Quadratic differentials and equivariant deformation theory of curves
aa r X i v : . [ m a t h . AG ] A p r Quadratic Differentials and EquivariantDeformation Theory of Curves
B. K¨ock and A. KontogeorgisNovember 12, 2018
Abstract
Given a finite p -group G acting on a smooth projective curve X over an algebraically closed field k of characteristic p , the dimensionof the tangent space of the associated equivariant deformation functoris equal to the dimension of the space of coinvariants of G acting onthe space V of global holomorphic quadratic differentials on X . Weapply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when G is cyclic or when theaction of G on X is weakly ramified. Moreover we determine certainsubrepresentations of V , called p -rank representations. Let X be a non-singular complete curve of genus g X ≥ k of characteristic p and let G be a subgroup of theautomorphism group of X . The equivariant deformation problem associatedwith this situation is to determine in how many ways X can be deformed toanother curve that also allows G as a group of automorphisms. More pre-cisely, we are led to study the deformation functor D which maps any localArtinian k -algebra A to the set of isomorphism classes of lifts of ( G, X ) toan action of G on a smooth scheme ˜ X over Spec( A ), see [1] for a detaileddescription of D . The dimension of the tangent space of D (in the senseof Schlessinger [16]) can be thought of as a (crude) answer to the above-mentioned equivariant deformation problem. In [1], Bertin and M´ezard haveshown that the tangent space of D is isomorphic to the equivariant cohomol-ogy H ( G, T X ) of ( G, X ) with values in the tangent sheaf T X of X . A more1etailed analysis of the above-mentioned equivariant deformation problemalso involves determining obstructions in H ( G, T X ) and studying the struc-ture of the versal deformation ring associated with D , see [1] and [4].In the classical case, i.e. when k = C (see e.g. [6, Section V.2.2]) or, moregenerally, when the action of G on X is tame (see [1, p. 206] and [10, for-mula (40)]), the dimension of the tangent space of D is known to be equal to3 g Y − r where g Y denotes the genus of the quotient curve Y = X/G and r denotes the cardinality of the branch locus Y ram ; note that 3 g Y − g Y and each branch pointadds one further degree of freedom in our deformation problem, as expected.In the case of wild ramification, the computation of the dimension of thetangent space of D turns out to be a difficult problem. In this paper, wedisregard the known case of tame ramification and assume that the charac-teristic p of k is positive and that G is a p -group.Formulas for the dimension of H ( G, T X ) can be derived from results in [1]and [4] when in addition G is a cyclic group or X is an ordinary curve (or,more generally, the action of G on X is weakly ramified), see Remarks 2.4and 3.2. Furthermore, when G is an elementary abelian group, the second-named author computes this dimension in [10] using the Lyndon-Hochschild-Serre spectral sequence.The object of this paper is to pursue the following alternative method forcomputing the dimension of H ( G, T X ). In [11], the second-named author hasshown that this dimension is equal to the dimension of the space H ( X, Ω ⊗ X ) G of coinvariants of G acting on the space of global holomorphic quadraticdifferentials on X . We will use known results on the Galois module structureof Riemann-Roch spaces H ( X, O X ( D )) for certain G -invariant divisors D on X to determine the dimension of H ( X, Ω ⊗ X ) G .We now fix some notations. Let P , . . . , P r denote a complete set of represen-tatives for the G -orbits of ramified points. The corresponding decompositiongroups, ramification indices and coefficients of the ramification divisor aredenoted by G ( P j ), e ( P j ) and d ( P j ), j = 1 , . . . , r , respectively.Section 2 of this paper will deal with the case when G is cyclic. Generalizinga result of Nakajima [14, Theorem 1] from the case when G is cyclic of order p to the case when G is an arbitrary cyclic p -group, Borne [3, Theorem 7.23]has explicitly determined the Galois module structure of H ( X, O X ( D )) forany G -invariant divisor D on X of degree greater than 2 g X −
2. Although theformulation of Borne’s theorem requires quite involved definitions and is inparticular difficult to state, its consequence for the dimension of H ( X, Ω ⊗ X ) G
2s simple: dim k H ( X, Ω ⊗ X ) G = 3 g Y − r X j =1 (cid:22) d ( P j ) e ( P j ) (cid:23) , see Corollary 2.3. This result can also be derived from core results in [1] andbecomes [1, Proposition 4.1.1] if G is cyclic.In Section 3 we assume that the action of G on X is weakly ramified, i.e.that the second ramification group G ( P ) vanishes for all P ∈ X , and provethe formuladim k H ( X, Ω ⊗ X ) G = 3 g Y − r X j =1 log p | G ( P j ) | + ( r if p > r if p = 2 or 3 , see Theorem 3.1. This formula matches up with results of Cornelissen andKato [4, Theorem 4.5 and Theorem 5.1(b)] for the deformation of ordinarycurves, see Remark 3.2. The proof of our result proceeds along the followinglines. Let W denote the space of global meromorphic quadratic differentialson X which may have a pole of order at most 3 at each ramified point andwhich are holomorphic everywhere else. A result of the first-named author[9, Theorem 4.5] implies that W is a free k [ G ]-module and that its rank andhence the dimension of W G is equal to 3( g Y − r ), see Proposition 3.4.The difference between this dimension and the dimension of H ( X, Ω ⊗ X ) G can be expressed in terms of group homology of G acting on the space ofglobal sections of a certain skyscraper sheaf and can be computed using aspectral sequence argument, see Proposition 3.6 and Lemma 3.7.In Section 4 we first show that there exists an effective G -invariant canonicaldivisor D on X , see Lemma 4.2. This allows us to split the k [ G ]-module H ( X, Ω ⊗ X ) ∼ = H ( X, Ω X ( D )) into its semisimple and nilpotent part withrespect to the corresponding Cartier operator [20]: H ( X, Ω X ( D )) ∼ = H s D ⊕ H n D . Little seems to be known about the k [ G ]-module structure of the nilpotentpart H n D . We use a result of Nakajima [13, Theorem 1] to show that thesemisimple part H s D is a free k [ G ]-module and to compute its rank and hencethe dimension of ( H s D ) G provided D satisfies further conditions, see Theo-rem 4.3. In cases the k [ G ]-module structure of H ( X, Ω ⊗ X ) is known (suchas Section 2) we then also get information about H n D , see Corollary 4.5.Section 5 is an appendix and gives an account of a structure theorem (Theo-rem 5.2) for weakly ramified Galois extensions of local fields. This structuretheorem is used in Section 3, but only in the case p = 2. We finally derive a3eature of the Weierstrass semigroup at any ramified point of X if p = 2 andthe action of G on X is weakly ramified, see Corollary 5.3. Acknowledgments:
The second-named author would like to thank GuntherCornelissen and Michel Matignon for pointing him to the work of Niels Borne,Bernhard K¨ock and Nicolas Stalder.
Notations.
Throughout this paper X is a connected smooth projectivecurve over an algebraically closed field k of positive characteristic p and G is a finite subgroup of the automorphism group of X whose order is a powerof p . We also assume that the genus g X of X is at least 2. For any point P ∈ X let G ( P ) = { g ∈ G : g ( P ) = P } denote the decomposition group and e ( P ) = | G ( P ) | the ramification index at P . The i th ramification group at P in lower notation is G i ( P ) = { g ∈ G ( P ) : g ( t P ) − t P ∈ ( t i +1 P ) } where t P is a local parameter at P . Let e i ( P ) denote its order. Note that e ( P ) = e ( P ) because G is a p -group. Let π : X → Y := X/G denote the canonical projection, and let g Y denote the genus of Y . We use thenotation X ram for the set of ramification points and Y ram = π ( X ram ) for theset of branch points of π . We fix a complete set P , . . . , P r of representativesfor the G -orbits of ramified points in X ; in particular we have r = | Y ram | .The sheaves of differentials and of relative differentials on X are denoted byΩ X and Ω X/Y , respectively. Furthermore for any divisor D on X we use thenotation Ω X ( D ) for the sheaf Ω X ⊗ O X O X ( D ). If D = P P ∈ X n P [ P ] is aneffective divisor, we write D red := X P ∈ X : n P =0 [ P ]for the associated reduced divisor. Let R = X P ∈ X d ( P )[ P ]denote the ramification divisor of π ; then R red = P P ∈ X ram [ P ] is the reducedramification divisor. Note that d ( P ) = ∞ X i =0 ( e i ( P ) − K Y standsfor a canonical divisor on Y ; the divisor K X := π ∗ K Y + R is then a G -invariant canonical divisor on X by [7, Proposition IV 2.3]. For any realnumber x , the notation ⌊ x ⌋ means the largest integer less than or equal to x .4 The Cyclic Case
In this section we assume that the group G is cyclic of order p ν and computethe dimension dim k H ( G, T X ) = dim k H ( X, Ω ⊗ X ) G of the tangent space ofthe deformation functor associated with G acting on X , see Corollary 2.3.We derive Corollary 2.3 from Theorem 2.1 which in turn is a consequence ofa theorem of Borne [3, Theorem 7.23]. While a lot of notations and expla-nations are needed to formulate Borne’s very fine statement, Theorem 2.1 iseasy to state and sufficient to derive the important Corollary 2.3. A differentway of formulating Corollary 2.3 is given in Lemma 2.5 and a different wayof proving it is outlined in Remark 2.4.Let σ denote a generator of G . Let V denote the k [ G ]-module with k -basis e , . . . , e p ν and G -action given by σ ( e ) = e and σ ( e l ) = e l + e l − for l =2 , . . . , p ν . It is well-known that the submodules V l := Span k ( e , . . . , e l ), l =1 , . . . , p ν , of V form a set of representatives for the set of isomorphism classesof indecomposable k [ G ]-modules. In particular for every finitely generated k [ G ]-module M there are some integers m l ( M ) ≥ l = 1 , . . . , p ν , such that M ∼ = p ν M l =1 V L m l ( M ) l ;let Tot( M ) = P p ν l =1 m l ( M ) denote the total number of direct summands. Theorem 2.1.
Let D = P P ∈ X n P [ P ] be a G -invariant divisor on X suchthat deg( D ) > g X − . Then we have Tot( H ( X, O X ( D ))) = 1 − g Y + X Q ∈ Y (cid:22) n ˜ Q e ( ˜ Q ) (cid:23) where ˜ Q denotes any point in the fiber π − ( Q ) .Remark . Our global assumption that g X ≥ Proof.
Borne defines certain divisors D ( l ), l = 1 , . . . , p ν , on Y and provesthat m l ( H ( X, O X ( D ))) = deg( D ( l )) − deg( D ( l + 1)) for l = 1 , . . . , p ν − m p ν ( H ( X, O X ( D ))) = 1 − g Y + deg( D ( p ν )) , p −
1) + ( p − p + . . . + ( p − p ν − is the p -adicexpansion of p ν − H ( X, O X ( D ))) = p ν X l =1 m l ( H ( X, O X ( D ))) = deg( D (1)) + 1 − g Y . We now explain the definition of D (1). For each µ = 0 , . . . , ν let H µ denotethe unique subgroup of G of order p µ and set X µ := X/H µ . Furthermorelet π µ : X µ − → X µ denote the canonical projection. For each divisor E on X µ − set ( π µ ) ∗ ( E ) := j p ( π µ ) ∗ ( E ) k where ⌊·⌋ denotes the integral part of adivisor, taken coefficient by coefficient. Then D (1) is defined as D (1) = ( π ν ) ∗ . . . ( π ) ∗ ( D ) , see [3, Definition 7.22 and Theorem 7.23]. Now Theorem 2.1 follows fromthe formula D (1) = X Q ∈ Y (cid:22) n ˜ Q e ( ˜ Q ) (cid:23) [ Q ] . We prove this formula by induction on ν . It is obvious if ν = 0. So let ν ≥ π ν − ) ∗ . . . ( π ) ∗ ( D ) = X R ∈ X ν − (cid:22) n ˜ R | G ( ˜ R ) ∩ H ν − | (cid:23) [ R ]where, as above, ˜ R denotes any point in the fiber ( π ν − ◦ . . . ◦ π ) − ( R ). Thus,if Q ∈ Y is not a branch point of π ν , the coefficient of the divisor D (1) at Q is equal to (cid:22) n ˜ Q | G ( ˜ Q ) ∩ H ν − | (cid:23) = (cid:22) n ˜ Q | G ( ˜ Q ) | (cid:23) = (cid:22) n ˜ Q e ( ˜ Q ) (cid:23) , as desired. Otherwise we have | G ( ˜ Q ) ∩ H ν − | = | H ν − | = p ν − and thecoefficient of D (1) at Q is equal to j n ˜ Q p ν − k p ( ∗ ) = (cid:22) n ˜ Q p ν (cid:23) = (cid:22) n ˜ Q e ( ˜ Q ) (cid:23) , again as stated. (To see ( ∗ ) write n ˜ Q in the form ap ν + bp ν − + c with some a ≥ b ∈ { , . . . , p − } and c ∈ { , . . . , p ν − } .)6 orollary 2.3. We have dim k H ( X, Ω ⊗ X ) G = 3 g Y − r X j =1 (cid:22) d ( P j ) e ( P j ) (cid:23) . Proof.
Let K Y = P Q ∈ Y m Q [ Q ] be a canonical divisor on Y and let K X denote the canonical G -invariant divisor π ∗ K Y + R on X . As dim( V l ) G = 1for all l = 1 , . . . , p ν we havedim k H ( X, Ω ⊗ X ) G = Tot( H ( X, Ω ⊗ X ))= Tot( H ( X, O X (2 K X ))) = Tot( H ( X, O X (2 π ∗ K Y + 2 R ))) . For any Q ∈ Y the coefficient of the divisor 2 π ∗ K Y + 2 R at any point˜ Q ∈ π − ( Q ) is 2 e ( ˜ Q ) m Q + 2 d ( ˜ Q ). Thus from Theorem 2.1 we obtaindim k H ( X, Ω ⊗ X ) G = 1 − g Y + 2 X Q ∈ Y m Q ! + X Q ∈ Y $ d ( ˜ Q ) e ( ˜ Q ) % = 3 g Y − r X j =1 (cid:22) d ( P j ) e ( P j ) (cid:23) , as stated. Remark . The following chain of equalities sketches a different approachto prove Corollary 2.3 based on core results of the paper [1] by Bertin andM´ezard.dim k H ( X, Ω ⊗ X ) G = dim k H ( G, T X )= dim k H ( Y, π G ∗ ( T X )) + r X j =1 dim k H ( G ( P j ) , ˆ T X,P j )= 3 g − r X j =1 (cid:24) d ( P j ) e ( P j ) (cid:25) + r X j =1 (cid:18)(cid:22) d ( P j ) e ( P j ) (cid:23) − (cid:24) d ( P j ) e ( P j ) (cid:25)(cid:19) = 3 g Y − r X j =1 (cid:22) d ( P j ) e ( P j ) (cid:23) . The statements (together with some notations) used in the above chain ofequalities can be found in [11], [1, pp. 205-206], [10, formula (40)], [1, Propo-sition 4.1.1]. 7he following lemma gives a different interpretation of the term ⌊ d ( P j ) e ( P j ) ⌋ inCorollary 2.3. To simplify notation we fix one of the points P , . . . , P r andwrite just P for this point and just d and e , e , . . . for d ( P ) and e ( P ), e ( P ) , . . . , respectively. Furthermore, N and M denote the highest jumps inthe lower ramification filtration and in the upper ramification filtration of G ( P ), respectively. Lemma 2.5.
We have (cid:22) de (cid:23) = 2(1 + M ) + (cid:22) − N ) e (cid:23) . Proof.
Let k := log p e . By the Hasse-Arf theorem (see the example onpage 76 in [18]) there exist positive integers a , a , . . . , a k − so that the se-quence of jumps in the lower ramification filtration is i = a o , i = a + pa , . . . , i k = a + pa + . . . + p k − a k − and the sequence of jumps in the upper ramification filtration is a , a + a , . . . , a + . . . + a k − . We therefore obtain d = i X i =0 ( e i −
1) + i X i = i +1 ( e i −
1) + . . . + i k X i = i k − +1 ( e i k − i X i =0 ( p k −
1) + i X i = i +1 ( p k − −
1) + . . . + i k X i = i k − +1 ( p − i )( p k −
1) + ( i − i )( p k − −
1) + . . . + ( i k − i k − )( p − a )( p k −
1) + ( pa )( p k − −
1) + . . . + ( p k − a k − )( p − a + . . . + a k − ) p k − (1 + a + pa + . . . + p k − a k − )= (1 + M ) p k − (1 + N )and hence (cid:22) de (cid:23) = (cid:22) M ) p k − N ) p k (cid:23) = 2(1 + M ) + (cid:22) − N ) e (cid:23) , as stated. 8 The Weakly Ramified Case
In this section we assume that the cover π : X → Y is weakly ramified, i.e.that G i ( P ) is trivial for all P ∈ X and all i ≥
2, and prove the followingexplicit formula for the dimension dim k H ( G, T X ) = dim k H ( X, Ω ⊗ X ) G ofthe tangent space of the deformation functor associated with G acting on X . Theorem 3.1.
We have dim k H ( X, Ω ⊗ X ) G = 3 g Y − r X j =1 log p | G ( P j ) | + ( r if p > r if p = 2 or . Remark . Notice, that if the curve X is ordinary, then the cover π isweakly ramified [15, Theorem 2(i)]. In this case, Theorem 3.1 can be proved,similarly to Remark 2.4, using a result of G. Cornelissen and F. Kato ondeformations of ordinary curves [4, Theorem 4.5]. In fact, the argumentsused in the proof of [4, Theorem 4.5] also work in the weakly ramified caseand we thus obtain an alternative proof of Theorem 3.1 in the general case. Example . If we moreover assume that G is cyclic then the group G ( P j ) iscyclic of order p for all j = 1 , . . . , r and the formula in Theorem 3.1 becomesdim k H ( X, Ω ⊗ X ) G = 3 g Y − ( r if p > r if p = 2 or p = 3 , which is the same as in Corollary 2.3. Proof (of Theorem 3.1).
Let Σ denote the skyscraper sheaf defined by theshort exact sequence 0 → Ω ⊗ X → Ω ⊗ X (3 R red ) → Σ → . Since deg(Ω ⊗ X ) = 2(2 g X − > g X −
2, we have H ( X, Ω ⊗ X ) = 0 ([7,Example IV 1.3.4]). By applying the functor of global sections we thereforeobtain the short exact sequence0 → H ( X, Ω ⊗ X ) → H ( X, Ω ⊗ X (3 R red )) → H ( X, Σ) → . As the k [ G ]-module H ( X, Ω ⊗ X (3 R red )) is projective (by Proposition 3.4below), its higher group homology vanishes. From the long exact group-homology sequence associated with the previous short exact sequence wethus obtain the exact sequence0 → H ( G, H ( X, Σ)) → H ( X, Ω ⊗ X ) G → H ( X, Ω ⊗ X (3 R red )) G → H ( X, Σ) G → . k [ G ]-module structure ofRiemann-Roch spaces in the weakly ramified case we will show in Propo-sition 3.4 below that the k [ G ]-module H ( X, Ω ⊗ X (3 R red )) is free and thenderive the dimension of H ( X, Ω ⊗ X (3 R red )) G . Furthermore we will explicitlydescribe the k [ G ]-module structure of H ( X, Σ) and determine (the differencebetween) the dimension of H ( X, Σ) G and of H ( G, H ( X, Σ)) using (unfor-tunately rather involved) homological computations (see Proposition 3.6 andLemma 3.7). Theorem 3.1 then immediately follows from the formulas ob-tained in Propositions 3.4 and 3.6.
Proposition 3.4.
The k [ G ] -module H ( X, Ω ⊗ X (3 R red )) is a free k [ G ] -moduleof rank g Y − r ) . In particular we have dim k H ( X, Ω ⊗ X (3 R red )) G = 3( g Y − r ) . Proof.
We will first show that the k [ G ]-module H ( X, Ω ⊗ X (3 R red )) is free. As G is a p -group it suffices to show that it is projective. Let D = P P ∈ X n P [ P ]be a G -invariant divisor on X . Theorem 2.1 in [9] tells us that the Riemann-Roch space H ( X, O X ( D )) is projective whenever both H ( X, O X ( D )) = 0and n P ≡ − e ( P ) for all P ∈ X ram . It therefore suffices to check thesetwo conditions for the divisor D = 2 K X + 3 R red . The first condition followsfrom [7, Example IV 1.3.4] because deg( D ) ≥ g X − > g X −
2. Thesecond condition follows from the formulas K X = π ∗ K Y + R and(1) R = X P ∈ X ram e ( P ) − P ] . Thus H ( X, Ω ⊗ X (3 R red )) is a free k [ G ]-module. We now determine its rank.As H ( X, Ω ⊗ X (3 R red )) vanishes (see above) we havedim k H ( X, Ω ⊗ X (3 R red )) = 2(2 g X −
2) + 3 | X ram | + 1 − g X = 3( g X − | X ram | )by the Riemann-Roch theorem [7, Theorem IV 1.3]. Furthermore the Rie-mann-Hurwitz formula [7, Corollary IV 2.4] and formula (1) imply that2 g X − | G | (2 g Y −
2) + X P ∈ X ram e ( P ) − . By combining the previous two equations we obtaindim k H ( X, Ω ⊗ X (3 R red ))= 3 | G | ( g Y −
1) + X P ∈ X ram ( e ( P ) −
1) + | X ram | ! = 3 | G | ( g Y − r ) .
10n particular the rank of H ( X, Ω ⊗ X (3 R red )) over k [ G ] is 3( g Y − r ) anddim k ( H ( X, Ω ⊗ X (3 R red )) G = 3( g Y − r ) , as stated. Remark . A slightly different approach to Proposition 3.4 is to first checkthe two conditions for the divisor D = 2 K X + 3 R red as above but then to useTheorem 4.5 in [9] which computes the isomorphism class of H ( X, O X ( D ))directly. Proposition 3.6. If p > we have dim k H q ( G, H ( X, Σ)) = r for q = 0 r P j =1 log p | G ( P j ) | for q = 1 . If p = 3 we have dim k H q ( G, H ( X, Σ)) = r for q = 0 r P j =1 (log | G ( P j ) | − for q = 1 . If p = 2 we have dim k H ( G, H ( X, Σ)) − dim k H ( G, H ( X, Σ)) = 2 r − r X j =1 log | G ( P j ) | . Proof.
As Σ is a skyscraper sheaf, H ( X, Σ) is the direct sum of the stalksof Σ: H ( X, Σ) ∼ = M P ∈ X ram Σ P ∼ = r M j =1 Ind GG ( P j ) (Σ P j ) . We therefore have H q ( G, H ( X, Σ)) ∼ = r M j =1 H q ( G ( P j ) , Σ P i ) for q ≥ P , . . . , P r and write just P for this point. Let t and s be local parameters at P and π ( P ),respectively. Let m denote the multiplicity of a canonical divisor K Y at π ( P ).By formula (1) the multiplicity of the canonical divisor K X = π ∗ K Y + R at P
11s then equal to me ( P ) + 2 e ( P ) −
2. Hence the multiplicity of 2 K X + 3 R red is equal to 2 me ( P ) + 4 e ( P ) −
1, and we obtainΣ P ∼ = ( t − e ( P )(2 m +4) ) / ( t − e ( P )(2 m +4) ) ∼ = ( t ) / ( t )where the latter isomorphism is given by multiplication with the G ( P )-invariant element s m +4 . We have g ( t ) ≡ t mod ( t ) for all g ∈ G ( P ),since G ( P ) = G ( P ). Let the maps α and β from G ( P ) to k be defined bythe congruences(2) g ( t ) ≡ t + α ( g ) t + β ( g ) t mod ( t ) ,g ∈ G ( P ). The map α is obviously a homomorphism from G ( P ) to theadditive group of k and it is injective because G ( P ) is trivial. In particular G ( P ) is a non-trivial elementary abelian p -group. Let ω := t mod ( t ), ω := t mod ( t ) and ω := t mod ( t ). The congruences (2) imply that g ( ω ) = ω (3) g ( ω ) = ω + 2 α ( g ) ω (4) g ( ω ) = ω + α ( g ) ω + β ( g ) ω (5)for any g ∈ G ( P ). Furthermore we easily derive that the map β : G → k satisfies the condition(6) β ( hg ) = β ( h ) + 2 α ( h ) α ( g ) + β ( g )for all h, g ∈ G ( P ). Now Proposition 3.6 follows from the following homo-logical-algebra lemma which we formulate in a way that is independent fromthe context of this paper. In the case p = 2 we need to moreover use Propo-sition 5.2 which tells us that we can choose the local parameter t in sucha way that β = α ; in particular β is not a k -multiple of α unless G ( P ) iscyclic in which case 3 − | G ( P ) | = 1 = 2 − | G ( P ) | . Lemma 3.7.
Let k be a field of characteristic p > . Let G be a non-trivialelementary abelian p -group and let α and β be maps from G to k . We assumethat α is a non-zero homomorphism and that β satisfies the condition (6) forall h, g ∈ G . Furthermore let V be a vector space over k with basis ω , ω , ω .We assume that G acts on V by k -automorphisms given by (3), (4) and (5)(for any g ∈ G ). If p > , we have dim k H q ( G, V ) = ( for q = 0log p | G | for q = 1 . f p = 3 , we have dim k H q ( G, V ) = ( for q = 0log | G | − for q = 1 . If p = 2 , we have dim k H ( G, V ) − dim k H ( G, V )= ( − | G | if β = cα for some c ∈ k, − log | G | else.Proof. Let s denote the dimension of G when viewed as vector space over F p ,i.e. s = log p | G | , and let g , . . . , g s be a basis of G over F p such that α ( g ) = 0.For i = 1 , . . . , s the sequence . . . g i − −→ k [ h g i i ] g i + ... + g p − i −→ k [ h g i i ] g i − −→ k [ h g i i ] sum −→ k −→ k [ h g i i ]-projective resolution of the trivial k [ h g i i ]-module k , see [23, Sec-tion 6.2]. By the K¨unneth formula [23, Theorem 3.6.3], the tensor product ofthese sequences is a k [ G ]-projective resolution of the trivial k [ G ]-module k :(7) . . . → s M i =1 k [ G ] ! M M i
1, 0 and 1,respectively. It is obvious that the (first) dimension dim k E , − = dim k V /V is equal to 1. To determine the remaining five dimensions, let ∂ , ∂ , ∂ and ∂ denote the differentials as indicated in the above diagram of the E -page of our spectral sequence. We are now going to explicitly describe14hese differentials by using the fact that they are connecting homomorphismsassociated with the short exact sequences of complexes0 → F C. → F C. → F C./F C. → → F C./F C. → F C./F C. → F C./F C. → . For any a , . . . , a s ∈ k we have ∂ (( a ¯ ω , . . . , a s ¯ ω ))= ( g − a ω ) + . . . + ( g s − a ω )= ( a α ( g ) + . . . + a r α ( g s ))¯ ω . In particular, the differential ∂ is surjective (since the homomorphism α isnon-zero) and we obtain dim k E , − = 0 , as claimed above, and dim k E , − = s − . Similarly, for any a , . . . , a s ∈ k , we have ∂ (( a ¯ ω , . . . , a s ¯ ω )) = 2(( a α ( g ) + . . . + a s α ( g s ))¯ ω . In particular ∂ is surjective as well and we obtaindim k E , − = dim k E , − = 0 , as claimed above; hence the differential from E , − to E , − is zero and weconclude dim k E , − = dim k E , − = s − , as claimed above. As α ( g ) = 0, the s − y := ( α ( g )¯ ω , − α ( g )¯ ω , , . . . , y := ( α ( g )¯ ω , , − α ( g )¯ ω , , . . . , y s − := ( α ( g s )¯ ω , , . . . , , − α ( g )¯ ω ) . of L si =1 V /V are linearly independent over k and (hence) span the kernelof ∂ . For i = 1 , . . . , s − x i be the tuple of (cid:16) L si =1 V /V (cid:17) L (cid:16)L i 1. For i = 1 , . . . , s we therefore have(1 + g i + . . . + g p − i )( ω )= ω + ( ω + α ( g i ) ω + β ( g i ) ω ) + . . . + ( ω + α ( g p − i ) ω + β ( g p − i ) ω )= pω + (cid:18) p (cid:19) α ( g i ) ω + (cid:18)(cid:18) p (cid:19) β ( g i ) + (cid:18) p (cid:19) α ( g i ) (cid:19) ω and hence (1 + g i + . . . + g p − i ) ω = 0since char( k ) = p > 3; similarly we have (1 + g i + . . . + g p − i ) ω = 0 and(1 + g i + . . . + g p − i ) ω = 0. Therefore e ((˜ x, e ( x ) = e ((0 , ˆ x )).We may assume that ˆ x = ( c ij ω ) i 17s desired.We now turn to the case p = 3. The above proof shows that the first fivedimensions are the same as in the case p > 3. However the (final) dimensiondim k E , is equal to 0 in the case p = 3, as we are going to prove now. Asabove we have dim k E , = dim k (coker( ∂ )) = 1 . In order to prove that dim k E , = 0 we will show that the differential ∂ : E , − = ker( ∂ ) → coker( ∂ ) = E , is surjective. Using the same calculation as in the case p > g + g )( ω ) = 2 α ( g ) ω . Since α ( g ) = 0, the tuple (2 α ( g ) ω , , . . . , 0) does not lie in the image of ∂ (use the same reasoning as in the case p > ∂ issurjective and we obtain dim k E , = 0 , as claimed above.We finally prove Lemma 3.7 in the case p = 2. We may and will assumethat not only α ( g ) = 0 but α ( g i ) = 0 for all i = 1 , . . . , s : if α ( g i ) = 0 forsome i > 1, we replace g i by g i g . For brevity, we will write just α i and β i for α ( g i ) and β ( g i ), respectively. As p = 2, the map β is a homomorphismas well. Also, the group G acts trivially on both ω and ω and hence on V .Moreover the norm element 1 + g i is equal to g i − g i − ω ) = α i ω + β i ω for all i = 1 , . . . , s . In particular all differentials in F C. and C./F C. arezero and the long exact sequence associated with the short exact sequence ofcomplexes 0 → F C. → C. → C./F C. → . . . / / (cid:18) s L i =1 V /V (cid:19) L L i 1, so the rank of the total matrix is 2 s − k H ( G, V ) − dim k H ( G, V ) = dim k H ( C. ) − dim k H ( C. )= dim k V /V + dim k V − dim k s M i =1 V /V ! − dim k coker( ∂ )= ( − s − (2 s − s ) = 3 − s if β = cα for some c ∈ k, − s − (2 s − (2 s − − s else,19s claimed above. Finally, to prove the above claim, we first observe thatthe rank of the submatrix in the lower right-hand corner is at most s − s − s − i := 1; as β is not a k -multiple of α , i.e. as the vector ( β , . . . , β s ) is not a k -multiple of the vector( α , . . . , α s ), there exists i ∈ { , . . . , s } such that α β i + α i β = 0; if α i β i + α i β i = 0 for all i ∈ { , . . . , s }\{ i , i } , let i , . . . , i s run through allindices in { , . . . , s }\{ i , i } ; otherwise choose i ∈ { , . . . , s }\{ i , i } suchthat α i β i + α i β i = 0; continuing this way we end up with indices i , . . . , i s such that { i , . . . , i s } = { , . . . , s } and such that there exists an index l ∈{ , . . . , s − } with the property that α i m β i m +1 + α i m +1 β i m is not equal to 0for m = 1 , . . . , l and is equal to 0 for m = l + 1 , . . . , s − 1; in particular wehave α i l β i m + α i m β i l = 0 as well for m = l + 1 , . . . , s − 1. We now consider the s − ^ ( i , i ) , . . . , ^ ( i l − , i l ) , ^ ( i l , i l +1 ) , . . . , ^ ( i l , i s ) where the notation ] ( i, j )stands for the pair ( i, j ) if i < j and for ( j, i ) else. Then the s − C , . . . , C s − corresponding to these s − a , . . . , a s − ∈ k such that a C + . . . + a s − C s − = 0, then by successivelylooking at the i th1 , . . . , i th l − component we see that a = 0 , . . . , a l − = 0 and byfinally looking at the i th l +1 , . . . , i th s component we see that a l = 0 , . . . , a s − = 0,as desired. p -rank representation. In this section we assume that p > π is not unramified. Let D be a G -invariant effective canonical divisor on X . We will see in Lemma 4.2below that such a divisor always exists. As the divisor D is canonical wehave an isomorphism between the space H D := H ( X, Ω X ( D ))and the space H ( X, Ω ⊗ X ), this paper’s main object of study. As D is effectivewe furthermore have the decomposition H D = H s D ⊕ H n D where H s D and H n D are the spaces of semisimple and nilpotent differentialswith respect to the Cartier operator on H D , see [13], [20] or [22]. Notethat this decomposition depends on the actual divisor D rather than juston its equivalence class; this will become apparent in Theorem 4.3 below,20or instance. We therefore work with the notation H D rather than with H ( X, Ω ⊗ X ) in this section.Since D is G -invariant the above decomposition is a decomposition of k [ G ]-modules. While little seems to be known about the k [ G ]-module H n D , the k [ G ]-module H s D has been studied by various authors ([2], [13], [20]) andis called the p -rank representation . As G is a p -group the only irreducible k [ G ]-module is the trivial representation k and has projective cover k [ G ] [17,15.6]. We conclude that H s D ∼ = core( H s D ) ⊕ k [ G ] b ( G,D,k ) where core( H s D ) denotes the direct sum of non-projective indecomposablesummands of H s D (see [20, Definition 2.3]) and b ( G, D, k ) is called the Borneinvariant corresponding to G , D and the trivial representation k (see [20, Def-inition 5.1]). The goal of this section is to compute the multiplicity b ( G, D, k )when we impose further conditions on the divisor D , see Theorem 4.3 below.Via the isomorphisms H D ∼ = H ( X, Ω ⊗ ) and H ( X, Ω ⊗ ) G ∼ = H ( G, T X ),Theorem 4.3 gives us some information on the space H ( G, T X ), see Corol-lary 4.4 below, or can be used to derive some information on the nilpotentpart H n D if the k [ G ]-module structure of H ( X, Ω ⊗ X ) is known, see Corol-lary 4.5.We begin with the following lemma which is of independent interest andwhich we therefore formulate in a way that is independent from the contextof this paper. Lemma 4.1. Let Z be a connected smooth projective curve of genus at least over an algebraically closed field k , and let S be a finite set of points on Z .Then there exists a global non-zero holomorphic differential on Z such thatnone of its zeroes belongs to S .Proof. If g Z = 1 then every non-zero holomorphic differential on Z is non-vanishing [19, III Proposition 1.5], and the result is obvious in this case.We therefore may and will assume that g Z ≥ 2. We will prove Lemma 4.1 byinduction on r := | S | . The case r = 0 is trivial. So let r ≥ P ∈ S .For the base step r = 1 we need to show thatdim k H ( Z, Ω Z ) > dim k H ( Z, Ω Z ( − [ P ])) . By the Riemann-Roch theorem [7, Theorem IV 1.3] the right-hand side isequal to (2 g Z − − 1) + 1 − g Z + l ([ P ]) = g Z − l ([ P ]) = g Z − g Z ≥ 2. Since the left-hand side is equal to g Z thisproves the base step r = 1.We now prove the inductive step. By the inductive hypothesis there exists aglobal non-zero holomorphic differential ω on Z such that none of its zeroesbelongs to S \{ P } . Furthermore, by the case r = 1 there exists a globalnon-zero holomorphic differential φ on Z that does not vanish at P . So foreach Q ∈ S at least one of the two differentials ω , φ does not vanish at Q .Hence the sets { ( λ, µ ) ∈ k : λω + µφ vanishes at Q } , Q ∈ S, are one-dimensional subspaces of k . Since k is infinite we can avoid thesefinitely many lines and hence find a pair ( λ, µ ) ∈ k such that none of thezeroes of λω + µφ belongs to S .For the following lemma we recall that the divisor of any non-zero meromor-phic differential is a canonical divisor [21, I.5.11]. Lemma 4.2. There exists a G -invariant effective canonical divisor D on X whose support contains X ram . Moreover, if g Y = 0 , we may choose D in sucha way so that its support is equal to X ram and, if g Y ≥ , we may choose D of the form D = div( π ∗ φ ) where φ is a non-zero holomorphic differential on Y whose zeroes do not belong to Y ram .Proof. If φ is a non-zero meromorphic differential on Y then π ∗ φ is a non-zero G -invariant meromorphic differential on X and its divisor div( π ∗ φ ) ishence a G -invariant canonical divisor on X . Furthermore we havediv( π ∗ φ ) = π ∗ div( φ ) + R by [21, Theorem 3.4.6]. We will choose φ in such a way so that D = div( π ∗ φ )is also effective and its support contains X ram .If g Y > 0, then by Lemma 4.1 there exists a non-zero global holomorphicdifferential φ on Y whose zeroes do not belong to Y ram . Then D = div( π ∗ φ )is certainly effective and its support contains X ram . If g Y = 0 and r = 1,we select a generator x of the function field K ( Y ) of Y ∼ = P k such that dx = − π ( P )] and put φ := dx . Then we havediv( π ∗ φ ) = π ∗ div( φ ) + R = − e ( P ) X P ∈ GP [ P ] + R = − ∞ X i =2 ( e i ( P ) − ! X P ∈ GP [ P ] . g X − − | G | + X P ∈ GP d ( P )[7, Corollary IV 2.4] together with the assumption that g X ≥ π is not weakly ramified. Hence the coefficient − P ∞ i =2 ( e i ( P ) − 1) ispositive because p > 3. Thus the divisor D = div( π ∗ φ ) is effective andits support is equal to X ram , as desired. If g Y = 0 and r ≥ x of K ( Y ) such that x ( π ( P )) = ∞ and x ( π ( P )) = ∞ and put φ = dx ( x − x ( π ( P )))( x − x ( π ( P ))) . Then we havediv( φ ) = − [ π ( P )] − [ π ( P )]and hencediv( π ∗ φ ) = − e ( P ) X P ∈ GP [ P ] − e ( P ) X P ∈ GP [ P ] + r X j =1 X P ∈ GP j d ( P j )[ P ]= X j =1 − ∞ X i =1 ( e i ( P j ) − ! X P ∈ GP j [ P ] + r X j =3 X P ∈ GP j d ( P j )[ P ] . Again we see that D = div( π ∗ φ ) is effective and its support is equal to X ram ,as desired.Let γ X and γ Y denote the p -ranks of X and Y , respectively. Furthermore,we recall that E red denotes the reduced divisor for any effective divisor E . Theorem 4.3. Let D be a G -invariant effective canonical divisor on X asin Lemma 4.2. Then the k [ G ] -module H s D is free of rank b ( G, D, k ) = ( γ Y − r if g Y = 0 γ Y − r + deg(div( φ ) red ) if g Y ≥ . Proof. By [22, Lemma 2.5], the k [ G ]-module H s D is the same as H s D red . Fur-thermore, if S is any nonempty set of Y containing Y ram and E is the effectivereduced G -invariant divisor on X corresponding to π − ( S ), then the semisim-ple part of H ( X, Ω X ( E )) is a free k [ G ]-module of rank γ Y − | S | by atheorem of Nakajima [13, Theorem 1]. As D red = ( R red if g Y = 0div( π ∗ φ ) red = ( π ∗ div( φ )) red + R red if g Y ≥ , D red corresponds to ( X ram = π − ( Y ram ) if g Y = 0 π − (supp(div( φ ))) ⊔ X ram = π − (supp(div( φ )) ⊔ Y ram ) if g Y ≥ k ( H n D ) G and, in the case g Y ≥ 1, also deg(div( φ ) red ) can be computed, we can alsocompute the dimension dim k H ( G, T X ) = dim k H ( X, Ω ⊗ X ) G of the tangentspace of the deformation functor associated with G acting on X : Corollary 4.4. Let D be as in Theorem 4.3. Then we have dim k H ( X, Ω ⊗ X ) G = ( dim k ( H n D ) G + γ Y − r if g Y = 0dim k ( H n D ) G + γ Y − r + deg(div( φ ) red ) if g Y ≥ . Proof. Obvious.For the second application we first introduce the following definition. If a k [ G ]-module M is isomorphic to a direct sum L li =1 M ⊕ m i i for some pairwisenon-isomorphic indecomposable k [ G ]-modules M , . . . , M l and some natu-ral numbers m , . . . , m l , we call m i the multiplicity of M i in M . In casesthe multiplicity m k [ G ] of the regular representation k [ G ] in the k [ G ]-module H ( X, Ω ⊗ X ) is known, we can also compute the multiplicity n k [ G ] of k [ G ] inthe nilpotent part H n D : Corollary 4.5. Let D be as in Theorem 4.3. Then we have n k [ G ] = m k [ G ] − ( γ Y − r if g Y = 0 γ Y − r + deg(div( φ ) red ) if g Y ≥ . Proof. Obvious.A formula for m k [ G ] is known if G is cyclic [3, Theorem 7.23] or if G iselementary abelian [8, Section 3]. For instance, if G is cyclic of order p , wecan derive from [14, Theorem 1] that m k [ G ] = 3 g Y − r X j =1 (cid:22) ( N j + 2)( p − p (cid:23) where N j is the highest (and single) jump in the lower ramification filtrationof G ( P j ). 24 Appendix This appendix gives an account of a structure theorem for finite weakly rami-fied Galois extensions of local fields of positive characteristic p in the case theGalois group is a p -group, see Proposition 5.2 below. This structure theoremis used at the end of the proof of Proposition 3.4 in the main part of thispaper, but only in the case p = 2. It appears as (part of) Proposition 1.4in [4] and Proposition 1.18 in [5]; in this appendix we give a self-containedand elementary proof. It implies a very explicit and simple description ofthe action of the Galois group on a local parameter, see Proposition 5.2. Inthe situation of Section 3 of the main part of this paper we finally derive acertain feature of pole numbers, if p = 2, see Corollary 5.3.Let K be a local field of characteristic p > 0; i.e. the field K is completewith respect to a discrete valuation v K : K × ։ Z and its residue field k isperfect; we assume that k is contained in K . Lemma 5.1. Let L/K be a totally ramified Galois extension of degree p .Then there exists an element y ∈ L whose valuation is coprime to p andnegative, say − m , such that y p − y ∈ K and L = K ( y ) . The greatestinteger M such that the higher ramification group G M of L/K does not vanishis then equal to m .Proof. Let σ be a generator of the Galois group G = Gal( L/K ). By theclassical Artin-Schreier Theorem there exist elements x ∈ K and y ∈ L suchthat L = K ( y ) and y p − y = x . We first show that the valuation v K ( x )of x is negative and that v L ( y ) = v K ( x ). Suppose that v K ( x ) ≥ 0. Thenalso v L ( y ) ≥ y and x in k by ¯ y and ¯ x ,respectively, we obtain( y − ¯ y ) p − ( y − ¯ y ) = ( y p − y ) − y p − y = x − ¯ x. In particular we have σ ( y − ¯ y ) = ( y − ¯ y ) + c for some c ∈ F × p and0 < v L ( y − ¯ y ) = v L ( σ ( y − ¯ y )) = v L (( y − ¯ y ) + c ) = 0 , which is a contradiction. Hence we have v K ( x ) < v L ( y ) < pv L ( y ) = v L ( y p ) = v L ( y p − y ) = v L ( x ) = pv K ( x ) , i.e. v L ( y ) = v K ( x ). Let m := − v L ( y ) = − v K ( x ). If p divides m , i.e. m = lp for some l ∈ N , we write x = u s lp + u s lp − + u s lp − + . . . u , u , u , . . . ∈ k and some local parameter s of K . Furthermore,as k is perfect, there is a v ∈ k such that v p = u . We now consider˜ y := y − v s l ∈ L and ˜ x := x − v p s lp + v s l ∈ K. Then we have L = K (˜ y ) and˜ y p − ˜ y = (cid:16) y − v s l (cid:17) p − (cid:16) y − v s l (cid:17) = ( y p − y ) − v p s lp + v s l = ˜ x. As above we obtain v L (˜ y ) = v K (˜ x ) < 0; furthermore we have v K (˜ x ) = v K (cid:18) x − v p s lp + v s l (cid:19) = v K (cid:16)(cid:16) u s lp − + u s lp − + . . . (cid:17) + v s l (cid:17) > − lp = − m = v K ( x ) . Continuing this way (if necessary) we obtain x ∈ K and y ∈ L such L = K ( y ), y p − y = x and p does not divide m = − v L ( y ) = − v K ( x ) > 0, asclaimed.Let r, l ∈ Z such that rp + lm = 1. Let s be a local parameter of K and put t := s r y − l ∈ L . Then we have v L ( t ) = rp + ( − l )( − m ) = 1, i.e. t is a localparameter of L . Furthermore we have σ ( t ) = σ ( s r y − l ) = s r σ ( y ) − l = s r ( y − c ) − l for some c ∈ F × p , hence σ ( t ) = s r (cid:18) y − − cy − (cid:19) l = t (1 + lcy − + . . . )and finally v L ( σ ( t ) − t ) = v L ( t ( lcy − )) = 1 + m, as was to be shown. Proposition 5.2. Let L/K be a finite weakly ramified Galois extension oflocal fields such that the residue field k of K is algebraically closed and theGalois group G is a p -group. Then there exist t ∈ K , t , . . . , t n ∈ L and c , . . . , c n − ∈ k × such that L = K ( t , . . . , t n ) , t := t n is a local parameterof L , t − p − t − = t − and t − pi − t − i = c i − t − i − for i = 2 , . . . , n . Furthermorefor each g ∈ G there is an a ∈ k (depending on g ) such that g ( t ) = t − at = t (1 + at + a t + . . . ) . roof. As L/K is weakly ramified, the Galois group G is in fact an elementaryabelian p -group, i.e. G = C × . . . × C n for some subgroups C , . . . , C n of G of order p . We proceed by induction on n . If n = 0 there is nothing toprove. So let n ≥ 1. We write just C for C n . By the inductive hypothesisthere exist c , . . . , c n − ∈ k × , t ∈ K and t , . . . , t n − in the fixed field L C such that L C = K ( t , . . . , t n − ), the element t n − is a local parameter of L C , t − p − t − = t and t − pi − t − i = c i − t − i − for i = 2 , . . . , n − 1. By Lemma 5.1 thereexist local parameters s and t of L C and L , respectively, such that L = L C ( t )and t − p − t − = s − . If n = 1 we redefine t to be equal to s and put t := t .If n > s − = c n − t − n − + e for some c n − ∈ k × and some e in thevaluation ring O L C of L C . As k is algebraically closed and O L C is Henselianwe can find f ∈ O L C such that f p − f = e . We finally define t n to be theinverse of t − − f and obtain t − pn − t − n = ( t − p − t − ) − ( f p − f ) = c n − t − n − , asdesired. To prove the last assertion we fix g ∈ G . By the inductive hypothesisthere exists b ∈ k such that g ( t n − ) = t n − − bt n − , i.e. g ( t − n − ) = t − n − − b . Thenwe have g ( t − ) p − g ( t − ) = g ( t − pn − t − n ) = g ( c n − t − n − ) = c n − t − n − − c n − b. Hence we have g ( t − ) = t − − a , i.e. g ( t ) = t − at , for some Artin-Schreierroot a of c n − b , as desired.We now return to the situation considered in the main part of this paper. Werecall that a natural number m is called a pole number at a point P ∈ X ifthere exists a meromorphic function f on X whose pole order at P is m andwhich is holomorphic everywhere else. The Riemann-Roch theorem (see [7,Theorem IV 1.3]) implies that every integer m > g X − Corollary 5.3. Suppose that p = 2 and that π : X → Y is weakly ramified.Let P ∈ X ram such that G ( P ) is not cyclic. Then the smallest odd polenumber m at P is congruent to modulo and m − is a pole number aswell. The following proof is a refinement of some arguments given in [12]. Proof. Let ˆ O X,P denote the completion of the local ring O X,P at P . Let t ∈ ˆ O X,P be a local parameter at P and let the maps α and β from G ( P ) to k be defined by the congruences g ( t ) ≡ t + α ( g ) t + β ( g ) t mod ( t ) , ∈ G ( P ). We recall that α is a monomorphism, see the proof of Propo-sition 3.6. We first observe that changing the local parameter t by multi-plication with a unit u ∈ ˆ O × X,P amounts to multiplying α and β with ¯ u − and ¯ u − , respectively, where ¯ u denotes the residue class of u in k × . FromProposition 5.2 we therefore conclude that β = α no matter which localparameter t we choose. We will prove below that we can choose t in such away that β = m +12 α + cα for some constant c ∈ k . As G ( P ) is not cyclic wetherefore obtain m +12 = 1 in F (and c = 0), i.e. m is congruent to 1 modulo4, as stated.Let f be a function in H ( X, O X ( m [ P ])) whose pole order at P is m . Thenthe function f − has a zero at P of order m . As ˆ O X,P is Henselian and m is odd there exists an m th root t of f − in ˆ O X,P . Obviously, t is a lo-cal parameter at P . Let m > m > . . . > m s = 0 be the sequence ofpole numbers at P smaller than m . Note that m , . . . , m s are even by as-sumption. Let f , f , . . . , f s be functions in H ( X, O X ( m [ P ])) of pole orders m , . . . , m s , respectively. Then there exist some units u , . . . , u s ∈ ˆ O × X,P suchthat f = u t m , . . . , f s = u s t ms ; without loss of generality we may assume that¯ u = . . . = ¯ u s = 1 in k × . As m [ P ] is a G ( P )-invariant divisor and f, f , . . . , f s form a basis of H ( X, O X ([ mP ])) (see [21, p. 34]), there exist maps c , . . . , c s from G ( P ) to k such that g ( f ) = c ( g ) f + c ( g ) f + . . . + c s ( g ) f s for all g ∈ G ( P ); i.e. we have(8) g (cid:18) t m (cid:19) = g ( f ) = c ( g ) t m + c ( g ) u t m + c ( g ) u t m + . . . + c s ( g ) u s t m s . On the other hand, using the geometric series and the binomial theorem wederive the following congruence for each g ∈ G ( P ): g (cid:18) t m (cid:19) = g ( t ) − m ≡ t − m (cid:0) α ( g ) t + β ( g ) t (cid:1) − m ≡ t − m (cid:0) − α ( g ) t − β ( g ) t + α ( g ) t (cid:1) m ≡ t − m (cid:18) m (cid:0) − α ( g ) t − β ( g ) t + α ( g ) t (cid:1) + (cid:18) m (cid:19) α ( g ) t (cid:19) ≡ t m + − mα ( g ) t m − + − mβ ( g ) + (cid:0) m +12 (cid:1) α ( g ) t m − mod (cid:18) t m − (cid:19) . Let c ∈ k be defined by the congruence u ≡ ct mod ( t ). 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