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1 SAMA- SUKU-RU Contents p-adic families of Eisenstein series modular form Hecke Eisenstein Eisenstein p The projection to the ordinary part The ordinary idempotent ordinary Eichler Γ 0 pp α Eichler ordinary part References : p :, Q C = Q p, p : Q p {0 p Q : p p = p 1, 4 p = 2, p := p p > 2, { N := { Γ 0 N := { Γ 1 N := a c a c a c b d b d b d c 0 mod N M 2 Z, a, N = 1, ad bc > 0 SL 2 Z c 0 mod N, SL 2 Z c 0, d 1 mod N, 1

2 2 H := {z C Imz > 0 :, ω : Z p Z p /pz p Z p : Teichmüller 1.1 : p Dirichlet, : Z p 1 + pz p ; x xωx 1, u := 1 + p Z p : 1 + pz p i.e. Z p = 1 + pzp ; x u x xx 1... x n + 1 := p n, n! χ : Z/NZ Q : N Dirichlet Dirichlet character, 1 N : Z/NZ {1 Q :N Dirichlet., N Dirichlet χ 1 N, r > 1 r, χr = 0 χ χ : Z C. a b 2 δ = N c d χδ := χa, N C p 3 A C, Q p or F p,. A[χ] := A[{χc c Z/NZ ] log p : Q p Q p p : 1 n s n log p 1 + s = n log p p = 0. s p < p x Z p /pz p, x Z p x = x mod pz p lim n xpn, x ω x p = 2, ω Z 2 /4Z 2 = {1, 1 Z 2

3 SAMA- SUKU-RU p-adic families of Eisenstein series 2.1. modular form Hecke. k 1, f : H C H, detγ > 0 a b γ = GL 2 R c d f k γz := detγ k 1 cz + d k fγ z., γ z, γ Γ 1 N, f k γ = f α SL 2 Z, L Z >0 1 L f k α z = f k αz + L = f k αz L L α Z >0, Fourier : f k αz = n Z an, f k αe 2πinz/Lα. 2.1, 2.1 k 1 Z, N Z >0 χ : Z/NZ Q Dirichlet., k, N, χ χγf k γ = f for γ Γ 0 N, f : H C M k Γ 0 N, χ := Fourier 2.1 : an, f k α = 0 for n < 0, α SL 2 Z, k, N, χ cusp form χγf k γ = f for γ Γ 0 N, f : H C S k Γ 0 N, χ := Fourier 2.1 : an, f k α = 0 for n 0, α SL 2 Z 2.2. M k Γ 0 N, χ S k Γ 0 N, χ C 2.3. f : H C γ Γ 1 N f k γ = f α SL 2 Z, L Z >0 1 L f k α z = f k αz + L = f k αz 0 1 f M k Γ 0 N, χ fz = an, fq n q = e 2πiz. n=0

4 4 q- C C[[q]] q M k Γ 0 N, χ, S k Γ 0 N, χ C[[q]] : 2.4. k, N, χ Z[χ] M k Γ 0 N, χ; Z[χ] := M k Γ 0 N, χ Z[χ][[q]], { m k Γ 0 N, χ; Z[χ] := f M k Γ 0 N, χ a1, f Q[χ], an, f Z[χ] for n > 0, S k Γ 0 N, χ; Z[χ] := S k Γ 0 N, χ Z[χ][[q]], A Z[χ], A,, M k Γ 0 N, χ; A := M k Γ 0 N, χ; Z[χ] Z[χ] A, m k Γ 0 N, χ; A := m k Γ 0 N, χ; Z[χ] Z[χ] A, S k Γ 0 N, χ; A := S k Γ 0 N, χ; Z[χ] Z[χ] A 2.5. M k Γ 0 N, χ; A, m k Γ 0 N, χ; A, S k Γ 0 N, χ; A A 2.6. Z[χ] A C, M k Γ 0 N, χ; A = M k Γ 0 N, χ A[[q]], { m k Γ 0 N, χ; A = f M k Γ 0 N, χ a1, f F raca, an, f A for n > 0, S k Γ 0 N, χ; A = S k Γ 0 N, χ A[[q]] Hecke l, Γ 0 N Γ 0 N 0 l 1 0 r Γ 0 N Γ 0 N = Γ 0 Nγ j 0 l. f M k Γ 0 N, χ Hecke T l r f T l := χγ j f k γ j, T 1 = id, m > 1 T l m j=1 j=1 T l m := T lt l m 1 χll k 1 T l m 2, l N, T l m := T l m, n Z >0, n = p m p m r r T n := r j=1 T p m j j, l 1, l 2, T l m 1 1 T lm 2 2, 2.9, T n well-defined.

5 SAMA- SUKU-RU q Hecke T q {γ j f T q M k Γ 0 N, χ Hecke q : 2.9. f M k Γ 0 N, χ. n Z >0 am, f T n = mn χbb k 1 a b 2, f. : 0<b m,n S k Γ 0 N, χ T q S k Γ 0 N, χ, M k Γ 0 N, χ; Z[χ] T q M k Γ 0 N, χ; Z[χ] k 1, χ N Dirichlet Z[χ] A H k Γ 0 N, χ; A := A[{T l 1 l ] End A m k Γ 0 N, χ; A, h k Γ 0 N, χ; A := A[{T l 1 l ] End A S k Γ 0 N, χ; A A Hecke Hecke : k 1, χ N Dirichlet, A Z[χ] m k Γ 0 N, χ, A H k Γ 0 N, χ; A A; f T a1, f T, S k Γ 0 N, χ, A h k Γ 0 N, χ; A A; f T a1, f T perfect pairing., : m k Γ 0 N, χ; A = Hom A H k Γ 0 N, χ; A, A, H k Γ 0 N, χ; A = Hom A m k Γ 0 N, χ; A, A, S k Γ 0 N, χ; A = Hom A h k Γ 0 N, χ; A, A, h k Γ 0 N, χ; A = Hom A S k Γ 0 N, χ; A, A Eisenstein. k 1, χ : Z/NZ Q N Dirichlet, : k, χ Eisenstein χ 1 = 1 k. E k z, χ := 2 1 L1 k, χ + = if k> σ k 1,χ nq n { N k k 1! 2τχ 1 2πi k m,n 0,0 χ 1 n mnz + n k,

6 6 Ls, χ : Dirichlet L- χn = n s for Res > 1, τχ 1 := N χ 1 re 2πir/N, r=1 σ k 1,χ n :=. k 2 χ 1 1, 2.14, k = 2, χ = 1 1, z = x + yi 0<d n χdd k 1 E k z, χ M k Γ 0 N, χ; Q[χ] E 2z := 1 8πy + E 2z, 1 1, E 2 z, γ Γ 01 E 2z 2 γ = E 2z π cotπz := πi eπiz + e πiz e πiz = πi 1 2 e 2πinz e πiz π cotπz = z 1 + n= { 1 z + n + 1 z n z, k 2, 1 z + n k = 2πik n k 1 e 2πinz k 1!, k > 2, { 2 1 L1 k, χ + σ k 1,χ nq n N k k 1! = 2τχ 1 2πi k k > 2., E k z, χ M k Γ 0 N, χ; Q[χ] m,n 0,0 χ 1 n mnz + n k k, χ N Eisenstein Ez, s, χ := y s c,d: χ 1 d cnz + d k cnz + d 2s

7 SAMA- SUKU-RU 7, s, s = 0 E k z, χ + δ χ,11 1 8πy Ez, 0, χ...[m], δ χ,11 Kronecker k 1 N Dirichlet χ, E k z, χ χpp k 1 E k pz, χ = χpp k 1 L1 k, χ +. k 1 N Dirichlet χ k 2, [M], Theorem E k z, χ χpp k 1 E k pz, χ M k Γ 0 p, 1 p 0<d n,p,d=1 χdd k 1 qn 2.3. Eisenstein p., a Dirichlet ω a : Z p /pz p Q if a 0, χ := 1 1 if a = s Z p Z p [[X]] 1 + X s :=, s m := s Z p, m Z. d, p = 1 d Z X = u k 1. : m=0 s m X m, ss 1 s m + 1. mm 1 1 s m Z p log p d A d X := d 1 log 1 + X p u, log A d u k 1 = d 1 k p d log u p u = d 1 d k = ωd k d k 1

8 p-adic L-function [Wa], Theorem 5.11, Proposition 7.6, Theorem η Dirichlet,, η 1 = 1. f η conductor f η = p c p cα N c = 0 or 1, α 0, N, p = 1. η 0 : Z/p c NZ Z/p c NZ 1 + p c Z/1 + p c p cα Z = Z/f η Z η Q., 2Z p [η][[x]] if η 0 1 p L p ηx c N, 2 X ηu+1 Z p[η][[x]] if η 0 = 1 p c N : Dirichlet ξ, f ξ = p d p dβ M d = 0 or 1, β 0, M, p = 1, ϵ : Z/f ξ Z 1 + p d Z/1 + p d p dβ Z Z/f ξ Z ξ Q, k 1, L p ηξuu k 1 = 1 ηϵω k pp k 1 L1 k, ηϵω k, ηϵ = ω k ηϵω k p = 1., n > 0, A χ n; X := 0<d n,p,d=1 χda d X EχX 1 X Z p[[x]][[q]] χ 1 1 Z p [[X]][[q]] : EχX := 2 1 L p χ + A χ n; Xq n. Eχ χ a 2.2, k a mod ϕp k 1, Eχu k 1 = E k z, 1 1 p k 1 E k pz, 1 1 M k Γ 0 p, 1 p ; Q ψ = ψ 0 ϵ : Z/pp α Z = Z/pZ 1 + pz/1 + pp α Z Q p α p Dirichlet, ψ 0 := ψ Z/pZ {1, ϵ := ψ {1 1+pZ/1+pp α Z, χ = ψ 0 α > 0 k a mod ϕp Eχψuu k 1 = E k z; ψω k M k Γ 0 p α p, ψω k ; Q[ψ]

9 2.20. χ a 2.2 SAMA- SUKU-RU 9 ψ = ψ 0 ϵ : Z/p α pz = Z/pZ 1 + pz/1 + p α pz Q p α p Dirichlet, ψ 0 := ψ Z/pZ {1, ϵ := ψ {1 1+pZ/1+p α pz, χ = ψ 0 k 1 Eψ 0 ψuu k ψuu k 1 1 Z p [ψ][[q]] if a = 0, 1 Z p [ψ][[q]] if a , The ordinary idempotent The projection to the ordinary part 3.1. R, A R x A, e := lim n xn! e 2 = e Proof. A R[x], A R, A R. A = R R m. x m e, e = 0. x / m, m x R/m m R/m m lim n x n! 1 R e e = R, M R, x End R M, e := lim n xn! e 2 = e x em, x em : em em 3.3. T p H k Γ 0 p α, χ; Z p [χ] e := lim n T p n! H k Γ 0 p α, χ; Z p [χ] 3.2 e Z p [χ] f M k Γ 0 p α, χ; Z p [χ] f: ordinary f e = f def

10 10, H ord k Γ 0p α, χ; Z p [χ] := eh k Γ 0 p α, χ; Z p [χ], h ord k Γ 0p α, χ; Z p [χ] := eh k Γ 0 p α, χ; Z p [χ], M ord k Γ 0 p α, χ; Z p [χ] := M k Γ 0 p α, χ; Z p [χ] e, S ord k Γ 0 p α, χ; Z p [χ] := S k Γ 0 p α, χ; Z p [χ] e 2.11 : Hom Zp [χ] H ord k Γ 0p α, χ; Z p [χ], Z p [χ] = M ord k Γ 0 p α, χ; Z p [χ], Hom Zp[χ] h ord k Γ 0p α, χ; Z p [χ], Z p [χ] = S ord k Γ 0 p α, χ; Z p [χ] ordinary χ : Z/p α pz Z p [χ] pp α Dirichlet, : E k χ T p = σ k 1,χ pe k χ = E k χ : E k χ T p = σ k 1,χ pe k χ = E k χ E k χ e = E k χ., E k χ ordinary. β α, M ord k Γ 0pp β, χ; Z p [χ] Zp Q p 3.5. χ pp α Dirichlet k 2, β α, : M ord k Γ 0pp β, χ; Z p [χ] Zp Q p = Q p [χ] E k χ Sk ord Γ 0 pp β, χ; Z p [χ] Zp Q p. 4. Eichler Γ, M Z[Γ] f C n Γ, M C n Γ, M := {f : Γ n M : d n : C n Γ, M C n+1 Γ, M n 1 d n fγ 1,..., γ n+1 := fγ 1, γ 2,..., γ n γ n r+1 fγ 1,..., γ n r γ n r+1,..., γ n+1 r=0 + 1 n+1 fγ 2,..., γ n+1

11 , n SAMA- SUKU-RU 11 d n+1 d n = 0, {C n, d n i n, 4.2. :, M H n Γ, M H n Γ, M Ext n Z[Γ] Z, M = H n Γ, M. Γ-Mod Ab; M M Γ := { x M xγ = x for any γ Γ n : 4.3. Γ, Γ Γ Γ. M Z[Γ], C n Γ, M C n Γ, M; f f Γ n res Γ Γ : Hn Γ, M H n Γ, M Hecke : 4.4 Hecke. Γ 1, Γ 2, Γ 3, Γ 1, Γ 2 Γ 3 σ Γ 3 Γ 1, σ 1 Γ 2 σ commensurable [Γ 1 : Γ 1 σ 1 Γ 2 σ], [σ 1 Γ 2 σ : Γ 1 σ 1 Γ 2 σ] <. M Z[Γ 1, Γ 2, σ], M Γ 1, Γ 2, σ, Γ 3,, [Γ 2 σγ 1 ] : H n Γ 2, M H n Γ 1, M Γ 2 \Γ 2 σγ 1 {σ j m j=1, γ Γ 1 Γ 2 σ j γ 1 = Γ 2 σ jγ σ jγ, [f] H n Γ 2, M f C n Γ 2, M, m [Γ 2 σγ 1 ][f] := γ i n i=1 f σ jγi γ i σj 1 n i=1, j=1 [f] [Γ2 σγ 1 ] := [Γ 2 σγ 1 ][f] σ j

12 12. Hecke well-defined , 4.4 [Γ 2 σγ 1 ] 0 M Γ M Γ ; x x M Z[Γ], m N Γ 2 N Γ 1 ; x xσ j N Z[Γ 1, Γ 2, σ] j= Γ 0 pp α Eichler-. Γ Γ 0 pp α [S], Chapter 8. { pp α a b := M 2 Z c d c ppα Z, a, p = 1, ad bc > 0 P 1 Q Γ, s P 1 Q Γ s M Γ HP 1 Γ, M := Ker res Γ Γ s : H 1 Γ, M H 1 Γ s, M s P 1 Q s P 1 Q M pp α l [ ] T l := Γ 0 pp α 1 0 Γ 0 pp α : H n Γ 0 pp α, M H n Γ 0 pp α, M, 0 l Γ 0 pp α l 0 Γ 0 pp α if l p, T l, l = 0 l 0 if l = p, T 1 = id, m > 1 T l m T l m := T lt l m 1 lt l, lt l m 2, n Z >0, n = p m p mr r r T n := T p m j j j=1, l 1, l 2, T l m 1 1 T lm T l m = [ Γ 0 pp α l m Γ 0 pp α ]

13 SAMA- SUKU-RU Ln, ψ; R. R, ψ : pp α R, pp α Ln, ψ; R n Ln, ψ; R := RX i Y n i ψ, i=0, R, R 2 n, fx, Y R 2 a b n γ = pp α, c d pp α fx, Y γ := ψγfax + by, cx + dy k 2, χ pp α Dirichlet γ Γ, z H, f M k Γ, χ S k Γ, χ c ϕ z fγ := γ 1 z. χ χ, z S k Γ, χ c := fwx wy k 2 dw Lk 2, χ; C { f z f S k Γ, χ, a b χ c d χ pp α = χa 4.10., ϕ z f C 1 Γ, Lk 2, χ; C : 4.11., ϕ z f Kerd 1 for z H, ϕ z f ϕ z f Imd 0 for z, z H, ϕ z f T n ϕ z f T n Imd 0 for n > 0. Φ : M k Γ, χ S k Γ, χ c H 1 Γ, Lk 2, χ; C; f ϕ z f z well-defined Z[{T l l ] 4.12 Eichler-. Φ ΦS k Γ, χ S k Γ, χ c = H 1 P Γ, Lk 2, χ; C, 4.13 [S], Proposition 8.6., Kerj. j : H 1 Γ, Ln, χ; Z[χ] H 1 Γ, Ln, χ; Q[χ] Imj Z[χ] Q[χ] = H 1 Γ, Ln, χ; Q[χ]

14 [LFE], Chapter 6, Proposition 1. + α. Z p [Γ] M H 2 Γ, M = 0 5. ordinary part k 2, pp α Dirichlet χ rank Zp [χ]h ord k Γ 0pp α, χω k ; Z p [χ] = rank Zp [χ]m ord k Γ 0 pp α, χω k ; Z p [χ] = rank Zp M ord 2 Γ0 p, χ 0 ω 2 ; Z p, rank Zp [χ]h ord k Γ 0pp α, χω k ; Z p [χ] = rank Zp [χ]sk ord Γ 0 pp α, χω k ; Z p [χ] = rank Zp S2 ord Γ0 p, χ 0 ω 2 ; Z p χ 0 χ 0 : Z/pZ Z/pZ 1 + pz/1 + pp α Z = Z/pp α Z : χ Q 5.2. R, ψ : pp α R., β α, eh 1 Γ 0 pp α, Ln, ψ; R res Γ 0 ppα Γ 0 pp β e := lim n T p n! 3.2. eh 1 Γ 0 pp β, Ln, ψ; R Proof. r β α 5.3: Γ 0 pp β p r Γ 0 pp α = Γ 0 pp β c Z/p r Z 1 c 0 p r 4.4 [ ] Γ 0 pp β p r Γ 0 pp α : H 1 Γ 0 pp β, Ln, ψ; R H 1 Γ 0 pp α, Ln, ψ; R 5.4: [ ] res Γ 0pp α Γ 0 pp β Γ 0 pp β p r Γ 0 pp α = T p r, [ ] Γ 0 pp β p r Γ 0 pp α res Γ 0pp α Γ 0 pp β = T pr.. e, 3.2

15 5.3. r β α 0, Γ 0 pp β p r Γ 0 pp α = : σ :=. SAMA- SUKU-RU p r [ res Γ 0pp α [ Γ 0 pp β Γ 0 pp β. c Z/p r Z Γ 0 pp β Γ 0 pp β \Γ 0 pp β σγ 0 pp α σ 1 Γ 0 pp β p r 1 c 0 p r ] p r Γ 0 pp α = T p r, ] Γ 0 pp α res Γ 0pp α Γ 0 pp β = T pr 5.5. ψ : pp α F p, ω : Γ 0 pp α ω Z p [χ] F p., Γ 0 pp α 5.6 i : Ln, ψ; F p L0, ψω n ; F p ; fx, Y f1, 0., i ordinary ei : eh 1 Γ 0 pp α, Ln, ψ; F p eh 1 Γ 0 pp α, L0, ψω n ; F p Proof. Γ = Γ 0 pp α. : 0 keri Ln, ψ; F p i L0, ψω n ; F p 0 H 1 Γ, keri H 1 Γ, Ln, ψ; F p i H 1 Γ, L0, ψω n ; F p H 2 Γ, keri. q = 1, 2 eh q Γ, keri = 0, q 1 0, σ := T p = [Γ 0 pp α σγ 0 pp α ] 0 p keri = X n 1 Y,..., Y n Fp keri σ 0. T p 4.4 q eh q Γ, keri = 0. ei 5.6. O Z p, ψ 1, ψ 2 : pp α O n 1 n 2, Hom Γ0 pp α Ln 1, ψ 1 ; O, Ln 2, ψ 2 ; O = {0 0

16 Proof. Z p [χ], ϖ. Hecke rank 2.11, 3.5, β cχ cχ χ p,, rank Zp[χ]S ord k rank Zp[χ]M ord k Γ 0 pp β, χω k ; Z p [χ] Γ 0 pp α, χω k ; Z p [χ] = rank Zp[χ]M ord k Γ 0 pp β, χω k ; Z p [χ] 1 = rank Zp M ord 2 Γ0 p, χ 0 ω 2 ; Z p Eichler- 4.12, 3.5, 4.13, k 2 pp β Dirichlet η rank Zp [η]eh 1 Γ 0 pp β, Lk 2, η; Z p [η] = 2rank Zp [η]m ord k Γ 0pp β, η; Z p [η] 1, n = k 2, ψ := χω n 2, Γ := Γ 0 p α p. 0 Ln, ψ; Z p [χ] ϖ Ln, ψ; Z p [χ] Ln, ψ; F p 0, ψ : Γ ψ Z p [χ] F p : H 0 Γ, Ln, ψ; F p H 1 Γ, Ln, ψ; Z p [χ][ϖ] 0. [ϖ] := { x ϖx = 0. Fp p 1 X n j Y j 1 c p = p j X + cy n j Y j X n 1 Y,..., Y n Fp if j = 0, = 0 p 0 if j > 0 c=0 c=1, , H 0 Γ, Ln, ψ; F p T p 2 = 0, eh 0 Γ, Ln, ψ; F p = 0 eh 1 Γ, Ln, ψ; Z p [χ][ϖ] = 0. eh 1 Γ, Ln, ψ; Z p [χ], 0 H 1 Γ, Ln, ψ; Z p [χ] Zp [χ] F p H 1 Γ, Ln, ψ; F p H 2 Γ, Ln, ψ; Z p [χ], H 2 Γ, Ln, ψ; Z p [χ] = ,. 5.5 dim Fp eh 1 Γ, Ln, ψ; F p = rank Zp[χ]eH 1 Γ, Ln, ψ; Z p [χ] 5.1 dim Fp eh 1 Γ, Ln, ψ; F p = dim Fp eh 1 Γ, L0, ψω n ; F p, Γ/Γ 1 pp β Γ 0 p/γ 1 p p ψω n Γ 0 p, ψω n = χ 0 ω 2 χ 0 : Γ 0 p χ 0 Z p [χ] F p, 5.2 dim Fp eh 1 Γ, L0, ψω n ; F p = dim Fp eh 1 Γ 0 p, L0, χ 0 ω 2 ; F p 5.2

17 SAMA- SUKU-RU n 0, Γ Γ 0 p, ψ ψω n = χ 0 ω 2, 5.2, dim Fp eh 1 Γ, L0, ψω n ; F p = rank Zp [χ]eh 1 Γ 0 p, L0, χ 0 ω 2 ; Z p [χ]., rank Zp [χ]eh 1 Γ, Ln, ψ; Z p [χ] = dim Fp eh 1 Γ, Ln, ψ; F p = dim Fp eh 1 Γ, L0, χ 0 ω 2 ; F p = rank Zp eh 1 Γ 0 p, L0, χ 0 ω 2 ; Z p References [LFE] H. Hida, Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts 26. Cambridge University Press, Cambridge, [M] T. Miyake, Modular Forms, Springer-Verlag, [S] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, [Wa] Lawrence C. Washington, Introduction to Cyclotomic Fields, Second Edition, Graduate Texts in Mathematics, ematics 83, Springer-Verlag, New York, 1997, xiv+487 pp.

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c

$2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c$ GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,