ON THE GALOIS COHOMOLOGY OF UNIPOTENT ALGEBRAIC GROUPS OVER LOCAL FIELDS NGUYE^N~ DUY TAN^ Abstract. In this paper, we give a necessary and sufficient condition for the finiteness of the Galois cohomology of unipotent groups over local fields of positive characteristic. AMS Mathematics Subject Classification (2000): 11E72. Introduction Let k be a field, G a linear k-algebraic group. We denote by ks the separable closure of k in ¯ 1 1 an algebraic closure k, by H (k; G) := H (Gal(ks=k);G(ks)) the usual first Galois cohomology set. It is important to know the finiteness of the Galois cohomology set of algebraic groups over certain arithmetic fields such as local or global fields. A well-known result of Borel-Serre states that over local fields of zero characteristic the Galois cohomology set of algebraic groups are always finite. A similar result also holds for local fields of positive characteristic with restriction to connected reductive groups, see [Se, Chapter III, Section 4]. Joseph Oesterl´ealso gives examples of unipotent groups with infinite Galois cohomology over local fields of positive characteristic (see [Oe, page 45]). And it is needed to give sufficient and necessary conditions for the finiteness of the Galois cohomology of unipotent groups over local function fields. In Section 1, we first present several technical lemmas concerning images of additive polynomials in local fields which are needed in the sequel, and we then give the definition of unipontent groups of Rosenlicht type. Then, in Section 2, we apply results in Section 1 to give a necessary and sufficient condition for the finiteness of the Galois cohomology of an arbitrary unipotent group over local function fields, see Theorem 10. It says roughly that a unipotent group over a local function field has the finite Galois cohomology set if and only if it has a decomposition series such that each factor is a unipotent group of Rosenlicht type. Section 3 deals with some calculations on unipotent groups of Rosenlicht type. We recall after Tits that a unipotent k-group G is called k-wound if every k-homomorphism (or even, k-morphism as in [KMT]) Ga ! G is constant. A polynomial P := P (x1; : : : ; xn) in n variables x1; : : : ; xn with coefficients in k is said to be additive if P (x + y) = P (x) + P (y), for any two elements j n n p x 2 k ; y 2 k . If this is the case, P is the so-called p-polynomial, i.e, a k-linear combination of xi . mi Denote by p the highest degree of xi appearing in P with coefficient ci 2 k n f0g. Then the sum n P pmi cixi is called the principal part of the p-polynomial P and denoted by Pprinc. i=1 1. Images of additive polynomials in local fields Let us first recall the notion of valuation independence (see [DK] or [Ku]). Let (K; v) be a valued field, L a subfield of K, and (bi)i2I a system of non-zero elements in K, with I 6= ;. This system is called L-valuation indepedent if for every choice of elements ai 2 L such that ai 6= 0 for only finitely many i 2 I, we have X v( aibi) = min v(aibi): i2I i2I If V is an L-subvector space of K, then this system is called a valuation basis of V if it is a basis of V and L-valuation independent. Supported by the ERC/Advanced Grant 226257. 1 2 NGUYE^N~ DUY TAN^ Lemma 1. Let k be a field of characteristic p > 0, v a discrete valuation on k, r r m−1 X pm X X pm−j P (T ) = ciTi + cijTi + T1 i=1 i=1 j=1 pm a p-polynomial with coefficients in k, where c1; : : : ; cr is a k -valuation basis of k. Take a 2 k and assume that the set fv(a − y) j y 2 im P g admits a maximum. Then there exist two constants α ≤ β depending only on P (not depending on a) such that this maximum is belong to the segment [α; β] or 1. Proof. Assume that y0 2 k such that v(a − y0) is the maximum of fv(a − y) j y 2 P (k × · · · × k)g. After replacing a by a − y0 we can assume that y0 = 0. Let c10 := c1 and I the set of indexes j, 0 ≤ j ≤ m − 1, such that c1j 6= 0. We set −v(c ) β := max 1j : j2I pm−j − 1 m−s p ∗ Any monomial of P (T1;:::;Tr) − Pprinc(T1;:::;Tr) is of the form λTj , λ 2 k ; 1 ≤ j ≤ r; s ≥ 1, and for such a monomial, we set m−s v(λ) − v(c ) a(λT p ) = a = j ; j λ,s,j pm − pm−s and set m−s α := minfv(λ) + p aλ,s,jg: λ,j;s −v(c1) It is trivial that 0 2 I, hence β ≥ pm−1 . On the other hand, for the polynomial T1 in P − Pprinc, v(1)−v(c1) −v(c1) −v(c1) the constant a(T1) = a1;m;1 = pm−1 = pm−1 , hence α ≤ pm−1 ≤ β. We shall show that α ≤ v(a) ≤ β or v(a) = 1 a) Suppose that 1 > v(a) > β. Then for all j 2 I, we have −v(c ) v(a) > β ≥ 1j ; pm−j − 1 and hence, pm−j m−j v(c1ja ) = v(c1j) + p v(a) > v(a): Now we set X pm−j y := c1ja + a = P (a; 0; ··· ; 0) 2 im P: j2I Then we get X pm−j pm−j v(a − y) = v( c1ja ) ≥ min v(c1ja ) > v(a); j2I j2I a contradiction. pm b) Now suppose that v(a) < α. Since c1; : : : ; cr is a k -valuation basis, we can write pm pm a = c1a1 + ··· + crar ; where ai 2 k for all i and pm v(a) = min v(ciai ): i pm−s For any monomial λTj of P − Pprinc, appearing in P (T1; :::; Tr), λ 2 k; 1 ≤ j ≤ r; s ≥ 1, if v(aj) < aλ,j;s then by the definition of aλ,s,j, we have pm−s m−s m v(λaj ) = v(λ) + p v(aj) > v(cj) + p v(aj) ≥ v(a): Also, if v(aj) ≥ aλ,j;s then by the definition of α, we have pm−s m−s v(λaj ) ≥ v(λ) + p aλ,j;s ≥ α > v(a): Thus, for all j, we always have pmj −s v(λaj ) > v(a): GALOIS COHOMOLOGY OF UNIPOTENT GROUPS 3 Hence v(a − P (a1; : : : ; ar)) = v(P (a1; :::; ar) − Pprinc(a1; :::; ar)) > v(a); a contradiction. Lemma 2. Let k be a local field of characteristic p > 0 and let P (T1;:::;Tr) be a separable p- Pr pm polynomial with coefficients in k with the principal part Pprinc(T1;:::;Tr) = i=1 ciTi : Assume that pm m c1; : : : ; cr is a k -basis of k and v(c1); : : : ; v(cr) are pairwise distinct modulo p . Then the quotient group k= im P is finite. Proof. We write r X pm P (T1;:::;Tr) = ciTi + (··· ) + a1T1 + ··· + arTr: i=1 Let I := fi j ai 6= 0g. Then I is a non-empty set since P is separable. Since v(c1); : : : ; v(cr) are m pairwise distinct modulo p , there exists the unique index i0 2 I such that −pm −pm v(ci0 ai ) = max v(ciai ): 0 i2I We may and shall assume that i0 = 1. Let us change the variables X1 := a1T1 + ··· + arTr, and Xi := Ti for all i > 1. Then the polynomial P becomes r X pm Q(X1;:::;Xr) = biXi + (··· ) + X1; i=1 −pm −pm pm where b1 = a1 c1, bi = ci − c1a1 ai , for all i > 1. Then im P = im Q. We also have v(b1) = m v(c1) − p v(a1), and v(bi) = v(ci) for all i ≥ 2. So v(b1); : : : ; v(br) are still pairwise distinct modulo m pm p . It is then clear that b1; : : : ; br is a k -valuation basis of k. By [DK], we know that the images of additive polynomials in (Fq((t)); vt) have the optimal ap- proximation property, that is, for every z 2 K and every additive polynomial F (X1;:::;Xn) with coefficients in Fq((t)) then the set fv(z − y) j y 2 im F g admits a maximum. Let α; β be the constants depending only on Q as in Lemma 1. Take a in K and suppose that a 62 im Q. Then by Lemma 1, there is y1 2 im Q such that v(a − y1) = m1, where α ≤ m1 ≤ β. There is some j1 2 Fq such that m1 v(a − y1 − j1t ) = m2; m1 where m2 > m1. Again by Lemma 1 applying for a − j1t , we have m2 ≤ β, or m2 = 1. If m2 6= 1 then there is some j2 2 Fq such that m1 m2 v(a − y1 − j1t − j2t ) = m3; where m3 > m2. Denote by [x] the biggest integer not exceeding x then since the segment [α; β] has at most u := [β −α]+1 natural numbers, there exist j1; : : : ; js 2 Fq, where s ≤ u, and m1 < ··· < ms, with mi 2 [α; β] \ N, such that m1 m2 ms v(a − y1 − j1t − j2t − · · · − jst ) = ms+1 > β: m1 m2 ms Again by Lemma 1, we have ms+1 = 1 and a − y1 − j1t − j2t − · · · − jst 2 im Q, hence X X a 2 jtm + im Q: j2Fq m2[α,β]\N Therefore, k= im P = k= im Q has at most q([β − α] + 1) elements.