Galois Deformation & L-Invariant: Study on Hilbert Modular Forms & Galois Reps, Study notes of Cryptography and System Security

Download Galois Deformation & L-Invariant: Study on Hilbert Modular Forms & Galois Reps and more Study notes Cryptography and System Security in PDF only on Docsity! GALOIS DEFORMATION AND L-INVARIANT HARUZO HIDA 1. Lecture 2 The notation is as in the first lecture (F : a totally real field, p > 2 is a fixed prime). For simplicity, we assume that p splits completely in F/Q. We start with a Galois representation ρF : Gal(Q/F ) → GL2(W ) associated to a Hilbert modular form (on GL(2)/F ) with coefficients in W . We assume the ordinarity of ρF : ρF |Dp ∼= ( βp ∗ 0 αp ) with βp 6= αp, βp|Ip = N k−1 and αp(Ip) = 1 on the decomposition group and the inertia group Ip ⊂ Dp ⊂ Gal(Q/F ) for all prime factor p of p in F . Here N (σ) ∈ Z×p is the p-adic cyclotomic character with exp(2πi pn )σ = exp(N (σ)2πi pn ) for all n > 0 and k > 1 is an integer. Again for simplicity, we assume that ρ is unramified outside p. We consider the universal nearly ordinary couple (R,ρ : Gal(Q/F ) → GL2(R)) considered in the first lecture where R is a pro-Artinian local K-algebra. The couple (R,ρ) is universal among Galois deformations ρA : Gal(Q/F ) → GL2(A) (for Artinian local K-algebras A with A/mA = K) such that (K1) unramified outside p; (K2) ρA|Gal(Qp/Fp) ∼= ( ∗ ∗0 αA,p ) with αA,p ≡ αp mod mA (and the local cyclotomy condition if p does not split completely in F ); (K3) det(ρA) = det ρF ; (K4) ρA ≡ ρF mod mA. Recall Γp = 1 + pZp = γ Zp p N−1 ↪→ Gal(Fp[µp∞]/Fp). Identify W [[Γp]] with W [[Xp]] by γp ↔ 1+Xp. Since ρ|Gal(Qp/Fp) ∼= ( ∗ ∗0 δp ), δpα−1p : Γp → R induces an algebra structure on R over W [[Xp]]. Thus R is an algebra over K[[Xp]]p|p. Here is the theorem we have seen in the first lecture: Theorem 1.1 (Derivative). Suppose R ∼= K[[Xp]]p|p. Then, if ϕ ◦ ρ ∼= ρF , for the local Artin symbol [p, Fp] = Frobp, we have L(IndQF Ad(ρF )) = L(Ad(ρF )) = det ( ∂δp([p, Fp]) ∂Xp′ ) p,p′ ∣∣∣ X=0 ∏ p logp(γp)αp([p, Fp]) −1. Greenberg proposed a conjectural recipe of computing the L–invariant. When V = Ad(ρF ), his definition goes as follows. Under some hypothesis, he found a Date: August , 2008. The second lecture (90 minutes) at TIFR on August 1, 2008. The author is partially supported by the NSF grant: DMS 0244401, DMS 0456252 and DMS 0753991. 1 GALOIS DEFORMATION AND L-INVARIANT 2 unique subspace H ⊂ H1(F,Ad(ρF )) of dimension e = |{p|p}| represented by cocycles c : Gal(Q/F ) → Ad(ρF ) such that (1) c is unramified outside p; (2) c restricted to Dp is upper triangular after conjugation for all p|p. By the condition (2), c|Ip modulo upper nilpotent matrices factors through the cyclo- tomic Galois group Gal(Qp[µp∞ ]/Qp) because Fp = Qp, and hence c|Dp modulo upper nilpotent matrices becomes unramified everywhere over the cyclotomic Zp-extension F∞/F ; so, the cohomology class [c] is in SelF∞(Ad(ρF )) but not in SelF (Ad(ρF )). Take a basis {cp}p|p of H over K. Write cp(σ) ∼ ( −ap(σ) ∗ 0 ap(σ) ) for σ ∈ Dp′ with any p′|p. Then ap : Dp′ → K is a homomorphism. His L-invariant is defined by L(Ad(ρF )) = det ( (ap([p, Fp′])p,p′|p ( logp(γp′) −1ap([γp′, Fp′]))p,p′|p )−1) . The above value is independent of the choice of the basis {cp}p. As we remarked in the first lecture, assuming the following condition: (ns) ρ = (ρ mod mW ) has nonsoluble image, by using basically a result of Fujiwara and potential modularity of Taylor (plus a very recent work of Lin Chen), we have R ∼= K[[Xp]]p|p. The following conjecture for the arithmetic L-function is almost a theorem except for the nonvanishing L(Ad(ρF )) 6= 0 (see [HMI] Theorem 5.27 combined with (5.2.6) there): Conjecture 1.2 (Greenberg). Suppose (ns). Let ? = arith,an. For L?p(s,Ad(ρF )) = Φarithρ (γ 1−s − 1), then L?p(s,Ad(ρF )) has zero of order equal to e = |{p|p}| and for the constant L(Ad(ρF )) ∈ K× specified by the determinant as in the theorem, we have lim s→1 L?p(s,Ad(ρF )) (s − 1)e = L(Ad(ρF )) ∣∣|SelQ(IndQF Ad(ρF )∗)| ∣∣−1/[K:Qp] p . If ? = arith, the identity is up to units. The factor E+(Ad(ρ)) does not show up in the above formula. If ρF is crystalline at p, writing SF (Ad(ρF ) ∗) for the Bloch-Kato Selmer group H1f (F,Ad(ρ) ∗), we have ∣∣|SelQ(IndQF Ad(ρF )∗)| ∣∣−1/[K:Qp] p = E+(Ad(ρF )) ∣∣|SF (Ad(ρF )∗)| ∣∣−1/[K:Qp] p up to units, and the value ∣∣|SF (Ad(ρF )∗)| ∣∣−1/[K:Qp] p is directly related to the primitive complex L-value L(1, Ad(ρF )) up to a period (see [MFG] page 284). In the following section, we describe the Selmer group and how to specify H. 1.1. Greenberg’s Selmer Groups. Write F (p)/F for the maximal extension un- ramified outside p and ∞. Put G = Gal(F (p)/F ) and GM = Gal(F (p)/M). Let V = Ad(ρF ). We fix a W -lattice T in V stable under G. Write D = Dp ⊂ G for the decomposition group of each prime factor p|p. Choosing a basis of ρF so that ρF |D is upper triangular, we have a 3-step filtration: (ord) V ⊃ F−p V ⊃ F+p V ⊃ {0},