a}. This gives us the following exact sequences 1 0 → Sel(K,T ) → Sela(K,T ) → ⊕v|aHS,F (Kv,T ) → 0
a ∗ ∗ 1 ∗ 0 → Sel (K,T ) → Sel(K,T ) → ⊕v|aHf,F (Kv,T ) → 0 We have 1 ∼ 1 ∗ ∨. We can put the two sequences above together as follows ⊕v|aHs,F (Kv,T ) = ⊕v|aHf,F ∗ (Kv,T )
M 1 ∗ ∨ a ∗ ∨ 0 → Sel(K,T ) → Sela(K,T ) → Hs,F (K,T ) → Sel(K,T ) → Sel (K,T ) → 0 v|a Proposition 3.4. The above sequence is exact. The exactness of this sequence yields
1 ∗ ∨ a ∗ ∨ 0 → ⊕v|aHs,F (Kv,T ) im(Sela(K,T )) → SelF ∗ (K,T ) → Sel (K,T ) → 0
3.3 Bounding Sela + − From now on p 6= 2. If τ ∈ GK is an involution, we have a decomposition T = T ⊕ T , where T = {t ∈ T : τ(t) = t}. We say that τ is non-scalor if T + 6= 0, T − 6= 0. From now on, T will be assumed to be irreducible. L/K = nite Galois extension such that the action of GK on T factors through Gal(L/K). If 1 nite dimensional -subspace a 1 and . We can S ⊆ H (K,T ) Fp S = {s ∈ S : sv ∈ Hf,F (Kv,T ) (sV = 0∀v|a)} nd a nite Galois extension M/L such that S ⊆ inf(H1(M/K, T )). Assume that τ ∈ Gal(L/K) is a non-scalar involution and that it extends to an involution in Gal(M/K). Let {γ1, . . . , γr} be a set of group generators for Gal(M/L).
4 Proposition 3.5. Let ω1, . . . , ωr be places of M such that FrobM/K (wi) = τγi. Let vi be the restricted places to a 1 K and set a = v1 ··· vr. Then we have S ⊆ inf(H (L/K, T )). Let be an elliptic curve over without CM, . , 1 . E Q T = E[p] ρ : GK GL2(Fp) H (Q(E[p])/Q,E[p]) = 0 Let L0/K be an extension such that GK factors through Gal(L0/K), and we assume that K is a quadratic extension of K0. We also assume that the action Gal(L0/K) is a restriction of a Gal(L0/K0)-action on T . We have M
L
L0
K 2
K0 1 S will be a nite dimensional subspace of H (M/K, T ) and τ will be a non-scalor involution in Gal(M/K0) which 1 projects to the non-trivial element of Gal(K/K0). By the action of τ on H (M/K, T ), we have the decomposition H1(M/K, T ) = H1(M/K, T )+ ⊕ H1(M/K, T )− 1 −1 −1 We further assume that S ⊆ H (M/K, T ) for some ∈ {±}. Let σ ∈ Gal(M/L0) be such that τστ = σ . We again let {γ1, . . . , γr} be a set of generators of Gal(M/L).
Proposition 3.6. Let w1, . . . , wr be places of M such that FrobM/K (wi) = τσγi and let vi be the restrictions of wi a 1 to K. Set a = v1 ··· vr, then S ⊆ inf(H (L/K, T ) ).
4 Calculations in Galois Cohomology
Let E be an elliptic curve over Q. K/Q an imaginary quadratic extension where all primes dividing Cond(E) split. Fix yk ∈ E(K) a Heegner point Theorem 4.1. Let p ≥ 3 be such that 1. E has good reduction at p 2. ∼ Gal(Q(E[p])/Q) = GL2(Fp)
3. yk ∈/ pE(K) The is of dimension and generated by . Sel(K,E[p]) Fp 1 κ(yk) Condition 2.) implies ∼ Gal(K(E[p])/K) = GL2(Fp) Fact. Under complex conjugation yk 7→ yk (up to torsion) where is the global root number. (⇒ yk ∈ (E(K)/pE(K)) ) Recall: Sela(K,E[p]) restricted. Sela(K,E[p]) relaxed. this gives the exact sequence
M 1 ∨ a ∨ Sela (K,E[p]) → Hs (Kv,E[p]) → Sel(K,E[p]) → Sel (K,E[p]) → 0 v|a Complex conjugation commutes with the maps so splits into a + part and the − part. Let L0 = K(E[p]). For z such that pz = yk let L = L0(z). Lemma 4.2. There exists an ideal a of K such that Sela(K,E[p])± ⊆ H1(L/K, E[p])±. Moreover if we x σ ∈ −1 −1 Gal(L/L0) such that τστ = σ . We can choose such that is divisible by primes lying above primes of with . a a Q FrobL/Q ∼ τσ
5 4.1 Preliminaries
Lemma 4.3. Let ` be a prime of Q such that 1. E has good reduction at ` 2. ` 6= p 3. FrobK(E[p])/Q` ∼ τ Then 1 ± is a 1-dimensional -vector-space. HS(Kλ,E[p]) Fp Proof. K(E[p]) is the xed eld of the kernel of the mod p representation. Then Neron-Ogg-Shar implies that is unrami ed at and so all primes above in are unrami ed in . , the residue E[p] ` ` K K(E[p]) FrobK/Q` ∼ τ 6= id class of ` in K/Q is 2, and the residue class degree of ` in K(E[p])/K is 1. Hence ` splits completely in K(E[p])/K. So K(E[p])λ = K`, i.e., E[p] ⊆ E(K`).
unr 1 ∼ 1 GK HS(K`,E[p]) = H (I`,E[p]) `
= HomGunr (I`,E[p]) K` = HomGunr (I`/pI`,E[p]) K` ∼ unr 1/p unr ∼ . 1 ∼ . Everything done so far commutes with I`/pI` = Gal(K` (` )/K` ) = µp HS(K`,E[p]) = Hom(µp,E[p]) ± ∓ complex conjugations. We also get Hom(µp,E[p]) → E[p] .
4.2 − eigenspace i Lemma 4.4. H (L0/K, E[p]) = 0 for all i Lemma 4.5. 1 is -dimensional -vector-space generated by . H (L/K, E[p]) 1 Fp κ(yK ) ± Lemma 4.6. Gal(L/L0) is isomorphic to E[p] as Gal(L0/K)-modules. And Gal(L/L0) is 1-dimensional.
Consider Gal(L/L0) ,→ E[p] de ned by σ : {z 7→ z + q} 7→ q. This map is actually the image of κ(yK ) ∈ 1 1 Gal(L0/K) H (K,E[p]) under the restriction to H (L0,E[p]) . Proposition 4.7. Let ` be a prime of with Frob ∼ τ which does not split completely in L/L . Then there Q L0/Q 0 exists ± such that s in 1 . c(`) ∈ Selλ(Kλ,E[p]) c(`)λ 6= 0 Hλ(Kλ,E[p]) Theorem 4.8. Sel(K,E[p])− = 0.
−1 a − 1 − Proof. 1 6= σ ∈ Gal(L/L0) such that τστ = σ . There exists a such that Sel (K,E[p]) ⊆ H (L/K, E[p]) . But the second thing is 1-dimensional and is generated by κ(yK ). If we pick a such that it is divisible only by primes lying over ` , . . . , ` ∈ with Frob L ∼ τσ. ⊕ H1(K ,E[p])− has dimension r, but we use the fact there 1 r Z L/Q i li|a i `i exists such that s with linear independent images. Hence the cokernel is . c(`1) c(`i)λ 6= 0 0 5 Finishing the proof 5.1 Notation and Recap Notation.
• E an elliptic curve of conductor N
• φ : X0(N) → E a modular parametrization • K an imaginary quadratic eld with discriminant D, in which every prime dividing N splits.
6 • p a prime at which E has good reduction. Assume ∼ . • Gal(Q(E[p])/Q) = GL2(Fp) • τ complex conjugation
• Kn the ring class eld of K of conductor n, which (among other properties) is an abelian extension Kn/K unrami ed away from n. Let be a prime such that is inert in and splits in . Setting , splits • ` ∈ Z ` K/Q K(E[p])/K λ = `OK λ completely in Kn and is totally rami ed in Kn`. In the case n and ` are coprime we get:
λn` Kn` tot ramified
λn Kn K`
K tot splits 1
λ K inert 2 ` Q Note that . • Kn,λn = Kλ the reduction map . • redλn E(Kn,λn ) → E(Fλ) We have the short exact sequence 1 1 1 , with 1 0 → Hf (K,E[p]) → H (K,E[p]) → Hs (K,E[p]) → 0 Hf (K,E[p]) = imκ. un 1 G un We also recall that if E has good reduction at `, then H (K ,E[p]) ∼ Hom(I ,E[p]) Kλ , with G = s λ = λ Kλ un . Gal(Kλ /Kλ)
Theorem 5.1. For any integer n not dividing ND there exists a point yn ∈ E(Kn) such that then ` - ND Tr`yn` = a`yn ∈ E(Kn) and is inert in , ` - ND K redλn` (yn`) = redλn (FrobKn/K λn · yn) There exists σ ∈ Gal(Kn/K) such that τyn = σyn in E(K)/E(K)tors.
• L0 = K(E[p]), For z such that pz = yk let L = L0(z). This 5 weeks have been building towards proving Theorem. Let p be an odd prime such that: • E has good reduction at p ∼ • Gal(Q(E[p])/Q) = GL2(Fp)
• yK ∈/ pE(K)
Then Sel(K,E[p]) has order p; it is generated by the image of yK under the Kummer map. This was done by using the cohomology tools that Chris and Pedro set up, to bound Sel(K,E[p])± using the a restricted and relaxed Sel (K,E[p]) and Sela(K,E[p]). Alex proved this last week under the assumption of the following two proposition, which we will prove this week. Proposition. Assume that y ∈/ pE(K). Let ` ∈ be a prime with Frob ` ∼ τ and ` not splitting completely k Q L0/Q in . There there exists − with s in 1 L/L0 c(`) ∈ Selλ(K,E[p]) c(`)λ 6= 0 H (Kλ,E[p]) Proposition. Assume that . Let be a prime with and not splitting completely in yk ∈/ pE(K) ` ∈ Q FrobL/Q` ∼ τ ` 0 . There there exists with s in 1 . L /L c(q`) ∈ Selλ(K,E[p]) c(q`)λ 6= 0 H (Kλ,E[p]) We will prove this using the Heegner points that Marc introduced in the rst week to construct a class in H1(K,E[p]) that satisfy those conditions.
7 5.2 The derivative operator Let be the set of square free integers, and if then . This implies that R gcd(n, pND) = 1 `|n FrobK(E[p])/Q` ∼ τ ` is inert in K/Q and splits in K(E[p])/K, and that E has good reduction at `. Lemma 5.2. For all primes , and ` ∈ R p|` + 1 p|a` = ` + 1 − #(F`) Proof. As on E[p], complex conjugation and Frob are conjugate, their characteristic polynomial must be the same mod p
Recall that, for ` ∈ R, G` = Gal(K`/K1) is cyclic (Marc's talk) of order ` + 1 (as ` is inert in K). Fix a generator σ` and for a prime ` de ne P` i, • D` = i=1 iσ` Pl i. • T` = i=0 σ` For we de ne Q (using the fact that ∼ Q ) • n ∈ R Dn = `|n D` Gn = `|n G` Let be a set of coset representative for in .T P ,T−1 P −1. • γi Gn Gal(Kn/K) = i γi = i γi
Gn Lemma 5.3. For any n ∈ R we have Dnγn ∈ (E(Kn)/pE(Kn)) . Proof. As Q , we just need to show that . This is calculations using the identity Gn = `|n G` (σ` − 1)Dnyn ∈ pE(Kn) (σ` − 1)D` = ` + 1 − T`, Theorem 5.1 and Lemma 5.2.
We de ne Pn = TDnyn ∈ E(Kn). If we consider Pn in E(Kn)/pE(Kn), then by the above it is Gal(Kn/K)-invariant and independent of the choice made for T. 1 Consider the following, we want to get an element in H (K,E[p]) from Pn ∈ E(Kn)/pE(Kn)
E(Kn)/pE(Kn) κ 1 H (Kn,E[p])
1 1 Gal(Kn/K) H (K,E[p]) res/ H (Kn,E[p])
1 Gal(Kn/K) As the Kummer map is equivariant, κ(Pn) ∈ H (Kn,E[p]) . Lemma 5.4. The restriction map is an isomorphism
i Gal(Kn/K) Proof. Using long exact sequence have that the kernel and cokernel of res is H (Kn/K, E[p] ) for i = 1, 2 respectively. Using the non-trivial fact that E has no Kn-rational p-torsion for n ∈ R, these two groups are trivial. Hence the restriction map is an isomorphism
1 So we let c(n) ∈ H (K,E[p]) be such that c(n) = κ(Pn). Recap: At this point we have constructed a class c(n) ∈ H1(K,E[p]) which comes from an Heegner point yn ∈ E(Kn). We now show it lies in one of the + or − space
k Lemma 5.5. Let n ∈ R have k prime factors. Then c(n) ∈ H1(K,E[p])(−1) .
8 k Proof. As τ commutes with κ and res, we show τPn = (−1) Pn in E(Kn)/pE(Kn). Y τPn = τT D`yn Definition `|n −1 Y −1 = T τD`yn by τT = T τ `|n −1 Y = T (`T` − σ`D`) τyn `|n −1 Y = T (`T` − σ`D`)σyn Lemma 5.1 `|n −1 Y = T (−σ`D`)σyn since T`yn = a`yn/` = 0 in E(Kn)/pE(Kn) `|n k Y −1 = (−1) σ σ`T Dnyn `|n
Then using the fact that Dnyn is Gn-invariant, and Pn is Gal(Kn/K)-invariant, this collapse down to what we want. Lemma 5.6. Fix , then n = `1 . . . `rR c(n) ∈ Selλ1...λr (K,E[p])
Proof. Let ν be a place of K distinct from λi such that E has good reduction at ν (The proof of ν has bad reduction is more involved and uses tools we have not developed). Instead of showing 1 , we c(n)ν ∈ Hf (Kν ,E[p]) Gunr show s in 1 . We have 1 Kν . Let be a place of above c(n)ν = 0 Hs (Kν ,E[p]) Hs (Kν ,E[p]) = Hom(Iν ,E[p]) w Kn ν and note that Kn,w/Kν is unrami ed. Hence its inertia group is also Iν , and using the exact sequence we get the following commuting diagram:
κ 1 E(Kν )/pE(Kν ) / H (Kν ,E[p]) / Hom(Iν ,E[p]) res 1 E(Kn,w)/pE(Kn,w) κ / H (Kn,w,E[p]) / Hom(Iν ,E[p])
unr unr G GK where we use Hom(Iν ,E[p]) Kν ⊆ Hom(Iν ,E[p]) n,w ⊆ Hom(Iν ,E[p]). Then it is jut a matter of diagram chasing. By de nition , so s . Hence s . resc(n)ν = κ(Pn) (resc(n)ν ) = 0 c(n)ν = 0
As Pn` ∈ E(Kn`) is Gal(Kn`/K)-invariant in E(Kn`)/pE(Kn`), and G` = Gal(K`/K1) ≤ Gal(Kn`/K), we have (σ` − 1)Pn` ∈ pE(Kn`). Setting ` + 1 a Q = TD γ − ` P (†) n,` p n n` p n
(which makes sense by Lemma 5.2) we see pQn,` = (σ` − 1)Pn` (using Lemma 5.3). Note that by the fact that E has no Kn`-rational p-torsion point, Qn,l is the unique point in Kn` with that property. Lemma 5.7. is trivial in if and only if redλn` (Qn,`) E(Fλ) Pn ∈ pE(Kλ) Proof. First we show for all . This is true by Theorem 5.1 redλn` (σyn`) = redλn (FrobKn/K λn · σyn) σ ∈ Gal(Kn/K) −1 for σ = 1, and we use Theorem 5.1 with the ideal σ λn.
−1 −1 −1 redσ λn` (yn`) = redσ λn (FrobKn/K (σ λn) · yn) −1 redλn` (σyn`) = redλn (σFrobKn/K (σ λn) · yn) but as −1 −1 we are done. By the de nition of and T, we have FrobKn/K (σ λn) = σ FrobKn/K λnσ Dn T redλn` (TDnλn`) = redλn (FrobKn/K λn · Dnyn)
9 Hence combining this with (†) we get
` + 1 a red (Q ) = red Frob λ − ` P λn` n,` λn p Kn/K n p n
Claim: annihilates . Note that + − as is an (` + 1)FrobKn/K λn − a` E(Fλ) E(Fλ) = E(Fλ) ⊕ E(Fλ) FrobKn/K λn involution. But + which by de nition has order . Then by the Weil conjectures − E(Fλ) = E(F`) ` + 1 − a` E(Fλ) has order l + 1 + a`. Now is the reduction of a complex conjugation, so , which by the proof of FrobKn/K λn FrobKn/K λnPn = τPn Lemma 2.4 k in , so for some and . τPn = (−1) Pn E(Kn)/pE(Kn) FrobKn/K λnPn = νPn +pQ ν ∈ {±1} Q ∈ E(Kn)
(` + 1)ν − a red (Q ) = ` red (P ) ∈ E( )ν /pE( )ν λn` n,` p λn n Fλ Fλ We now claim that the -primary part of ν is cyclic. First we note that , so we have at least p E(Fλ) p|(l + 1) ± a` -torsion point in ±. Suppose we had more than -torsion point in one of ±, then we would p p E(Fλ) p p E(Fλ) have 2 of them, but 2 + − which is a contradiction (as is a good prime, p p > |E(Fλ) [p]| + |E(Fλ) [p]| = |E(Fλ)[p]| ` 2). We can conclude that ν ν is cyclic of order . E(Fλ)[p] = p E(Fλ) /pE(Fλ) p Hence if and only if . But the kernel of is pro- , which redλn` (Qn,`) = 0 redλn (Pn) ∈ pE(Kn,λn ) = pE(Kλ) redλn ` is coprime to p, so Pn ∈ pE(Kλ) Lemma 5.8. Let with prime. Then s if and only if . n` ∈ R ` c(n`)λ = 0 Pn ∈ pE(Kλ)
Gun Proof. Since we have 1 Kλ so we can view s as a homomorphism . ` ∈ R Hs (Kλ,E[p]) = Hom(Iλ,E[p]) c(n`)λ Iλ → E[p] We shall show this factors as follows:
Iλ / E[p] O ( ∼ Iλ/Iλn` = G` To see this we consider the diagram
1 H (Kλ,E[p]) res 1 E(Kn`)/pE(Kn`) κ / H (Kn`,λn` ,E[p]) / Hom(Iλnl ,E[p])
Then lies in the image of the Kummer map, hence . c(n`) c(n`)(Iλn` ) = 0 Looking at the diagram in the introduction (since ), we see that is the inertia group of at ` - n G` Gal(Kn`/K) λ. Hence considering the diagram
Kλ
Iλn Kun n`,λn`
Iλ G Kn`,λn` `
un G` Kλ
Kλ = Kn,λn we have ∼ . Hence s if and only if . Iλ/Iλn` = G` c(n`)λ = 0 c(n`)(σ`) = 0 Fix such that , then we claim that the cocycle given by 1 Q ∈ E(K) pQ = Pn` GK → E[p] σ 7→ (σ − 1)Q − p (σ − represents (where 1 is the unique point in which is a th root of ). This 1)Pn` c(n`) p (σ − 1)Pn` E(Kn`) p (σ − 1)Pn`
10 expression is easy to see is in E[p]. To see it represents c(n`) one checks that resc(n`) = κ(Pn`), but κ(Pn`) is just the cocycle and for all . σ 7→ (σ − 1)Q σ ∈ GKn` (σ − 1)P = 0 Now c(n`)(σ`) = (σ` − 1)Q − Qn,` (as we had pQn,` = (σ` − 1)P ). Fix a prime λ of K over λ and consider red : E(K) → E(kλ). As E has good reduction at p, this map is injective on p-torsions, so c(n`)(σ`) = 0 if and only if red((σ` − 1)Q − Qn,`) = 1. Note that red((σ` − 1)Q) and red(Qn,`) are -torsion (we had 1 ), and lies in inertia and hence acts trivially, we have that p red(Qn,`) = p ((l + 1)ν − a`)red(Pn) σ` red((σ` − 1)Q) = 0. So red((σ` − 1)Q − Qn,`) = −red(Qn,`). Hence c(n`)(σ`) = 0 if and only if red(Qn,`) = 0 if and only if Pn ∈ pE(Kλ) Proposition. Assume that y ∈/ pE(K). Let ` ∈ be a prime with Frob ` ∼ τ and ` not splitting completely k Q L0/Q in . There there exists − with s in 1 L/L0 c(`) ∈ Selλ(K,E[p]) c(`)λ 6= 0 H (Kλ,E[p])
Proof. We let c(`) be the class de ned by the Heegner point yk. Then Lemma 5.5 and 5.6 shows that c(`) ∈ −. Lemma 5.8 tells us that s if and only if . Selλ(K,E[p]) c(`)λ 6= 0 P1 ∈/ pE(Kλ) But P1 ∈ E(K) is the point yk by de nition. Since by de nition L is the minimal extension of L0 which yk is divisible by p, we have that yk is divisible by p in E(Kλ) if and only if λ splits completely in L/L0. By assumption it does not, hence yk ∈/ pE(Kλ). For the Sel(K,E[p]) space we need to reintroduce some work done by Alex. Let q be a prime satisfying the previous proposition, so − and s . We let 0 be the smallest extension of such that c(q) ∈ Selq(K,E[p]) c(q)q 6= 0 L L c(q) ∈ H1(K,E[p]) is de ned, i.e., c(q) ∈ H1(L0,E[p]). Lemma 5.9. Let be a prime with . Then if and only if splits completely in 0 . ` FrobL/Q` ∼ τ Pq ∈ pE(Kλ) ` L /L Proposition. Assume that . Let be a prime with and not splitting completely in yk ∈/ pE(K) ` ∈ Q FrobL/Q` ∼ τ ` 0 . There there exists with s in 1 . L /L c(q`) ∈ Selλ(K,E[p]) c(q`)λ 6= 0 H (Kλ,E[p]) Proof. By the same argument as above we have c(q`) ∈ Selqλ(K,E[p]) . Then using the previous lemma and Lemma 5.8 we have that s in 1 . c(q`)λ 6= 0 Hs (Kλ,E[p]) It remains to show that or equivalently that s . As we have that c(q`) ∈ Selλ(K,E[p]) c(q`)q = 0 FrobL/Q` ∼ τ ` − splits in L/L0, hence γK ∈ pE(Kλ) using the proof of the above proposition. So considering c(`) ∈ Selλ(K,E[p]) , by Lemma 2.7, we have s and hence − as Alex proved last week. Hence c(`)λ = 0 c(`) ∈ Sel(K,E[p]) = 0 c(`) = 0 itself, and since by construction resc(`) = κ(P`) = 0, we must have P` to be 0 in E[K`]/pE[K`], i.e., P` ∈ pE(K`). So , hence by Lemma 2.7 s . P` ∈ pE(K`,q) c(q`)q = 0
6 __ (Angelos) 6.1 Notation
• p, ` are primes such that p is xed p 6= `
• K is a eld and Kv denotes the completion of K at v sep • GK = Gal(K /K) is a -module and ∗ . In general, is a free -module of nite rank then m • A GK A = Hom(A, µ∞) A Zp Am = A/p A ( , ∗ ∗ m A = A1) Am = A [p ] i i • H (K,A) = H (Gm,A)
Comment: Since, A may not be a discrete G -module, then it doesn't hold that Hi(G ,A) = lim Hi(G ,A ) for K K ←− K m i > 1.For A∗ everything is ne since it has discrete topology
11 6.2 Dualing
Theorem 6.1. Suppose A is nite GK -module, then the pair
1 1 ∗ ∪ 2 invKv H (GKv ,A) × H (GKv ,A ) → H (GKv , µ∞) → Q/Z is perfect
Remark. 1 m ⊥ 1 m Hf (Q`,E[p ]) = Hf (Q`,E[p ]) 1 If F is Selmer structure of A then we get a Selmer structure for Am just using the canonical map H (Kv,A) → 1 . This de nes a Selmer structure for ∗ and from that we de ne Selmer structure ∗ for ∗ by H (Kv,Am) Am F A 1 ∗ 1 ∗ H ∗ (K ,A ) = lim H ∗ (K ,A ). F v ←− F v m 1 1 1 ∗ 1 ∗ Proposition 6.2. It holds that H (K,A) = lim H (K,A ) and H ∗ (K,A ) = lim H (K,A ). F ←− F m F −→ F m De nition 6.3. If are Selmer structure of , we say that if 1 1 for all . F, G A G ⊂ F HG(Kv,A) ⊂ HF (Kv,A) v Note that if G ⊂ F then
1 1 • HG(K,A) ⊂ HF (K,A) •F ∗ ⊂ G∗
Theorem 6.4. Suppose F1, F2 are Selmer structure for a nite GK -module A and F1 ⊂ F2. Then
• 0 → H1 (K,A) → H1 (K,A) ⊕→res L H1 (K ,A)/H1 (K ,A) F1 F2 F2 v F1 v
1 1 ⊕res L 1 1 • 0 → H ∗ (K,A) → H ∗ (K,A) → H ∗ (Kv,A)/H ∗ (Kv,A) F2 F1 F1 F2 Summing over such that 1 1 . The images of are orthogonal complement of each other v HF (Kv,A) 6= HF (Kv,A) ⊕res with respect to the Tate pairing.1 2
From now on K will be Q.
De nition 6.5. The canonical Selmer structure Fcon for A is de ned to be • If v ∈ {∞, p} then H1 ( ,A) = H1( ,A) Fcon Qv Qv • If v∈ / {∞, p} then H1 ( ,A) = ker[H1( ,A) → H1( un,A) ⊗ ]. Fcon Qv Qv Qv Qp Proposition 6.6. If for m and , then 1 m 1 m ∗ 1 m F = Fcon A = E[p ] ` 6= p HF (Q`,E[p ]) = HF ∗ (Q`,E[p ] ) = Hf (Q`,E[p ])
Proposition 6.7. If F = Fcon . The following are the same
1 m 1 m 1 m 0 → Selpm (E/Q) → HF (Q,E[p ]) → HF (Qp,E[p ])/Hf (Qp,E[p ])
1 m 1 m 1 m 0 → HF ∗ (Q,E[p ]) → Selpm (E/Q) → Hf (Qp,E[p ])/HF ∗ (Qp,E[p ]) 6.3 The Hypotheses H.1 is an absolutely irreducible -module, not isomorphic to or A Fp[GQ] Fp µp H.2 There is such that on and is free of rank one over τ ∈ GQ τ = 1 µp∞ A/(τ − 1)A Zp H.3 1 1 ∗ ∗ where is the xed eld of the kernel of . H (Q(A)/Q, A) = H (Q(A )/Q, A ) = 0 Q(A) GQ → Aut(A) H.4 Either A =6∼ A∗ or p ≥ 5
12 H.5 Let Σ be a nite set of primes which contains ∞, p and ` such that A is rami ed at ` and all ` such that 1 1 . For every , 1 1 is torsion-free. HF (Q`,A) 6= Hun(Q`,A) ` ∈ Σ H (Qv,A)/HF (Qv,A)
Remark. Fcon satis es H.5
Let A = Tp(E) with Fcon Selmer structure. Assuming • p ≥ 5 the -adic representations ∞ is surjective (for this is the same as • p GQ → Aut(E[p ]) = GL2(Zp) p ≥ 5 GQ → Aut(E[p]) is surjective). Then Tp(E) satis es H.1 to H.5.
6.4 Euler - Kolyvagin Systems
Let Σ be a nite set of primes containing ∞, p and all primes where A rami es. If `∈ / Σ, P`(x) = det(1−Frob`x|A) ∈ . Let k . Zp[x] N = {np : n is square free product of primes `∈ / Σ, k ≥ 0} De nition 6.8. An Euler system for is a collection 1 such that for A F = {Fn ∈ H (Q(µn),A), n ∈ N } n` ∈ N ( P (Frob−1)F ` 6= p ` ` n . Let ES( ) denote the -module of Euler system of . Fn` = A Zp[GQ] A Fn otherwise
7 Kolyvagin Systems (Céline)
un Notation. K a non-archimedian local eld of characteristic 0. IK ⊂ GK , φ ∈ Gal(K /K) , k residue eld, |k| = q. A GK -Module
7.1 Transverse and Unrami ed cohomology groups
De nition 7.1. Suppose L/K is totally tamely rami ed extension of degree q − 1. Then there is a canonical isomorphism Gal(L/K) = k∗. In this talk, and . K = Q` L = Q`(µ`) We de ne the -transverse cohomology subgroup 1 1 to be 1 1 L Ht (K,A) ⊂ H (K,A) Ht (K,A) = ker[H (K,A) → H1(L, A)] = H1(L/K, AGL ).
Proposition 7.2. Suppose that A is a nite unrami ed GK -module such that (q − 1)A = 0. Then 1. 1 ∼ φ=1 Ht (K,A) = Hom(Gal(L/K),A ) 2. 1 ∼ φ=1 Ht (K,A) = A 3. Direct sum decomposition: 1 1 1 H (K,A) = Hu(K,A) ⊕ Ht (K,A) De nition 7.3. Suppose that , let . Suppose that is an unrami ed -module, free of n|q − 1 R = Z/mnZ A GK nite rank over R, det(1 − φ|A) = 0. Consider P (x) = det(1 − φ|Ax) ∈ R[x]. By assumption P (1) = 0 so P (x) = (x − 1)Q(x) with Q(x) ∈ R(x). By Cayley-Hamilton, P (φ−1) = 0, then Q(φ−1)A ⊂ Aφ=1. De ne the unrami ed-transverse comparison map by
−1 1 ∼ Q(φ ) φ=1 1 Hu(K,A) → A/(φ − 1)A → A → Ht (K,A) | {z } φut
Example. Under the same hypotheses: Aφ=1 is free of rank one if and only if A/(φ − 1)A is also as well. In this case φut is an isomorphism.
13 7.2 Kolyvagin Systems: De nition Let be a -module of nite rank with a continuous action of . Assume that is rami ed at only nitely A Zp GQ A many primes. For every positive integer , we have nite -module n . For a given Selmer structure m GQ An := A/p A for , let be a nite set of places of containing , , where rami es, all where 1 1 . F A Σ Q p ∞ ` A ` HF (Q`,A) 6= Hu(Q`,A) De nition 7.4. Let c be a positive integer such that c is not divisible by any prime in Σ. We de ne F(c) for A as ( H1 ( ,A) v c follows, 1 F Qv - . HF(c)(Qv,A) = 1 Ht (Qv,A) v|c Fix a -module and a Selmer structure for satisfying H.1 to H.5. If , let Zp[GQ] A F A `∈ / Σ ν(`) = max{m|l ≡ 1 m m . De ne be the set of square free product of primes mod p and Am/(φ` − 1)Am is free of rank 1 over Z/p Z} NA = `∈ / Σ. If n ∈ NA then we de ne ν(n) = min{ν(`)|`|n} and set ν(1) = ∞. n o De nition 7.5. A Kolyvagin System is a collection 1 such that if the κn ∈ HF(n)(Q,Aν(n))|n ∈ NA n` ∈ NA following commutes 1 κn ∈ HF(n)(Q,Aν(n))
1 Hu(Q`,Aν(n)) φut 1 res` 1 κn` ∈ HF(n`)(Q,Aν(n`)) / Ht (Q`,Aν(nl)) Let be the -module of the Kolyvagin system for . KS(A) Zp A 7.3 Kolyvagin system construction There exists a canonical map such that if then in 1 ES(A) → KS(A, Fcan) ξ → κ κ1 = ξ1 HF (Q,A) Construction: can For every , let . If , . With this identi cation Q . n ∈ NA Γn = Gal(Q(µn)/Q) n = n1n2 Γn = Γn1 × Γn2 Γn = `|n Γ` For , x a generator for , de ne the Kolyvagin's Derivative operator P`−2 i . If `∈ / Σ σ` Γ` D` := i=1 iσ` ∈ Z[Γ`] , Q . n ∈ NA Dn = `|n D` ∈ Z[Γn] Proposition 7.6. If , , then the image of under 1 1 lies ξ ∈ ES(A) n ∈ NA Dnξn H (Q(µn),A) → H (Q(µn),Aν(n)) in 1 Γn . H (Q(µn),Aν(n)) Proposition 7.7. If , , then the image of has a canonical inverse image in 1 ξ ∈ ES(A) n ∈ NA Dnξn H (Q,Aν(n)) under the restriction map 1 1 Γn . H (Q,Aν(n)) → H (Q(µn),Aν(n)) Denote 0 to be the canonical inverse image of as above. Let 0 0 . ξn Dnξn ξ = {ξn|n ∈ NA} Recall: 1 1 1 , 0 0 0 . H (Q`,Aν(n)) = Hu(Q`,Aν(n)) ⊕ Ht (Q`,Aν(n)) Res`(ξn), (ξn)`,u, (ξn)`,t For ξ0 to be a Kolyvagin system we need: 1. Res (ξ0 ) ∈ H1 ( ,A ) if ` n ` n Fcan Q` ν(n) - 2. 0 1 if Res(ξn) ∈ Ht (Q`,Aν(n)) `|n 3. ut 0 0 1 for φ ◦ Res`(ξn/`) = (ξn)`,t ∈ Ht (Q`,Aν(n)) `|n 0 satis es and , but 0 in general. But we can de ne a Weak Kolyvagin System by ignoring 2. and then ξ 1 3 (ξn)`,u 6= 0 re ning it to create a Strong Kolyvagin System.
7.4 Application Give a Kolyvagin system for , if , then 1 ∗ is nite and has length or equal to where A κ1 6= 0 HF ∗ (Q,A ) δ(K) j 1 . δ(K) = max{j|κ1 ∈ p HF (Q,A)}
14 8 Kato's Euler system
Let p ≥ 5 be prime. E/Q modular Elliptic Curve over Q conductor N. f newform weight 2, level N. T = TpE Euler system: 1 , (ζn ∈ H (Q(ζn),T ))n∈N Σ ⊃ {∞, p} ∪ primes(N) N = {squarefree product of ` ∈ Σ × pk} ( P (Frob−1)ζ ` 6= p Co ζ = ` ` n Q(ζn`)/Q(ζn) n` ζn ` = p
8.1 Overview of construction
(O(Y (n, L))∗)2 H1 (Y (n, L), (1))2 H2 (Y (N) , (1))2 / ´et Zp / ´et 1 Q(ζn) Zp H2 (Y (N) , (2)) ´et 1 Q(ζn) Zp
1 1 H ( (ζn),H (Y1(N) , p(2)) Q ´et Q Z
1 H (Q(ζn),T (1))
∗ Proposition 8.1. Let E/S be an elliptic curve, c prime to 6. Then there exists a unique cθE ∈ O(E\E[c]) such that
1. divisors c2(0) − E[c]
∗ ∗ 2. For all a coprime to c, [a]∗ : O(E\E[ac]) → O(E\E[c]) , [a]∗ cθE = cθE. Proof.
∗ a2 a2 a2−1 Uniqueness: Let f = ug, u ∈ O(S) , f = [a]∗f = [a]∗ug = u [a]∗g = u g, Hence u = 1 so a = 2, 3 and u = 1.
Existence: with ∗. If we let b2−1 a2−1, −3 . [a]∗f = uaf u ∈ O(S) ua = ub g = u2 u3f
( (E, e1, e2) E/S De nition 8.2. N ≥ 3, Y (N) modular curve over Q represent S → . e1, e2 Z/NZ − basis of N torsion Y (N)(C) =∼ (Z/NZ)∗ × Γ(N)\H. If N|N 0, there is a map Y (N 0) → Y (N) so O(Y (N)) ⊂ O(Y (N 0)). De nition 8.3. Let , coprime to . a b 2 . Then ∗ , N ≥ 3 c 6N (α, β) = ( N , N ) ⊂ (Q/Z) \{(0, 0)} cgα,β := Lα,β( cθE) Lα,β = ae1 + be2 : Y (N) → E[N].
GL2(Z/NZ) Y (N) Proposition 8.4.
1. , ∗ σ ∈ GL2(Z/NZ) σ cgα,β = cg(α,β)σ 2. if prime to : then Q . a c cgα,β = aα0=α,aβ0=β cgα0,β0 a b 1 0 MM De nition 8.5. Let M,N ≥ 3, M|L, N|L and de ne Y (M,N) := G\Y (L) where G = ≡ mod ⊂ c d 0 1 NN , . GL2(Z/LZ) S → {(E, e1, e2): E/S, e1 is M torsion,e2 is N torsion, (a, b) ∈ Z/NZ × Z/NZ 7→ ae1 + be2 is injective}
n De nition 8.6. étale sheaves: ∗ p ∗ . ∗ 1 , n ⊗k. Then 1 → µpn → OX → OX → 1 O(X) → Het´ (X, µpn ) (Z/p Z)(k) := (µpn ) i i ∗ 1 H (X, (k)) = lim H (X, µ n (k)). O(X) → H (X, (1)). ´et Zp ←− ´et p ´et Zp
15 i j 0 ∪ i+j 0 , ∗, 2 . H´et(X, Zp(k)) × H´et(X, Zp(k )) → H´et (X, Zp(k + k )) f, g ∈ O(X) {f, g} := κ(f) ∪ κ(g) ∈ H´et(X, Zp(2)) ∗ ∗ ∗ ∗ Lemma 8.7. f : U → V nite étale, u ∈ O(U) , v ∈ O(V ) , f∗{u, f v} = {f∗u, v}, f∗{f v, u} = {v, f∗u}.
2 De nition 8.8. M,N ≥ 3, (c, 6M) = 1 and (d, 6N) = 1. De ne c,dZM,N := { cg 1 , dg 1 } ∈ H (Y (M,N), p(2)). M ,0 0, N ´et Z Proposition 8.9. If we take M|M 0, N|N 0, (c, 6M 0) = 1 and (d, 6N 0) = 1 with primes(M) = primes(M 0) and 0 0 0 primes(N) = primes(N ). Then Y (M ,N ) → Y (M,N). The push-forwards of c,dZM 0,N 0 is c,dZM,N .
If ` - N ∼ 0 Y (M,N(`) / Y (M(`),N) T (`) ) u Y (M,N)
Proposition 8.10. If we take M|M 0, N|N 0, (c, 6M 0) = 1 and (d, 6N 0) = 1, there is a map Y (M`, N`) → Y (M,N) de ned by ∗ ∗ 1/` 0 1/` 0 Z 7→ 1 − T 0(`) + ` · Z c,d M`,N` 0 1 0 1/` c,d M,N
Let Y (N) = Y (N, 1), n, N ≥ 3, n|L, N|L and primes(L) = primes(nN). Then Y (N) ∼ G\Y (L). 1 1 Q(ζn) = ∗ ∗ G = mod N, det ≡ 1 mod n , Y (n, L) → Y (N) 0 1 1 Q(ζn)
De nition 8.11. 2 push-forward of c,dZ1,n,N ∈ H´et(Y1(N)Q(ζn), Zp(2)) c,dZn,N Proposition. ` - nN ∗ ` 0 Z 7→ 1 − T 0(`)σ−1 + σ−2` Z c,d 1,n`,N` ` 0 1/` ` c,d 1,n,N ∗ ` 0 where σ ∈ Gal( (ζ )/ ) and = σ ` Q m Q 0 1 `
De ne c,dζn :=map of c,dZ1,np,Np. Σ = primes(6cdNp).
Theorem 8.12. ( c,dζn) is an Euler system
9 Euler Systems and BSD
Notation. Let be an elliptic curve over , Tate module. nite set of prime containing , , and E Q T = TpE Σ = p ∞ all primes at which rami es. Euler system , 1 with corestriction conditions T {cn} cn ∈ H (Q(µn),T )} 9.1 Recap: What can we do with Euler systems? Mantra: Existence of Euler systems leads to bounds on Selmer groups. Pedro, Alex and Florian: Used Heegner points to bound Sel(Q,E[p]) Céline: Kolyvagin systems, , ∗ is nite. ζ ∈ KS(A) ζ1 6= 0 ⇒ Sel(Q,A ) Today we'll use a variant,
Theorem 9.1. Let be an elliptic curve, a Tate module, . If s 1 1 , E/Q T c ∈ ES(T ) locp(c1) 6= 0 ∈ H (Qp,T )/Hf (Qp,T ) then Sel(Q,E[p∞]) is nite. Proposition 9.2. There exists an exact sequence ∞ 0 → E(Q) ⊗ Qp/Zp → Sel(Q,E[p ]) → X(E/Q)p∞ → 0 Lemma 9.3. If s , then and are nite. locp(c1) 6= 0 E(Q) X(E/Q)p∞
16 9.2 Twisting
Recall: Aurel constructed an Euler System for the wrong module, T (1), i.e., T twisted by the cyclotomic character χp. Motivation: Bloeh-Kato conjecture: existence of 'nice' cohomology classes ↔ vanishing of L-values Euler Systems means lots of nice cohomology. So we might expect Euler systems to exists where there is a systematic vanishing of L-functions. Fact. For all elliptic curves over Q, L(E,S) has a simple zero at s = 0. (More generally, L(f, 0) = 0 for all modular form f) Twisting by characters of nite order. ker χ Let χ : G → ∗ of nite order. Let L := , L/ nite. For all n, G ⊂ ker(χ), hence the natural Q Zp Q Q LQ(µn) map on cocycles induces an isomorphism
∼ 1 = 1 cm H (LQ(µn),T ) ⊗ χ / H (LQ(µn),T ⊗ χ) O ⊗Zχ cor cn 1 1 χ cm ∈ H (LQ(µn),T ) H (Q(µn),T ) 3 cn
c1 Theorem 9.4. χ χ is an Euler Systems for P c = {cn} ∪ {` : `|cond(χ)}
Twisting by χp Problem: kerχ so corestriction doesn't exist. L := Q = Q∞ = ∪kQ(µpk ) Idea: /pn (1) is a trivial G -module, hence T/pk =∼ T/pk ⊗ /pk(1) =∼ T/pk(1) as G -modules. So Z Z Q(µpk ) Z Q(µpk ) 1 k ∼ 1 k H (Q(µpk ), T/p T ) = H (Q(µpk ), T/p T (1)) De nition. The Iwasawa cohomology group for over is 1 1 with respect T Q(µn) H∞(Q(µn),T ) = lim H (Q(µnpk ),T ) to corestriction. ←−k
1 1 k Proposition 9.5. H ( (µ ),T ) =∼ lim H ( (µ k ), T/p T ) ∞ Q n ←−k Q np Corollary 9.6. 1 ∼ 1 H∞(Q(µn),T ) = H∞(Q(µn),T (1)) Note: If , then for any , we can de ne 1 . c ∈ ES(T ) n cn,∞ = {cnpk }k≥0 ∈ H∞(Q(µn),T ) χ χp Theorem 9.7. c p = {cn } is an Euler System.
9.3 BSD
General Euler systems machinery doesn't require conditions at p. But: For the arithmetic applications, we'll need to be precise about conditions at p. Hence we will need p-adic Hodge Theory. Fact. There exists a speci c 1 1 , which is a -dimensional -vector space where , Hf (Qp,V ) ⊂ H (Qp,V ) 1 Qp V = T ⊗Qp such that its Selmer group is equal to the usual Selmer group. There exists an isomorphism, ∗ 1 , where is the regular dierential for . exp : Hs (Qp,V ) → Qp · ωE ωE E Theorem 9.8 ((Kato)). Let E/Q be an elliptic curve, T a Tate module, c the Euler system described by Aurel, −1 0 χ ∗ s 0 rE LNp(E,1)ωE c = c p . Then there exists rE ∈ \{0} such that exp (loc (c )) = , where ΩE = real period of E Z p 1 ΩE corresponding to ωE. Corollary 9.9. If , then and are nite. LNp(E, 1) 6= 0 E(Q) X(E/Q) Proof. Euler factors are non-zero at `|Np .
Theorem 9.10. Let L/Q be an abelian number eld, χ : Gal(L/Q) → C∗. If L(E, χ, 1) 6= 0 then E(L)χ = {P ∈ E(L) ⊗ C : σ(P ) = χ(σ)P ∀σ ∈ Gal(L/Q)} and X(E/L)χ are nite.
17 10 Euler Systems: Some Further Topics 10.1 Recap
Let E/Q be an elliptic curve. Theorem 10.1. If , then is nite, and is nite for many primes . L(E, 1) 6= 0 E(Q) Xp∞ (E/Q) p
Kolyvagin: Proof using Heegner points in E(Km) where K is imaginary quadratic eld, Km ring class eld (abelian extension of such that acts on as ) K Gal(K/Q) Gal(Km/K) −1 Kato: Proof using Siegel units - cohomology classes for TpE over cyclotomic elds. Kato can tell you about L(E, χ, 1) where χ is a Dirichlet character Kolyvagin: can tell you about L(E/K, ψ, 1) where ψ is a character of a ray class group of K such that ψσ = ψ−1. 1. Can we say something about arbitrary abelian extension of K (↔L-functions L(E/K), ψ, 1) for any ray class character ψ)?
2. Are there Euler systems for other Galois representations (not just TpE)?
10.2 An Euler System for two modular forms Theorem 10.2. Let be two modular forms of weight . Galois representations . Then there f, g 2 Tp(f),Tp(g) exists an Euler system for T = Tp(f) ⊗ Tp(g). Starting point: Siegel units, 2 cgα,β = c gα,β − gcα,cβ
w Y n a/N 2πib/N Y n −a/N −2πib/N ga/N,b/N = q 1 − q q e 1 − q q e n≥0 n≥1 where 1 a a2 . w = 12 − 2N + 2N 2 Not a modular form, it has poles at cusps (like the j-function). This lives in O(Y (N))∗. In particular can take , , ∗ 1 using a Kummer map. a = 0 b = 1 cg0,1/N ∈ O(Y1(N)) → H´et(Y1(N), Zp(1)) Consider diagonally. We get the pushfoward 1 ∆ : Y1(N) ,→ Y1(N) × Y1(N) ∆∗ : H´et(Y1(N), Zp(1)) → 3 2 1 for all weight level . Het(Y1(N) , Zp(2)) → H (Q,Tp(f) ⊗ Tp(g)) f, g 2 N This idea is due to Beilinson in 1984 (Beilinson - Flach elements)
Theorem 10.3 (Lei-Lo er-Zerbes). Can extend this to an Euler System as follows: for m ≥ 1, consider ∆m : 2 2 de ned by 1 . Then take . This is only de ned over Y1(m N) → Y1(N) z 7→ (z, z + m ) cm = (∆m)∗( cg0,1/m2N ) Q(µn) not Q. (40 pages later) Some version of norm-compatibility relation holds.
10.3 Twisting
r Recall: In Kato's setting, had to twist from Tp(E)(1) to Tp(E). This was possible because, modulo any power p of , ∼ as representations of . p Tp(E)(1) = Tp(E) Q(µpr ) Interesting cases for Tp(f) ⊗ Tp(g) are not the ones we've immediately get at. Their elements are related to L(f, g, 1) always being 0 (like L(E, 0) = 0 in Kato's case). Want to consider weight f is 2 and weight g is 1. r r Idea: The Galois representation of g mod p shows up in cohomology of Y1(Np ). Special case: K imaginary quadratic eld, ψ ray class character of K. ψ gives a weight 1 modular form θψ. This answers Q1.
18 10.4 More Euler systems?
Suppose we have some variety Xand we want Euler Systems in cohomology of X.We look for subvariety Y,→ X where Y is a modular curve (or product of modular curves). If you are lucky and pushforward of Siegel units lands in the right degree, then maybe that will give an Euler System. This is reasonable if X and Y are Shimura varieties (comes from Matrix groups, like modular curves come from over ) SL2 Q Eg,
• GL2 ,→ GL2 × GL2 (BF elts)
• SO2 ,→ GL2 (Heegner points)
• GL2 × GL2 ,→ GSp4 ∼ • SL2 = SU(1, 1) ,→ SU(2, 1) Problem: Very hard to show that these Euler Systems are not 0.
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