The Intrinsic Hodge Theory of p-adic Hyperbolic Curves

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The Intrinsic Hodge Theory of p-adic Hyperbolic Curves Shinichi Mochizuki Research Institute for Mathematical Sciences Kyoto University Kyoto 606-01, JAPAN motizuki@kurims.kyoto-u.ac.jp 1. Introduction 2. The Physical Approach in the p-adic Case 3. The Modular Approach in the p-adic Case 1

1. Introduction (A.) The Fuchsian Uniformization Hyperbolic Curve: smooth, proper connected genus g alg. curve r points, s.t. 2g 2+r>0 Over C: unif. by upper half-plane H X(C) =X = H/Γ = π 1 (X ) PSL 2 (R) =Aut(H) a p-adic analogue of this Fuchsian uniformization? Note: Fuchsian Schottky (cf. D. Mumford s theory), e.g., 2

Fuchsian unif. involves arithmetic, i.e., real analytic structures Frobenius at the infinite prime (B.) The Physical Interpretation alg. curve X SO(2)\PSL 2 (R)/Γ (physical/analytic obj.) π 1 (X )+ρ X top. + arith. str. Modular forms define first. 3

(C.) The Modular Interpretation ρ X : π 1 (X ) PSL 2 (R) PGL 2 (C) = π 1 (X ) P 1 C Algebraize quotient (H P 1 C )/π 1(X ) = (P X, P ) (P 1 -bundle + connection)... a (canonical) indigenous bundle Moduli of I.B. s: S g,r M g,r... algebraic Schwarz torsor (w.r.t. ΩofM g,r ), defined over Z[ 1 2 ]. can. I.B. = canonical real analytic s : M g,r (C) S g,r (C) 4

Teichmüller theory (Bers unif.) study of can. real an. sect. s study of quasi-fuchsian deformations of ρ X (D.) Intrinsic Hodge Theory alg. geom. topology + arith. Ex.: classical/p-adic Hodge theory: de Rham coh. sing./ét. coh.+gal. Here: alg. geom. = curve itself, moduli top. + arith. = theory of ρ X = intrinsic Not just philosophy; classical/p-adic Hodge theory techniques important. 5

2. The Physical Approach in the p-adic case (A.) The Arithmetic Fundamental Group K def = char. 0 field, Γ K def = Gal(K/K) X: hyp. curve/k, X def = X K K. 1 π 1 (X) π 1 (X) Γ K 1 π 1 (X) (geom. π 1 ): indep. of moduli (but in char. p, maydetermine moduli! A. Tamagawa) Grothendieck s anabelian philosophy: Extension should determine moduli. 6

(B.) The Main Theorem Theorem 1: K fin. gen. extn./q p, X: hyperbolic curve/k, S : smooth variety/k. X(S) dom Hom open Γ K (π 1 (S),π 1 (X)) i.e., alg. curve X phys./an. obj. Hom open Γ K (,π 1 (X)) Builds on work of: H. Nakamura, A. Tamagawa + G. Faltings, Bloch/Kato. Proof: Consider p-adic analytic diff. forms on (Z p [ T ](p) tame ) (maps to X) cf. mod. forms on H. 7

Remark: Alsopro-p, function field versions (cf. F. Pop). (C.) Comparison with the Case of Abelian Varieties Th.1 resembles Tate Conjecture, i.e., Hom(abelian varieties) Hom(Tate modules) But T. C. false over local fields! New point of view: Theorem 1 = p-adic version of physical aspect of of Fuchsian unif. 8

3. The Modular Approach in the p-adic case (A.) The Example of Shimura Curves a can. p-adic section of Sch. torsor: S g,r M g,r (cf. can. real. an. s)? Guide: theory of Shimura curves (cf. Y. Ihara s theory) Ex.: OverM 1,0, de Rham coh. of univ. ell. curve = can. ind. bun. Note: modp, p-curv. square nilpotent! N.B.: p-curvature def = [ Frob., ] 9

(B.) The Stack of Nilcurves (S g,r ) Fp N g,r : the stack of nilcurves (curves + I.B. with sq. nilp. p-curv.) Theorem 2: N g,r (M g,r ) Fp : finite, flat, local complete intersection, degree = p 3g 3+r, i.e., N g,r almost a section of Sch. torsor! Remarks: (1)N g,r = central object of study of p-adic Teichmüller theory. (2) natural, smooth substacks N g,r [d] N g,r,whered =degree of zero divisor (spikes) of p-curv. 10

(2) (cont d) (d = = dormant); N g,r [d] dim = 3g 3+r. (3) N g,r [0] affine; this= M g,r connected! (cf. Teich. th./c; ab. vars. (Oort)!) (4) molecule def = nilcurve s.t. curve is is tot. degen. ( P 1 s) analyze mol. s str. of N g,r at (5) atom def = toral nilcurve s.t. curve is P 1 {0, 1, }. {atoms} three radii F p /{±1} 11

(5) (cont d) reminiscent of: pants (top. = P 1 {0, 1, }) decomp. of hyp. Riemann surfaces (C.) Canonical Liftings N g,r (N ord g,r ) Fp def =ét. locus/(m g,r ) Fp 12

N ord g,r (M g,r ) Zp... étale morph. of p-adic formal stacks Theorem 3:!(s N : Ng,r ord S g,r ; Frob. lift. Φ N Ng,r ord ) s.t. I.B. def d by s N is invariant w.r.t. Frob. act. def d by Φ N, i.e., s N = desired can. sect. of Sch. torsor! Remarks: (1)(1/p) dφ N is isom., i.e., Φ N is ordinary Frobenius lifting (2) general theory of ord. F.L. s (a.) can. loc. iso. to Ĝm... Ĝm (b.) can. Witt vector liftings of 13

points/char. p perfect fields (cf. real analytic Kähler metrics) (3) Φ N Weil-Petersson metric can. mult. pars. Bers unif. (4) Serre-Tate Th. for ord. AV s arises from ord. F.L. Φ A (e.g., can. mult. coords. S.-T. pars., etc.) Th.3 = Serre-Tate theory for hyp. curves! (5) But Φ N,Φ A, respective ord s are not compatible! M g A g not isometric/c 14

(for WP metric, Siegel upper half-plane metric) (6) (Ng,r ord ) Fp N g,r [0]... for other d, can. lift. theory with Lubin-Tate (instead of Ĝm) uniformizations! In fact, the larger d the more Lubin-Tate the uniformization! (7) corresponding can. Gal. reps. ρ : arithmetic π 1 (curve) PGL 2 (large ring w/gal. act.)...the p-adic analogues of the can. rep. ρ X arising from the Fuchs. unif.! 15

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