The GL 2 main conjecture for elliptic curves without complex multiplication by Otmar Venjakob
Arithmetic of elliptic curves E elliptic curve over Q : E : y 2 + A 1 xy + A 3 y = x 3 + A 2 x 2 + A 4 x + A 6, A i ɛ Z. E(K) =? for number fields, local fields, finite fields K l any prime, Ẽ reduction of E mod l, #Ẽ(F l ) =: 1 a l + l Hasse-Weil L-function of E : L(E/Q, s) := l (1 a l l s +ɛ(l)l 1 2s ) 1, s ɛ C, R(s) > 3 2, where ɛ(l) := { 1 E has good reduction at l 0 otherwise 2
Mordell-Weil Theorem E(Q) is a finitely generated abelian group Birch & Swinnerton-Dyer Conjecture I. r := ord s=1 L(E/Q, s) = rk Z E(Q) II. lim(s 1) r #X(E/Q) L(E/Q, s) = Ω + R E s 1 (#E(Q) tors ) 2 l c l X(E/Q) R E = det(< P i, P j >) i,j ω Ω + = ω γ + c l = [E(Q l ) : E ns (Q l )] Tate-Shafarevich group regulator of E Néron Differential real period of E Tamagawa-number at l 3
The Selmer group of E Assumption: p 5 prime such that E has good ordinary reduction at p, i.e. #Ẽ(F p )[p] = p. For any finite extension K/Q we have the (p-primary) Selmer group Sel(E/K) 0 E(K) Z Q p /Z p Sel(E/K) X(E/K)(p) 0 Thus, assuming #X(E/K) <, it holds for the Pontryagin dual of the Selmer group that Sel(E/K) := Hom(Sel(E/K), Q p /Z p ), rk Z E(K) = rk Zp Sel(E/K) 4
Towers of number fields K n := Q(E[p n ]), 1 n, G n := G(K n /Q) G GL 2 (Z p ) G := G closed subgroup K i.e. a p-adic Lie group K n G G n Q X(E/K n ) := Sel(E/K n ) is a compact Z p [G n ]-module X := X(E/K ) := limsel(e/k n ) is a finitely generated Λ(G)-module, n where Λ(G) = lim n Z p [G n ] denotes the Iwasawa algebra of G, a noehterian possibly non-commutative ring. 5
Twisted L-functions Irr(G n ) irreducible representations of G n, realized over a number field C or a local field Q l R := {p} {l E has bad reduction at l} (ρ, V ρ ) ɛ Irr(G n ), n < L R (E, ρ, s) L-function of E ρ without Euler-factors of R, 6
From BSD to the Main Conjecture algebraic analytic X(E/K n ) L R (E/K n ) = Irr(G n ) L R(E, ρ, s) n ρ as G n -module p-adic families X(E/K ) (L R (E, ρ, 1)) ρ ɛ Irr(Gn ),n< p-adic L-functions F E := F X Characteristic Element L E analytic p-adic L-function Main Conjecture F E L E 7
Example (CM-case): What is new? E : y 2 = x 3 x End(E) = Z[i] Z, i.e. E admits complex multiplication (CM), thus is abelian. G = Z p 2 finite group Main conjecture is a Theorem of Rubin in many cases,i.e. the theory is rather well known! Example (GL 2 -case): E : y 2 + y = x 3 x 2 End(E) = Z, i.e. E does not admit complex multiplication, thus is not abelian. G o GL 2 (Z p ) open subgroup It was not even known how to formulate a main conjecture! New: existence of characteristic elements 8
Structure Theory G GL n (Z p ) compact p-adic Lie group without element of order p Classical: G = Z p n, Λ = Λ(G) = Z p [[X 1,..., X n ]] M torsion Λ-module, then up to pseudo-null modules M i Λ/Λf n i i, f i irreducible, F M := f n i i Now: G non-abelian, but still notion of pseudo-null Theorem (Coates, Schneider, Sujatha). For every torsion Λ-module M one has up to pseudo-null modules M i Λ/L i, L i (reflexive) left ideal Problems: (i) L i not principal in general, i.e. no characteristic element (ii) Euler characteristics do not behave well under pseudo-isomorphisms 9
Localization of Iwasawa algebras (joint work with: Coates, Fukaya, Kato and Sujatha) Assumption: H G with Γ := G/H = Z p (is satisfied in our application because K the cyclotomic Z p -extension Q cyc of Q) contains Λ := Λ(G) We define a certain multiplicatively closed subset T of Λ. Question Can one localize Λ with respect to T? In general, this is very difficult for non-commutative rings! If yes, the localisation with respect to T should be related - by construction - to the following subcategory of the category of Λ-torsion modules: M H (G) category of Λ-modules M such that modulo Z p -torsion M is finitely generated over Λ(H) Λ(G). 10
Characteristic Elements Theorem. The localization Λ T of Λ with respect to T exists and there is a surjective map : (Λ T ) K 0 (M H (G)) arising from K-theory. Definition. Any F M ɛ (Λ T ) with [F M ] = [M] is called characteristic element of M ɛ M H (G). Properties (i) Any f ɛ (Λ T ) can be interpreted as a map on the isomorphism classes of (continuous) representations ρ : G Gl n (O K ), [K : Q p ] < : ρ f(ρ) ɛ K { }. (ii) The evaluation of F M at ρ gives the generalized G-Euler characteristic χ(g, M(ρ)) F M (ρ) [K:Q p] p = χ(g, M(ρ)) if the Euler-characteristic is finite. 11
Numerical Example E = X 1 (11) : y 2 + y = x 3 x 2, A : y 2 + y = x 3 x 2 7820x 263580 p = 5 One can show: X ɛ M H (G), i.e. F X exists. G = G(Q(E(5))/Q) has 2 irreducible Artin Representations of degree 4 : induced by χ i, i = 1, 2. ρ i = Indχ i : G GL 4 (Z 5 ), Q(E[5]) Calculations show: i.e. up to Z 5. χ 1 χ(g, X(ρ i )) = Q(µ 5 ) Q χ 2 Q(A[5]) { 5 3 i = 1 5 i = 2, F X (ρ 1 ) 5 3, F X (ρ 2 ) 5
Analytic p-adic L-function Period - Conjecture: L R (E, ρ, 1) Ω(E, ρ) ɛ Q Conjecture (Existence of analytic p-adic L-function). Let p 5 and assume that E has good ordinary reduction at p. Then there exists L E ɛ (Λ(G) T ), such that for all Artin representations ρ of G one has L E (ρ) and L E (ρ) L R(E, ρ, 1) Ω(E, ρ) up to some modifications of the Euler factor at p. 13
Conjecture (Main Conjecture). Assume that p 5, E has good ordinary reduction at p, and X(E/K ) belongs to M H (G). Granted the existence of the p-adic L- function, L E is a characteristic element of X(E/K ) : [L E ] = [X(E/K )]. 14
Implications of the Main Conjecture and the main conjec- Assuming the existence of L E ture, one can show: 1) ) GL 2 main conjecture 1-variable main conjecture (over Q cyc ) 2) χ(g, X(ρ)) finite L R (E, ρ, 1) 0 In this case one has: χ(g, X(ρ)) = L E (ρ) m ρ p 3) If L(E, 1) 0, then by Kolyvagin: E(Q), X(E/Q) are finite and the p-part of the BSD-conjecture holds. 15
Evidence for Main Conjecture I CM-case Existence of L E follows from existence of 2-variable p-adic L-function (Manin-Vishik, Katz, Yager) If X ɛ M H (G), then the main conjecture follows from 2-variable main conjecture (Rubin,Yager) II GL 2 -case almost nothing is known! Only weak numerical evidence by calculations of T. and V. Dokchitser: E = X 1 (11), p = 5, ρ i, i = 1, 2, the two unique irreducible Artin representations of degree 4 χ(g, X(ρ i )) = L E (ρ i ) 1 p, i = 1, 2 16