11 Drinfeld. k( ) = A/( ) A K. [Hat1, Hat2] k M > 0. Γ 1 (M) = γ SL 2 (Z) f : H C. ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d Γ 1 (M))

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1 Drinfeld Drinfeld Drinfeld [Hat3] 1 p q > 1 p A = F q [t] A \ F q d > 0 K A ( ) k( ) = A/( ) A K Laurent F q ((1/t)) 1/t C Drinfeld Drinfeld p p p [Hat1, Hat2] 1.1 p k M > 0 { Γ 1 (M) = γ SL 2 (Z) γ ( ) } mod M 0 1 Γ 1 (M) k H = {z C Im(z) > 0} f : H C f ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d ( a c ) b Γ 1 (M)) d M k (Γ 1 (M)) C M 5 Z[1/M] X 1 (M) 1

2 X 1 (M) ω C X 1 (M) C ω C M k (Γ 1 (M)) C = H 0 (X 1 (M) C, ω k C ). X 1 (M) Γ 1 (M) Γ 1 (M) Z[1/M] R Γ 1 (M) k R R X 1 (M) R ω R M k (Γ 1 (M)) R = H 0 (X 1 (M) R, ω k R ) q f M k (Γ 1 (M)) R q f (q) R (Z[1/M][[q]]) Z[1/M] R S f M k (Γ 1 (M)) S M k (Γ 1 (M)) R f (q) R (Z[1/M][[q]]) S (Z[1/M][[q]]) q Hecke l Hecke T l l M U l l M M k (Γ 1 (M)) R f R M k (Γ 1 (M)) R p N 5 p N p f = (f i ) i Z 0 f i M ki (Γ 1 (N)) Q (f i ) (q) Q[[q]] Q p [[q]] Z p [[q]] p [Ser, 1.4 (b)] (k i ) i X = Z/(p 1)Z Z p p [Ser, 1.4 (c), Théorème 2]. Z X X p f N χ p Q p M χ (Np ) X (weight space) Q p W Q p W(Q p ) p Q p X 1 (N) Qp Q p X 1 (N) ord Q p X 1 (N) Qp X 1 (N) ord Q p X 1 (N) Qp ordinary locus 1.1. X 1 (N) p X 1 (N) ord Q p p 2

3 χ X X 1 (N) ord Q p ω χ Q p M χ (Np ) = H 0 (X 1 (N) ord Q p, ω χ Q p ) χ = k Z ω χ Q p = ω k X1 (N) ord M k (Γ 1 (N)) Qp M χ (Np ) Qp p q p M χ (Np ) Hecke p Hecke U p q n 0 a n q n n 0 a pn q n 1.2. canonical subgroup p p M χ (Np ) Hecke T p U p M χ (Np ) p p p 2 q p p p ) p X p W Q p X = W(Q p ) Z p Z p W( Q p ) Z p O Qp Coleman Coleman-Mazur eigencurve eigenvariety p m 1 v p v p (p) = 1 p M k (Γ 1 (Np m )) C U p v p slope Q 0 {+ } U p 3

4 S k (Γ 1 (2)) C U 2 [Buz, Introduction] (k 2)/2 2-newform 2-oldform k 1 2-oldform k 12 3,8 14 6,6 16 3,7, ,8, ,9,9, ,10,10, ,7,11,16, ,12,12,12, k 3, 6, 3, 4, 3, 5, 3, 4 S k (Γ 1 (2)) C [Eme, Theorem 1.1, Lemma 1.2] Gouvêa-Mazur Buzzard-Kilford-Coleman-Mazur Coleman U p p 1.2 Drinfeld Drinfeld Drinfeld Drinfeld A = F q [t] m { Γ 1 (m) = γ GL 2 (A) γ ( ) } mod m 0 1 Drinfeld Ω Ω = C \ F q ((1/t)) Ω F q ((1/t)) Ω GL 2 (A) ( ) a b γ = GL 2 (A), z Ω γ(z) = az + b c d cz + d k Z m Z/(q 1)Z Γ 1 (m) k m Drinfeld Ω f : Ω C f ( ) az + b = (cz + d) k (ad bc) m f(z) ( z Ω, γ = cz + d ( a c ) b Γ 1 (m)) d 4

5 [Gos1, Gek] M k,m (Γ 1 (m)) C Drinfeld 1.3. Drinfeld C \ R Drinfeld Drinfeld Drinfeld 2 Drinfeld 1.4. S A S Drinfeld S L V (L) A S Zariski V (L) G a = Spec(O S [X]) F q A O S t [t](x) = tx + a 1 X q + a 2 X q2 a 2 O S (S) Drinfeld Γ 1 (m) X 1 (m) A[1/m] m q 1 Γ 1 (m) Γ 1 (m) [Hat3, 4.1] X1 (m) A[1/m] X1 (m) Drinfeld ω M k,m (Γ 1 (m)) C H 0 (X1 (m) C, ω k C ) F q A[1/m] R R Drinfeld M k,m (Γ 1 (m)) R x Drinfeld f M k,m (Γ 1 (m)) R x f (x) R A (A[1/m][[x]]) t f x x [Hat3, Proposition 5.1] Hecke A v Hecke T v v m U v v m M k,m (Γ 1 (m)) R Drinfeld n A \ F q n Drinfeld Drinfeld f = (f i ) i Z 0 f i M ki,m i (Γ 1 (n)) Fq (t) (f i ) (x) F q (t)[[x]] K[[x]] 5

6 O K [[x]] [Vin, Definition 2.5] [Gos2, Definition 5] Goss Drinfeld S = Z/(q d 1)Z Z p [Gos2, Definition 3] S p Drinfeld f = (f i ) i χ S χ S χ Drinfeld Drinfeld p Eisenstein p [Gos2, Theorem 2] W W(Q p ) = Z/(p 1)Z Z p Goss S Drinfeld W(K) = S W W O K Frobenius p [Jeo, Lemma 2.1] W Drinfeld U p = 2 q = 2 = t Γ 1 (t) k Drinfeld U t U t Bandini-Valentino [BV, (6)] k 5 1, , 2, 3, , 3, , 2, 4 3, , 5 2 2, , 2, 3, 5 3, + 3 4, , 3, 5, , 2, 4, 5, 6 3, + 4 k 13 1, 5 2 2, 5, , 2, 3, 6, 7 5, , 3, 7 2 2, 7, , 2, 4 3, 8 5, , 5 2 2, 9 2 2, , 2, 3, 5 3, 9 5, , 3, 5, , 9, , 2, 4, 5, 6, 9, 10 7, , 2, 3, 2 8 3, 7 2 2, 4 3, 9 2 2, 3, 5, 4, 5 6

7 q = 2 k 60 q = 4 k 82 q = 8 k 126 q = 9 k 108 p Emerton [Eme] [Eme] X 0 (2) 0 [BV] 0 Drinfeld > 0 p Drinfeld [Hat3, Theorem 5.9] i = 1, 2 f i M ki,m i (Γ 1 (n)) OK x (f 1 ) (x) (f 2 ) (x) 0 mod n k 1 k 2 mod (q d 1)p log p (n) Drinfeld S 2. [Hat3, Proposition 5.10] χ S X1 (n)ord K ωχ K n χ Drinfeld H 0 (X1 (n)ord K, ωχ K ) 3. [Hat3, Theorem 5.11] n q 1 f Γ 1 (n) Γ 0 ( ) k m Drinfeld x A ( ) f n k Drinfeld Drinfeld Drinfeld ω χ K S Drinfeld Drinfeld B k( ) Ē B Drinfeld Ē ordinary [Sha, Definition 2.11]. Hodge Hasse Drinfeld Hasse p 1 Eisenstein 7

8 E p 1 p q d 1 Drinfeld-Eisenstein g d E p 1 [Gek, (6.8)] B O K E B Drinfeld E ordinary reduction E B/ B Ē B E n E[ n ] B q dn C n (E) Ē n q d Frobenius [Hat3, Lemma 3.5] E n 3.2 Hodge-Tate- 1 p [Kat, Corollary 4.4.2] [Kat] Cartier Hodge- Tate G S ω G G Cartier Car(G) Car(G) = Hom S (G, G m ) S G m = Spec(O S [X, X 1 ]) dx/x Hodge-Tate HT G : Car(G) ω G E p Z p R E[p n ] C n Cartier HT Cn Car(C n ) R/p n R ω E R R/p n R de Rham [Kat, Corollary 4.4.2] Drinfeld E Cartier Hodge-Tate v finite v-module A [Tag, 4] S A C S Carlitz C 1 Drinfeld C = V (O S ) = G a = Spec(O S [Z]) t [t](z) = tz+z q E C = Hom Fq,S(C, G a ) = n 0 O SZ qn E (q) C S q Frobenius E C C q Frobenius t O S φ C : E (q) C E C, ψt C : E C E C O S v C : E C E (q) C, Zqn Z qn 1 (t qn t) + Z qn 1 ψ C t t = φ C v C ψ C t v v-structure S v S A G O S E G v v G : E G E (q) G 8

9 v O K A 0 G O K v O K v X 1 (n) O K S v G G D G D = Hom v,s (G, C) S v Hom v A C dz Hodge-Tate- HTT G : G D ω G B O K E B Drinfeld C n (E) HTT Cn (E) C n (E) D A B/ n B ω E B B/ n B [Hat3, Proposition 3.8] Cartier 3.3 Riemann-Hilbert X ord n X1 (n) O K /( n ) ω Xn ord ω n ord Xord n Drinfeld n C n,n Yn ord C n,n v Hodge-Tate- HTT : C D n,n A O Y ord n ω n ord Y ord n Frobenius Tate-Drinfeld HTT C n,n Xn ord v C n,n Katz Riemann-Hilbert [Kat, Proposition 4.1.1] [Hat3, Lemma 5.6] HTT Frobenius C n,n D Frobenius ωord n Riemann-Hilbert [Hat3, Proposition 5.8] f i f 1 /f 2 ( ω n ord ) k 1 k 2 Frobenius Frobenius ( ω n ord ) k 1 k 2 [Hat3, Corollary 5.7] Riemann-Hilbert C n,n D k 1 k 2 C n,n D π 1(Xn ord ) (A/ n A) [Hat3, Lemma 5.3] k 1 k 2 (A/ n A) exponent (q d 1)p log p (n) 9

10 [BV] A. Bandini and M. Valentino: On the diagonalizability of the Atkin U-operator for Drinfeld cusp forms, preprint, arxiv: [Buz] K. Buzzard: Questions about slopes of modular forms, Automorphic forms I, Astérisque 298 (2005), [Eme] M. Emerton: 2-adic modular forms of minimal slope, thesis (1998). [Gek] E.-U. Gekeler: On the coefficients of Drinfeld modular forms, Invent. Math. 93 (1988), no. 3, [Gos1] D. Goss: π-adic Eisenstein series for function fields, Compos. Math. 41 (1980), no. 1, [Gos2] D. Goss: A construction of v-adic modular forms, J. Number Theory 136 (2014), [Hat1],, 61. [Hat2], Coleman-Mazur, [Hat3] S. Hattori, Duality of Drinfeld modules and -adic properties of Drinfeld modular forms, preprint, arxiv: v2. [Jeo] S. Jeong: On a question of Goss, J. Number Theory 129 (2009), no. 8, [Kat] N. M. Katz: p-adic properties of modular schemes and modular forms, Modular functions of one variable III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp Lecture Notes in Mathematics, Vol. 350, Springer, Berlin, [Ser] J.-P. Serre: Formes modulaires et fonctions zêta p-adiques, Modular functions of one variable III (Proc. Internat. Summer School, Univ. Antwerp, 1972), pp Lecture Notes in Math., Vol. 350, Springer, Berlin, [Sha] S. M. Shastry: The Drinfeld modular Jacobian J 1 (n) has connected fibers, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 4, [Tag] Y. Taguchi: A duality for finite t-modules, J. Math. Sci. Univ. Tokyo 2 (1995), no. 3, [Vin] C. Vincent: On the trace and norm maps from Γ 0 (p) to GL 2 (A), J. Number Theory 142 (2014),

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