FORMAL GROUPS OVER DISCRETE VALUATION RINGS. Contents

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1 FORMAL GROUPS OVER DISCRETE VALUATION RINGS GEUNHO GIM Cotets 1. The ivariat differetial 1 2. The formal logarithm 2 3. Formal groups over discrete valuatio rigs 3 Refereces 5 1. The ivariat differetial Throughout this sectio, F is a formal group over a rig R with formal group law F (X, Y ). Defiitio 1.1. A ivariat differetial o a formal group F is a differetial form satisfyig ω(t ) = P (T )dt R[[T ]]dt ω F (T, S) = ω(t ) (i.e., P (F (T, S))F 1 (T, S) = P (T ) where F 1 (X, Y ) is the partial derivative of F with respect to its first variable.) A ivariat differetial is said to be ormalized if P (0) = 1. Propositio 1.2. There exists a uique ormalized ivariat differetial o F/R give by the formula ω(t ) = F 1 (0, T ) 1 dt. Furthermore, every ivariat differetial is of the form aω for some a R. Proof. We first show that ω(t ) = F 1 (0, T ) 1 dt is a ormalized ivariat differetial. Sice F (T, S) T + S mod (T 2, T S, S 2 ), F 1 (0, S) = T F (T, S) T =0 1 mod (S) Thus, F 1 (0, T ) R[[T ]] is ivertible ad F 1 (0, T ) 1 1 mod (T ). We differetiate the equality F (U, F (T, S)) = F (F (U, T ), S) with respect to U ad get by chai rule. Puttig U = 0 gives us F 1 (U, F (T, S)) = F 1 (F (U, T ), S)F 1 (U, T ) F 1 (0, F (T, S)) = F 1 (F (0, T ), S)F 1 (0, T ) = F 1 (T, S)F 1 (0, T ) 1

2 ad F 1 (0, F (T, S)) 1 F 1 (T, S) = F 1 (0, T ) 1 Therefore, ω is a ormalized ivariat differetial. Suppose P (T )dt is a ivariat differetial satisfyig P (T ) = P (F (T, S))F 1 (T, S). Puttig T = 0 gives us P (0) = P (F (0, S))F 1 (0, S) = P (S)F 1 (0, S). Thus every ivariat differetial is a costat multiple of F 1 (0, T ) 1 dt depedig o P (0). Example 1.3. ω(t ) = dt is the ormailzed ivariat differetial for Ĝa. ω(t ) = dt 1+T = (1 T + T 2 T 3 + )dt is the ormalized ivariat differetial for Ĝm. Corollary 1.4. Let F ad G be formal groups over a rig R with ormalized ivariat differetials ω F ad ω G. Let f : F G be a homomorphism. The ω G f = f (0)ω F where f (T ) is the formal derivative of f(t ). Proof. Let F(X,Y) ad G(X,Y) be the formal group laws for F ad G. Let ω G (T ) = P G (T )dt ad (ω G f)(t ) = P G (f(t ))df(t ) = P G (f(t ))f (T )dt = P (T )dt. Sice f(0) = 0, we have P (0) = f (0). By Prop 1.2, it suffices to show that ω G f is a ivariat differetial for F. (ω G f)(f (T, S)) = ω G (f(f (T, S)) = ω G (G(f(T )), G(f(S))) = ω G (f(t )) = (ω G f)(t ) shows that ω G f is ideed a ivariat differetial for F. Corollary 1.5. Let p Z be a prime. The there are power series f(t ), g(t ) R[[T ]] with f(0) = g(0) = 0 such that [p](t ) = pf(t ) + g(t p ) Proof. Note that [p](t ) pt mod (T 2 ). Let ω(t ) = P (T )dt be the ormalized ivariat differetial for F. Above corollary gives pω(t ) = [p] (0)ω(T ) = (ω [p])(t ) = P ([p](t ))[p] (T )dt Sice P ([p](0)) = P (0) = 1, P ([p](t )) R[[T ]] ad [p] (T ) pr[[t ]]. If R has characteristic p, the [p] (T ) = 0, ad there is a power series g(t ) R[[T ]] such that [p](t ) = g(t p ). Otherwise, let [p](t ) = a i T i, the the coditio above implies p ia i for all i, which explais the existece of such f(t ) ad g(t ). 2. The formal logarithm Throughout this sectio, F is a formal group over a torsio-free rig R, ad K = R Q. Note that the atural map R R Q = K is ijective. 2

3 Defiitio 2.1. Let ω(t ) = (1 + c i T i )dt be the ormalized ivariat differetial o F. The formal logarithm of F is the power series log F (T ) = ω(t ) = (1 + c i T i )dt = T + c i i + 1 T i+1 K[[T ]] The formal expoetial of F is the uique power series exp F (T ) K[[T ]] satisfyig log F exp F (T ) = exp F log F (T ) = T. Example 2.2. logĝa (T ) = T = expĝa (T ). dt logĝm (T ) = 1 + T = ( 1) i 1 T i (= log(1 + T )). i expĝm (T ) = T i i! (= et 1). Propositio 2.3. log F (T ) = a 1 = b 1 = 1. a i i T i ad exp F (T ) = b i i! T i with a i, b i R ad Proof. The form of log F follows from the defiitio. Let f(t ) = log F (T ) = a i i T i ad g(t ) = exp F (T ) = b i i! T i with a i R ad b i K. Repeated differetiatio o f(g(t )) = T gives f (g(t ))g (T ) = 1, f (g(t ))g (T ) + f (g(t ))g (T ) 2 = 0, etc. The first equatio evaluated at T = 0 gives b 1 = 1 a 1 = 1. By usig those equalities, we ca show that b R[a 1,, a, b 1,, b 1 ] R iductively. Propositio 2.4. The map log F : F Ĝa gives a isomorphism of formal groups over K. Proof. Let ω(t ) be the ormalized ivariat differetial o F. Itegratig the equality ω(f (T, S)) = ω(t ) with respect to T gives us log F F (T, S) = log F (T ) + C(S) for some C(S) K[[S]]. Puttig T = 0 shows that C(S) = log F F (0, S) = log F (S). Thus log F is a homomorphism betwee two formal groups F ad Ĝa with a iverse exp F. 3. Formal groups over discrete valuatio rigs Throughout this sectio, R is a complete discrete valuatio rig with maximal ideal M, residue field k ad ormalized valuatio v. I this case, the group F(M) associated to F/R is well-defied. Let char R = 0 ad char k = p > 0. Sice [m] : F F is a isomorphism for p m, the torsio subgroup of F(M) is a p-group. 3

4 Theorem 3.1. Let x F(M) be a elemet of order p for some 1. The v(x) p p 1 Proof. From Cor 1.5, we have [p](t ) = pf(t ) + g(t p ) for some f(t ), g(t ) R[[T ]]. We use a iductio o. First assume that [p](x) = pf(x) + g(x p ) = 0. We have i R[[T ]] ad px(1 + ) = pf(x) = g(x p ) = x p ( ) + v(x) = v(px) = v(pf(x)) = v( g(x p )) v(x p ) = pv(x) Hece v(x). Now assume that the theorem is true for, ad let x be a elemet of order p +1. Note that [p](x) has order p. By iductio hypothesis, p p 1 v([p](x)) = v(pf(x) + g(xp )) mi{ + v(x), v(x p )} Note that the first expressio caot be greater tha +v(x) because v(x) > 0. Thus, v(x) = 1 p v(xp ) 1 p p p = 1 p +1 p Remark 3.2. Let R = Z p. There is o elemet x R satisfyig v(x) p p = 1 1 p p uless p = 2 ad = 1. Therefore F(pZ p) is torsio-free for p > 2 ad the 1 torsio group of F(2Z 2 ) cosists of elemets of order 2. Lemma 3.3. For all, Proof. ( [ ] ) v(!) = sice p [log p ] 1. p i Lemma 3.4. (a) Let f(t ) = the f(x) coverges i R. (b) Let g(t ) = coverges i R. v(!) [log p ] i=1 ( 1) p = (1 p [logp] ) i ( 1) a i i T i R[[T ]] with a i R. If x R satisfies v(x) > 0, b i i! T i R[[T ]] with b i R. If x R satisfies v(x) >, the g(x) 4

5 Proof. (a) We have a x v = v(a ) + v(x) v() v(x) (log p ) a x Thus v goes to ifiity as goes to ifiity. Sice v is oarchimedia ad R is complete, f(x) coverges i R. (b) We have ( b x ( 1) v = v(b )+v(x) v(!) v(x) = v(x)+( 1) v(x) )! by Lem 3.3. By assumptio, v(x) > 0 ad v b x goes to ifiity as goes! ifiity. Thus g(x) coverges i R. Theorem 3.5. Let K be a field of fractios of R. The formal logarithm iduces a homomorphism betwee groups log F : F(M) (K, +) Let r > be a iteger. The the formal logarithm iduces a isomorphism of groups log F : F(M r ) Ĝa(M r ) Proof. Lem 3.4 gives covergece of log F (x) for x M. By Prop 2.4, log F iduces a homomorphism betwee F(M) ad K. Also by Lem 3.4, exp F (x) coverges for x M r sice v(x) r >. Thus log F gives a isomorphism betwee F(M r ) ad Ĝa(M r ) with a iverse exp F. Remark 3.6. If the residue field k is fiite, the F(M) cotais a subgroup of fiite idex isomorphic to R. This is because each quotiet i the series F(M) F(M 2 ) F(M r ) = Ĝa(M r ) is of fiite idex ad Ĝa(M r ) = (R, +) (π r y y for a uiformizer π). Remark 3.7. logĝm : Ĝm(2Z 2 ) Ĝa(2Z 2 ) is ot a isomorphsm. For x 2Z 2, we ca check that v(logĝm (x)) 2. This meas the map logĝm is ot surjective. Refereces [1] C. Erickso, Oe dimesioal formal groups, at erickso/pdfs/ formal_groups.pdf. [2] J. Silverma, The Arithmetic of Elliptic Curves, Spriger,