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On p-rank representations

The p-rank of an algebraic curve X over an algebraically closed field k of characteristic p>0 is the dimension of the first etale cohomology vector space H^1(X,Z/pZ). We study the representations of finite groups G of automorphisms of X induced on the base extension of this vector space to k, and obtain two main results: First, the sum of the nonprojective direct summands of the representation, i.e. its core, is determined explicitly by local data given by the fixed point structure of the group acting on the curve. As a corollary, we derive a congruence formula for the p-rank. Secondly, the multiplicities of the projective direct summands of quotient curves, i.e. their Borne invariants, are calculated in terms of the Borne invariants of the original curve and ramification data. In particular, this is a generalization of both Nakajima's equivariant Deuring-Shafarevich formula and a previous result of Borne in the case of free actions.


Introduction
We fix an irreducible, smooth and complete curve X over an algebraically closed field k of positive characteristic p. The etale cohomology group H 1 (X et , F p ) is a finite-dimensional vector space over F p . Its dimension, the p-rank of X, is a global invariant of the curve.
If we fix a finite group G of automorphisms of the curve X, then H 1 (X et , F p ) becomes a finite-dimensional representation of G over F p . Moreover, H 1 (X et , F p )⊗ Fp k is a finite-dimensional representation of G over k.
First results on determining this representation up to isomorphism by local invariants of the curve and of the group action have been obtained by Shoichi Nakajima [Nak], under the assumption that G is a p-group, and by Niels Borne [Bor], under the assumption that G operates without fixed points. We continue this tradition, with no assumptions on either the group G or its action.
The local invariants cannot determine the representation completely, as the example of an elliptic curve E over a field of characteristic = 2 shows: Such a curve always allows an automorphism of order 2, which stabilizes exactly 4 points (and the projection to the quotient curve is tamely ramified in these points). Namely, if the curve is given by the equation y 2 = f (x), consider the mapping given by (x, y) → (x, −y). However, the p-rank of E can be 0 or 1, depending on whether this curve is supersingular or not.
Accordingly, our results must be incomplete. Using the language of modular representation theory, what we do determine completely is the core of the representation (i.e., its "non-semisimple" part, cf. Section 1); this is the content of Theorem 4.8. In a sense, this result is surprising, since generally the non-semisimplicity of representations is what makes modular representation theory more difficult than representation theory in characteristic zero. Now the representation is determined completely by its core and the multiplicities of the indecomposable projective summands, which we call Borne invariants of the curve and introduce in Section 5. However, it is impossible to determine these by local invariants, as the above example shows.
The content of Theorem 5.4 is to determine explicitly in terms of local data the Borne invariants of quotient curves X/N with respect to the quotient group G/N , for any normal subgroup N ⊂ G. This gives a procedure for calculating the Borne invariants for those representations of G induced by quotient groups, in terms of the "smaller" curve X/N , and may thus be regarded as a partial solution to the problem of determining Borne invariants. In particular, if N is a p-group this approach gives all Borne invariants, and if G itself is a p-group we recover Nakajima's equivariant Deuring-Shafarevich formula.
This article is a digest of my diploma thesis at ETH Zürich; it is a pleasure to thank my advisor Richard Pink for his support and guidance. In particular, the proof of Proposition 5.2 is entirely due to him.

Modular representation theory of finite groups
It is customary to call a (finite dimensional) representation of a (finite) group a modular representation if the characteristic of the field divides the order of the group. In this situation, the notions of simple and indecomposable module no longer coincide, as would be the case in characteristic 0 by Maschke's theorem. This makes for a richer representation theory, which we will now review.
In the following we shall fix an algebraically closed field k of characteristic p > 0, a finite group G, and denote by k[G] the group ring of G over k. All modules under consideration will be finitely generated left k[G]-modules, and we identify finite dimensional representations of G over k with such modules. All homomorphisms are assumed to be k[G]-linear.
2.1. Definition. A representation is simple (or irreducible) if it is nontrivial and has no proper submodules. We denote the set of isomorphism classes of simple modules by Irr G. A representation is indecomposable if it is nontrivial and admits no proper direct summands. It is projective if the functor Hom(P, −) is exact.
2.2. Theorem (Krull-Schmidt). If M is a representation, and M ∼ = ⊕ m i=1 M i ∼ = ⊕ n j=1 N i are two decompositions with indecomposable summands, then m = n and, after suitable renumbering, M i ∼ = N i for all i.
This theorem allows us to speak of "the" indecomposable direct summands of a given module. To study modules in terms of these summands, we must introduce cores, projective covers, and loop spaces.
2.3. Definition. The (isomorphism class of the) direct sum of the non-projective indecomposable summands of a given representation M is called the core of M , and will be denoted by core(M ). If we have M ∼ = core(M ), we call M itself a core. The (isomorphism class of the) direct sum of the projective indecomposable summands is called the projective part of M .

Definition.
(i) A homomorphism of modules is called essential if it is surjective and its restriction to every proper submodule of its domain is not surjective. (ii) A projective cover of a module M consists of a projective module P and an essential map π : P −→ M .
2.5. Theorem. Any module has a projective cover, which is again finitely generated and unique up to (non-unique) isomorphism. The projective cover of a direct sum is the direct sum of the individual projective covers.
We may thus speak of "the" projective cover P G (M ) of a module. It is known that the number of isomorphism classes of simple modules is finite [Ser77, Chapter 18, Corollary 3]. By contrast, there are in general infinitely many isomorphism classes of indecomposable modules [Ben,Theorem 4.4.4]. However, the projective indecomposable modules are easily described by the following lemma.
2.6. Theorem. The operation "projective cover" induces a bijection between the set Irr G of isomorphism classes of simple modules and the set of isomorphism classes of projective indecomposable modules.
It follows from the above theorem that any module M has a decomposition for unique integers b(M, S) ≥ 0. To know the isomorphism class of M is to know its core and to know the value of these integers.
The core of a module is the degree zero case of a concept of "loop spaces" developed to understand modules "up to projectives". Other authors write Ω 0 G (M ) := core(M ). We will need the degree one case: = Ω G (Ω i−1 G (M )) for i > 1. What follows are some technical lemmas. The reader only interested in the statements of our theorems now has the necessary notation, and may skip the rest of this subsection.

Proposition. Given a module M and a simple module S, we have
Proof. We apply the functor Hom G (−, S) to the exact sequence The lemma follows from the equation i * = 0. Assume that i * = 0, then there exists , this is a contradiction to the fact that π M is essential.
2.9. Corollary. If we write Proof. We fix S ∈ Irr G and calculate by means of the previous lemma: By Schur's lemma, the dimension of Hom G (T, S) is 0 or 1, depending on whether T and S are isomorphic or not. Thus the corollary follows by counting dimensions.
2.10. Proposition. Given a module M , the following are equivalent: Proof. The equivalence of (i) and (ii) follows from [Ben,Prop. 1.6.2] and [Ben,Prop. 3.1.2]. Clearly, (i) and (iii) are equivalent by definition. Since Ω G (M ) = 0 if and only if P G (M ) −→ M is an isomorphism, (i) and (v) are equivalent. The equivalence of (iv) and (v) follows from the claim that Ω G (M ) is a core, which we now prove: Assume that P ⊂ Ω G (M ) ⊂ P G (M ) is a non-zero projective submodule. Then (by the equivalence of (i) and (ii)) P G (M ) decomposes as a direct sum P G (M ) ∼ = P ⊕ Q, and the image of Q in M is all M . This is a contradiction to the fact that The following proposition is well-known; we give a proof here since it will be a central component in the proof of our Theorem 4.8.
Proof. We construct the following commutative diagram The middle vertical arrow exists because P is projective; it is surjective because is projective, so is Q. By the snake lemma, the first vertical arrow is surjective, and its kernel is isomorphic to Q. Since Q is injective (Proposition 2.10), we have an isomorphism which proves the second claim. Since Q is projective, and Ω G (M ) is a core (Proposition 2.10), Ω G (M ) is the core of N .
2.12. Proposition. Let p n be the p-part of the order of G, i.e. |G| = p n k with k ∈ N and p ∤ k. Then the dimension of every projective module is divisble by p n .
2.13. Proposition. Let N ⊂ G be a normal subgroup, and consider the group H := G/N . There is an inclusion Irr H ⊂ Irr G. Given S ∈ Irr G, we have Proof. [Bor,Lemma 2.7].
In the last section we will need the following statement about group cohomology.
2.14. Proposition. Let K ⊂ G be a subgroup with p ∤ [G : K]. Then for any representation M of G, and for all i ≥ 0, the restriction map Proof. [Ben,Corollary 3.6.18].

The p-rank of curves
We continue to assume as given an algebraically closed field k of characteristic p > 0. In this article, a curve signifies a complete, smooth, connected, 1-dimensional variety over k. The (absolute) Frobenius morphism F of such a curve X is the (canonical) morphism which is the identity on topological spaces, and the p-power map on sections of the structure sheaf. It induces maps on the (Zariski) cohomology groups H i (X, O X ). These are additive, but not k-linear maps: They are p-linear, meaning that The only nontrivial case for curves is the induced map on H 1 (X, O X ).
For this, let us review some material on p-linear maps. There is a category of p-linear maps, with objects the pairs (V, F ) consisting of a finite-dimensional vector space V and a p-linear endomorphism F of V . The morphisms in this category are the linear maps on the underlying vector spaces which commute with the given p-linear endomorphisms. Given such an object (V, F ), we set The integer h = dim k V s is called the stable rank of F . The vector space V s is often called the semisimple part of V .
3.1. Proposition. In the above situation, we have (v) F restricted to V s is bijective, F restricted to V n is nilpotent.
(vi) (−) s is an exact functor on the category of p-linear maps.
Proof. See [HW] or [Ser58] for (i) to (v). The last statement is clear, since we assume the maps in the category to be compatible with the respective p-linear maps F .
On the dual vector space V * = Hom k (V, k) we can define a map C by setting C(ψ)(v) := ψ(F (v)) p for v ∈ V and ψ ∈ V * . This map is additive and 1/p-linear, i.e. we have C(λψ) = λ 1/p C(ψ). The decomposition V = V s ⊕ V n corresponds to a decomposition of V * , and C has the same stable rank as F . Since any 1/plinear map can be viewed as the dual of a p-linear map, the structure theory of the previous proposition can be translated to 1/p-linear maps.
3.2. Definition. The p-rank h X of a curve X is the stable rank of the Frobenius morphism on H 1 (X, O X ).
It is clear that we have estimates 0 ≤ h X ≤ g X , where g X = dim k H 1 (X, O X ) is the genus of X.
We would like to know explicitly the dual map of the Frobenius morphism on H 1 (X, O X ). Recall that a rational function t ∈ k(X) is called separating if the field extension k(X)/k(t) is separable. Given a meromorphic differential ω = f · dt, where f ∈ k(X), we may write The Cartier operator on differentials is defined by setting This is well-defined and independent of the choice of t [Ser58].
3.3. Proposition. The dual vector space of H 1 (X, O X ) is, by Serre duality, the vector space H 0 (X, Ω X ). Under this identification, the Cartier operator is the dual map of the Frobenius morphism.
The geometric meaning of the p-rank is the following: There are p hX unramified Galois coverings of the curve X with Galois group F p , up to isomorphism of the covering curve together with the action of F p . More precisely, the group Hom(π et 1 (X), F p ) classifies such covers, and there are natural isomorphisms , compatible with the operation of automorphisms of X on the respective vector spaces. For proofs and further background, we refer to the survey in [Bou]. In this article, we will avoid rationality questions in representation theory by studying . Also, we will study the dual representation H 0 (X, Ω X ) s instead of H 1 (X, O X ) s to simplify computations.

The cores of p-rank representations
Consider a curve X, and a finite subgroup G ⊂ Aut(X). If X is of genus g X ≥ 2, then Aut(X) itself is finite, but even in that case we wish to allow ourselves the freedom of choosing a smaller group.
Given a point x ∈ X, we use the notation v x (−) for the function which assigns to a function, differential or divisor its order at x. 4.1. Proposition. Let D be an effective divisor on a curve X. The Cartier operator C operates on the sheaf Ω X (D). If D is G-invariant, then G also operates on this sheaf, and the two operations commute.
In particular, the vector space H 0 (X, Ω X (D)) s of semisimple differentials with respect to C is a (finite dimensional) representation of G.

Proof.
Consider an open set U ⊂ X and a differential ω ∈ Ω X (D)(U ). For x ∈ U choose a local parameter t at P and write (4.1) ω = (f p 0 + f p 1 · t + · · · + f p p−1 · t p−1 )dt = f · dt as in Section 3, noting that t is separating. Setting n = v x (D) ≥ 0, the assumptions imply that v x (f ) = v x (ω) ≥ −n. Thus the estimate where ⌈y⌉ signifies the least entire number greater than y. Therefore, we have C(ω) ∈ Ω X (D)(U ).
Choose g ∈ G. We have C(ω) g = (f p−1 dt) g = f g p−1 (dt) g . On the other hand, if t is separating, so is s = t g , thus if we write ω g = (· · · + (f g p−1 ) p · s p−1 )ds, we have C(ω g ) = f g p−1 ds = C(ω) g , since the definition of C does not depend on the choice of separating variable.

4.2.
Definition. The module V D := H 0 (X, Ω X (D)) s of the previous proposition is the p-rank representation of G associated to the divisor D.
We introduce the notion D red for the reduced effective divisor associated to D.
The following observation will prove to be helpful: 4.3. Proposition. If D is an effective divisor on X, then the p-rank representation does not depend on the multiplicities of D, i.e. V D = V D red .
Proof. The claim is that semisimple differentials have poles of order at most one. By Proposition 3.1(iii), it is sufficient to prove this claim for differentials of the form ω = C(ω).
If v x (ω) = −n < 0, then as in the proof of Proposition 4.1, we have v x (C(ω)) ≥ ⌈ 1−p−n p ⌉. It is elementary to prove that In the following, we will always assume that D and D are are effective and G-invariant reduced divisors. 4.4. Definition. We will call D sufficiently large with respect to G if it is non-empty, and contains all points of X with nontrivial stabilizer in G. (Nakajima). If D is sufficiently large with respect to G, the prank representation V D is a projective k[G]-module.

Proposition
Proof. Let P ⊂ G be a p-Sylow subgroup of G. By [Nak,Theorem 1] we know that V D is k[P ]-free. This is equivalent to the fact that V D is k[G]-projective [Ben,Corollary 3.6.10].
We will present the core of a p-rank representation as a loop space of the following ramification module. 4.6. Definition. Given a G-invariant effective reduced divisor D as above, we choose a sufficiently large divisor D ⊃ D. The ramification module of V D (with respect to D) is the following: where, for any reduced effective divisor E, by k[E] := x∈E k · x we denote the affine coordinate ring of the reduced subvariety of X associated to E. The core module of V D is the loop space

4.7.
Remark. We note that the module C D does not depend on the choice of D, since enlarging D corresponds to adding to R G,D, D direct summands isomorphic to k[G], and such free summands are annihilated by the loop space operator. Furthermore, is a sum of induced representations of the trival representation.
4.8. Theorem. The core of the p-rank representation associated to a G-invariant effective divisor D (not necessarily reduced) is given by the following formula: Proof. By Proposition 4.3 we may assume that D is reduced. We choose D ⊃ D sufficiently large. Then D D is also reduced and G-invariant, and the residue map induces an exact sequence which is invariant under the operation of G. It induces a long exact sequence which terminates at H 1 (X, Ω X ( D)) = 0 since D = 0. Clearly, we have ker δ = R G,D, D ; hence there is an exact sequence of k[G]-modules In order to extract from this an exact sequence of semisimple parts, we define a 1/plinear map on k[ D D] = d∈ D D k · d by letting it operate on the standard basis {d} d∈ D D as the identity. This induces a 1/p-linear map on R G,D, D , compatible with the operation of G. Since we know that Res(Cω) p = Res(ω) by [Ser58], the above sequence is an exact sequence in the category of 1/p-linear maps. Thus, by the exactness of (−) s , we obtain the exact sequence By Proposition 4.5 the middle term is a projective module, and Proposition 2.11 gives the desired result.
4.9. Remark. If G has no fixed points and D = ∅, the core of the associated p-rank representation is core(V ∅ ) = Ω 1 G (R G,∅, D ) = Ω 2 G (k), since k[ D] ∼ = k[G] r for some r ≥ 1, which implies that core(R G,∅, D ) = Ω 2 G (k). This particular core has been calculated by Borne in [Bor]. 4.10. Remark. Since a projective representation is determined up to isomorphism by its composition factors [Ser77,Chapter 14,Corollary 3], the local invariants used in Theorem 4.8 and the modular character of a p-rank representation determine such a representation up to isomorphism. 4.11. Corollary. Consider a curve X and a finite group G of automorphisms of X. Let r be the number of fixed points of G on X, and let p n be the p-part of the order of G. Then: Proof. We choose a minimal sufficiently large divisor D ⊃ ∅, and set R := R G,∅, D .
Since h X is the dimension of V ∅ and by Theorem 4.8 this module differs from its core only by projective summands, Proposition 2.12 implies that h X ≡ dim Ω G (R) (mod p n ).
Similar reasoning applies to Ω G (R) and R, which have dimensions adding up to the dimension of the projective module P G (R), and shows that If G has fixed points, then dim R = dim k[ D] − 1 = r − 1, if G has no fixed points, then dim R = dim k[ D] − 1 = p n − 1 ≡ r − 1 (mod p n ); hence we can combine the above congruences to obtain the corollary.

Borne invariants of quotient curves
In addition to the notation and conventions of the previous section, we consider a normal subgroup N of G, and the short exact sequence A representation of H lifts to a representation of G, and we obtain an inclusion Irr H ⊂ Irr G of the set of irreducible representations.
Let Y := X/N be the quotient curve, and let π : X −→ Y be the canonical projection. There is a natural induced operation of H on Y . The notation of the last section will sometimes have to be decorated by subscripts X or Y . 5.1. Definition. The Borne invariants b(G, D, S) of the curve X (with respect to G and D) are the multiplicities of the projective indecomposable modules in the p-rank representation of G with respect to D. Thus, we have an isomorphism We simplify notation, setting b(G, S) := b(G, ∅, S).

Proposition (Pink). Let D be an N -invariant reduced effective divisor on X.
There is a natural isomorphism of sheaves for an effective divisor E on Y , which commutes with the Cartier operator and the operation of G. We have E red = π(D) red ∪ {y ∈ Y | π is wildly ramified over y} .
Proof. Pulling back differentials from Y to X via π induces an injective sheaf homomorphism The target of this homomorphism is a torsion-free, coherent sheaf of rank 1; hence there is a unique effective divisor E on Y such that the above homomorphism extends to an isomorphism Ω Y (E) −→ π * Ω X (D) N . By construction of the Cartier operator, this is Cartier-equivariant.
We now proceed to determine E. If R is a local ring, we denote its completion by R. Choose y ∈ Y , we then have for any x ∈ π −1 (y).
= Ω X (D) x Nx Choose x ∈ π −1 (y), and denote again by x and y local parameters at x and y respectively. We have we may express y as y = x n + terms of higher order in x.
It follows that m := v x ( dy dx ) = n − 1 if p ∤ n, and m ≥ n if p | n. Let us set d := v x (D) and write d + m = an + b, for integers a, b ≥ 0 with b ≤ n − 1. The following are equivalent: We now see that Proof. This is a lengthy calculation, which we divide into several steps. We choose a sufficiently large divisor D on X with respect to G, and set E := π * ( D) red ; this is a sufficiently large divisor on Y with respect to H.
Step 1: Since D contains all ramified points, wild or not, Proposition 5.2 implies that (V X, D ) N = V Y, E .
In particular, since V X, D is projective by Proposition 4.5, we may apply Proposition 2.13 to its indecomposable summands to obtain (5.1) b(G, D, T ) = b(H, E, T ) for T ∈ Irr H.
Step 3: On the other hand, let us consider S ∈ Irr G and the short exact sequence Using the fact that the alternating sum of dimensions in an exact sequence is 0, the equality T G = T H and, for y = π(x), the similar equalities V Gx = V Hy , we have Step 4: Finally, combining equations (5.1) and (5.2) (for X and Y ), and (5.3) gives the result. 5.5. Remark. If N is a p-group, then it is known that Irr G = Irr H [Bor, Remark after Definition 2.5]. Thus, in this case, the Borne invariants of Y determine all the Borne invariants of X. In this sense, Theorem 5.4 generalizes the equivariant Deuring-Shafarevich formula of Shoichi Nakajima [Nak], which is the special case of N = G being a p-group. 5.6. Remark. If the operation of G on X is tame, that is if p ∤ |G x | for all x ∈ X, then all higher cohomology groups of the stabilizers G x vanish by Proposition 2.14. Thus d(G, X, S) = dim H 1 (G, S), which proves the conjecture that Niels Borne states in [Bor] after Proposition 2.4.