. Mono-anabelian Reconstruction of Number Fields. Yuichiro Hoshi 2014/07/09
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1 .. Mono-anabelian Reconstruction of Number Fields Yuichiro Hoshi RIMS 2014/07/09 Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 1 / 21
2 1 Mono-anabelian Reconstruction of Number Fields Question: Can one reconstruct a number field [i.e., a finite extension of Q] from the associated absolute Galois group? Definition For a topological group G, G: of NF-type def G = the abs. Gal. gp of a number field Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 2 / 21
3 Theorem [Neukirch-Uchida] {, } F : a global field [i.e., a fin. ext. of Q or F p (t)] F : a separable closure of F def G F = Gal(F /F ) = The natural map Isom(F /F, F /F ) Isom(G F, G F ) is bijective. In particular, F = F G F = GF. Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 3 / 21
4 Mochizuki s Mono-anabelian Philosophy Give a(n) [functorial group-theoretic ] algorithm G F F /F. A reconstruction as in Theorem [N-U] is called bi-anabelian reconstruction. In the case where char(f ) > 0, the proof mono-anab n rec n, char(f ) = 0, the proof mono-anab n rec n. Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 4 / 21
5 Rough Statement of Main Theorem A functorial group-theoretic algorithm G : of NF-type F (G) : an algebraically closed field G which satisfies certain conditions. E.g., every G Gal(F /F ) determines (F (G) G) (F Gal(F /F )). Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 5 / 21
6 Notation F : a number field O F F : the ring of integers of F V F : the set of nonarchimedean primes of F If v V F, then O (v) F : the localization of O F at v m (v) O (v) : the maximal ideal of O (v) κ(v) def = O (v) /m (v) : the residue field at v def U (v) = 1 + m (v) O (v) ord v : F Z: the surjective valuation at v Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 6 / 21
7 Outline of the Proof def G F = Gal(F /F ) (a) By Neukirch s work, def G F G F V F = { nonarch. primes of F } V F = VF /G F. (b) By Class Field Theory + Local Rec n Results, the multiplicative groups F v V F F v, where F v is the completion of F at v. Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 7 / 21
8 M F def = (F, O F, V F, {U (v) } v VF ), where the monoid F with respect to, the submonoid O F F, the set V F, and the subgroups U (v) F for v V F. (c) M F 2 + of F, i.e., the field structure of F. Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 8 / 21
9 2 Mono-anabelian Reconstruction of the Additive Structures of Number Fields Theorem (= Uchida s Lemma for Number Fields) A functorial algorithm for reconstructing from [a collection of data which is isomorphic to] M F def = (F, O F, V F, {U (v) } v VF ) [the map corresponding to] the additive structure of F F F F ; (a, b) a + b. Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 9 / 21
10 2.1 The General Case I (1) 0 F : the unique a F s.t. ax = a ( x F ) F = F \ {0}, O F = O F \ {0} (2) 1 F : the unique a F s.t. ax = x ( x F ) (3) 1 F : the unique a F s.t. a 1 but a 2 = 1 Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 10 / 21
11 v V F U (v) O (v) κ(v) 1 1 U (v) F F /U (v) 1 ord v Z Z 1 1 Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 11 / 21
12 (4) κ(v) = (F /U (v) ) tor κ(v) = κ(v) {0} (5) char(κ(v)): the unique prime p s.t. p κ(v) (6) the {±1}-orbit of ord v = { F F /U (v) (F /U (v) )/κ(v) ±1 = Z } (7) ord v : the unique element of this orbit which maps O F F to Z 0 Z Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 12 / 21
13 (8) For a F, Supp(a) def = { v V F ord v (a) 0 } V F (9) O (v) = Ker(ord v) (10) O (v) κ(v) as O (v) O (v) /U (v) ( F /U (v) ) Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 13 / 21
14 2.2 In the Case of Q Suppose that F = Q. (11) 2Z (resp. 3Z; 5Z) V F : the unique v V F s.t. char(κ(v)) = 2 (resp. 3; 5) (12) 2 O F : the unique a O F s.t. Supp(a) = {2Z}, ord 2Z (a) = 1, a U (3Z) (13) 2 = ( 1) 2 O F Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 14 / 21
15 (14) 3 O F : the unique a O F s.t. Supp(a) = {3Z}, ord 3Z (a) = 1, 2 a U (5Z) a O F \ { 2, 1, 0, 1, 2} (15) {a 1, a + 1} = { b OF Supp(a) Supp(b) =, a b 1 U (v) ( v V F ) } OF Note: a b 1 U (pz) a b 1 1 mod p a b mod p p a b Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 15 / 21
16 (16) Suppose: Supp(a) {2Z} a + 1 OF : the unique b {a 1, a + 1} s.t. b U (v) ( v Supp(a)) (17) Suppose: Supp(a) {2Z} ( a: even Supp(a ± 1) {2Z}) a + 1 OF : the unique b {a 1, a + 1} s.t. a b + 1 Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 16 / 21
17 (18) a O F Next(a) def = 1 a = 2 0 a = 1 1 a = 0 2 a = 1 3 a = 2 a + 1 a / { 2, 1, 0, 1, 2} Next: O F O F : a bijection Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 17 / 21
18 (19) The map O F O F O F ; (a, b) a + b by the bijection Next (We omit the proof.) (20) For v V F, the map κ(v) κ(v) κ(v); (a, b) a + b by a + b = a + b Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 18 / 21
19 2.3 The General Case II (21) Q = { a F a char(κ(v)) 1 U (v) for all but finitely many v V F } (by Chebotarev s density theorem) (22) Q = Q {0}, Z = Q O F (23) V Q = V F /, where v w def char(κ(v)) = char(κ(w)) (24) For v V Q, U (v) = Q U (v) Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 19 / 21
20 M Q = (Q, Z, V Q, {U (v) } v VQ ) 2.2 The add. str. of κ(v) ( v V Q ), i.e., the add. str. of κ(v) for v V F s.t. κ(v) = char(κ(v)) Recall: v V F s.t. κ(v) = char(κ(v)) (by Chebotarev s density theorem) Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 20 / 21
21 (25) a, b F a = 0 a + b = b b = 0 a + b = a a b 1 = 1 a + b = 0 otherwise a + b F : the unique c F which satisfies the following: For infinitely many v V F s.t. κ(v) = char(κ(v)); a, b, c O (v), a + b = c in κ(v) Yuichiro Hoshi (RIMS) Mono-anabelian Reconstruction 2014/07/09 21 / 21
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