title: Automorphism group schemes of bielliptic and quasi-bielliptic surfaces
authors: Gebhard Martin
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  First-principles electronic structure, phonon properties, lattice thermal conductivity and prediction of figure of merit of FeVSb half-Heusler 
      Sudhir K. Pandey
      Received: date / Accepted: date
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  § INTRODUCTION
  
  
  We are working over an algebraically closed field k of characteristic p ≥ 0.
  Bielliptic and quasi-bielliptic surfaces form one of the four types of minimal smooth projective surfaces of Kodaira dimension 0. Each bielliptic surface X is a quotient π: E × C → (E × C)/G = X, where E and C are elliptic curves and G ⊆ E is a finite subgroup scheme of E that acts faithfully on C via
  α: G →_C.
  Moreover, the image of α is not entirely contained in the group of translations C. This latter condition guarantees that X is not an Abelian surface. All possible combinations of E,C,G and α have been determined: if p = 0 by Bagnera and de Franchis in <cit.>, and if p ≠ 0 by Bombieri and Mumford in <cit.>.
  
  Similarly, quasi-bielliptic surfaces, which exist if and only if p ∈{2,3}, are obtained by replacing C by a cuspidal plane cubic curve and by imposing on α the condition that the cusp of C is not a fixed point of the group scheme α(G). As in the bielliptic case, it is possible to determine all combinations of E,C,G and α. We refer the reader to <cit.>, but note that not all cases listed there actually occur (see Remark <ref> and Remark <ref>).
  
  Bielliptic and quasi-bielliptic surfaces come with two natural fibrations: one of them is the Albanese map f_E: X → E/G =: E', which is quasi-elliptic if X is quasi-bielliptic, and elliptic if X is bielliptic. All closed fibers of f_E are isomorphic to C, since this holds after pulling back along the faithfully flat morphism E → E/G. The second fibration f_C: X → C/α(G) =: C' ≅ℙ^1 is always elliptic, but has multiple fibers. 
  
  The purpose of this article is to determine the automorphism group scheme _X of X. If p = 0, this has been carried out by Bennett and Miranda in <cit.>. By Proposition <ref>, the actions of the centralizers C__E(G) and C__C(α(G)) on the first and second factor of E × C, respectively, descend to X and we consider them as subgroup schemes of _X via these actions.
   Then, the following theorem is the key result of this article.
  
   
  Let X = (E × C)/G be a bielliptic or quasi-bielliptic surface. Then, there is a short exact sequence of group schemes
  
      1 → (C__E(G) × C__C(α(G)))/G π_*→_X → M → 1,
  
  where G is embedded via id×α and M is a finite and étale group scheme. In particular, _X is of finite type.
  
  
  We refer the reader to Theorem <ref> for a refined statement including a description of the group schemes _X/E' and _X/C' of automorphisms of X over E' and over C', respectively. While the part of _X coming from the centralizers is straightforward to calculate and understand, the part M is more elusive. In particular, we note that M can be non-trivial even in characteristic 0, contrary to what is claimed in <cit.>. Even though we do not see an a priori reason for this, M always comes from automorphisms of E × C. Then, a posteriori, Theorem <ref> (<ref>) gives a complete description of M (see Remark <ref>). By Proposition <ref> and Lemma <ref> (<ref>), we have the following corollary of our analysis.
  
  
  Let X = (E × C)/G be a bielliptic or quasi-bielliptic surface. Then, 
  
      _X ≅ N__E×_C(G)/G,
  
  where N__E×_C(G) is the normalizer of G in _E ×_C.
  
  
  By Corollary <ref>, we have E ≅ (_X^∘)_ and (_X^∘)_ is normal in _X, so we can write the quotient _X/E as an extension of M by (C__E(G)/E) × (C__C(α(G))/α(G)). These group schemes can be calculated explicitly and this will be carried out in Section <ref>.
   In the following Tables <ref>, <ref>, and <ref>, the groups S_n,A_n, and D_2n are the symmetric, alternating, and dihedral group (of order 2n), respectively, and M_2 is the p-torsion subscheme of a supersingular elliptic curve. The stars and daggers in Table <ref> will be explained in Remark <ref>.
  
  
  Let X = (E × C)/G be a bielliptic or quasi-bielliptic surface. Then, depending on the group scheme G and the j-invariants j(E) and j(C), the group schemes C__E(G)/E, C__C(α(G))/α(G) and M are as in Table <ref>, <ref>, and <ref>.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   
  Let us explain the meaning of the stars and daggers in Table <ref>. We denote by O ∈ E the neutral element with respect to the group law on E:
  
  
    * Stars: If p ≠ 2,3 and j(E) = 1728, then every automorphism h_E of (E,O) of order 4 fixes a unique cyclic subgroup of E of order 2. Similarly, if p ≠ 2,3 and j(E) = 0, then every automorphism h_E of (E,O) of order 3 fixes a unique cyclic subgroup of E of order 3. A star after a j-invariant in Table  <ref> denotes that the translation subgroup of G or α(G) coincides with this cyclic subgroup. By Lemma <ref>, this implies that h_E is in the corresponding centralizer. We note that such special 2 and 3-torsion points do not exist if p = 2,3, because (E,O) has more automorphisms in these characteristics.
  
    * Daggers: A dagger after j(E) denotes that the special 2 or 3-torsion points described above maps to a translation in α(G). In these cases, the automorphism (h_E,h_C), where h_E is an automorphism of order 4 or 3 of (E,O) and h_C is translation by a suitable 4 or 3-torsion point, respectively, normalizes the G-action on E × C and hence descends to X. Since (h_E,h_C) does not centralize the G-action, it induces a non-trivial element of M. See the proof of Proposition <ref> for a precise description of the automorphism (h_E,h_C) in these cases.
  
  These cases seem to be missing from <cit.>, since they were not listed in <cit.>, which is why <cit.> differs from our Table <ref>. 
  
  
  
  In the quasi-bielliptic case in characteristic 2, the action of G on E × C sometimes depends on a parameter λ∈ k and so does _X. For an explicit description of λ, see Section <ref>. The parameter λ should be thought of as a replacement for the j-invariant of the curve C.
  
  
  Recall that the space H^0(X,T_X) is the tangent space of _X at the identity. Since E ≅ (_X^∘)_, _X is smooth if and only if h^0(X,T_X) = 1. A careful inspection of Tables <ref>, <ref>, and <ref>, and of the orders of the canonical bundle ω_X determined in <cit.> and <cit.> shows the following.
  
  
  Let X be a bielliptic or quasi-bielliptic surface. Then, the following hold:
  
  
    * h^0(X,T_X) ≤ 3.
  
    * If X is bielliptic or p ≠ 2, then h^0(X,T_X) ≤ 2.
  
    * h^0(X,T_X) = 1 if and only if ω_X ≇_X if and only if _X is smooth.
  
  
  
  
  
   §.§ Acknowledgements
  
  I would like to thank Daniel Boada de Narváez, Christian Liedtke, and Claudia Stadlmayr for helpful comments on a first version of this article and Curtis Bennett and Rick Miranda for interesting discussions. I am indebted to Azniv Kasparian and Gregory Sankaran for pointing out a mistake in an earlier version of this article. I am grateful to the anonymous referee for thorough comments and very helpful suggestions that helped me to improve the exposition and to fix inaccuracies.
  Finally, I would like to thank the Department of Mathematics at the
  University of Utah for its hospitality while this article was written. 
  
  
  
  
  
  
  § NOTATION AND GENERALITIES ON AUTOMORPHISM GROUP SCHEMES
  
  
  Let π: Y → X be a morphism of proper varieties over an algebraically closed field k. There are several k-group schemes of automorphisms associated to π. We follow the notation of <cit.>, which we recall for the convenience of the reader. Throughout, T is an arbitrary k-scheme.
  
  
  
    * The automorphism group scheme _X of X is the k-group scheme whose group of T-valued points _X(T) := (X ×_k T) is the group of automorphisms of X_T := X ×_k T over T. By <cit.>, _X is a group scheme locally of finite type over k. The identity component of _X is denoted by _X^∘.
  
    * The automorphism group scheme _π of the morphism π is the k-group scheme such that _π(T) consists of pairs (g,h) ∈_Y(T) ×_X(T) making the diagram
  
      Y_T [r]^g [d]^π_T    Y_T [d]^π_T
      
      X_T [r]^h    X_T
  
  commutative. In particular, _π(-) is a closed subfunctor of _Y(-) ×_X(-), hence _π is representable by a group scheme locally of finite type over k.
  
    * The group scheme _π comes with projections to _Y and _X.
  If π is faithfully flat, then the first projection _π→_Y is a closed immersion and we will use this to consider _π as a subgroup scheme of _Y. We denote the second projection by π_*: _π→_X.
  
    * The automorphism group scheme _Y/X of Y over X is the k-group scheme whose group of T-valued points _Y/X(T) consists of automorphisms g ∈_Y(T) such that π_T ∘ g = π_T. 
  By definition, there is an exact sequence realizing _Y/X as a subgroup scheme of _π:
  
      1 →_Y/X→_ππ_*→_X
  
  
    * Given a closed subgroup scheme G ⊆_Y, the normalizer N__Y(G) of G in _Y is the k-group scheme whose group of T-valued points is
  
      N__Y(G)(T) = { h ∈_Y(T) | h_T'∘ g ∘ (h_T')^-1∈ G(T')  for all  T' → T  and  g ∈ G(T') }.
  
  The centralizer C__Y(G) of G in _Y is the group scheme whose T-valued points satisfy the stronger condition h_T'∘ g ∘ (h_T')^-1 = g instead. 
  By <cit.>, both N__Y(G) and C__Y(G) are closed subgroup schemes of _Y.
  
  
  
  The notation _Y/X is also a standard notation for the group functor on the category of X-schemes that associates to an X-scheme Z the automorphism group of Y ×_X Z over Z. Since these relative automorphism group functors do not occur in this article, we decided to use the notation introduced above instead of more cumbersome, albeit more precise, notation such as _Y/X/k.
  
  
  
  
  
  § AUTOMORPHISM GROUP SCHEMES OF QUOTIENTS
  
  
  In this section, we study Sequence (<ref>) in the case where π: Y → X is a finite quotient.
  
   
  If G is a finite group scheme acting freely on a proper variety Y such that the geometric quotient π:Y → Y/G =: X exists as a scheme, then we have _Y/X = G and _π = N__Y(G) as subgroup schemes of _Y. In particular, Sequence (<ref>) becomes
  
      1 → G → N__Y(G) π_*→_X.
  
  
  
  First, we show that _Y/X = G. By <cit.>, there is a G-equivariant isomorphism _Y/X≅ Hom(Y,G), where the T-valued points of the latter are Hom(Y × T,G) and G is embedded as G = Hom( k, G).
  Since Y is a proper variety and taking global sections commutes with flat base change, we have H^0(Y × T,_Y × T) = k ⊗_k H^0(T,_T) = H^0(T,_T) for every affine k-scheme T. As G is affine, this implies Hom(Y × T,G) =  Hom(T,G) = G(T), which is what we had to show. 
  
  Next, we show _π = N__Y(G). For this, let h ∈_Y(T) be an automorphism of Y_T. Then, h ∈_π(T) if and only if there is h' ∈_X(T) such that the following diagram commutes
  
      Y_T [r]^h [d]^π_T    Y_T [d]^π_T
      
      X_T [r]^h'    X_T.
  
  Comparing degrees, it is easy to check that the geometric quotient of Y_T by the induced free action of G coincides with π_T, so the morphism π: Y → X is a universal geometric quotient of Y, hence also a universal categorical quotient by <cit.>. Therefore, the automorphism h' exists if and only if π_T ∘ h is G-invariant, that is, if and only if for every T-scheme T' we have π_T'∘ h_T'∘ g = π_T'∘ h_T' for all g ∈ G(T'). This is equivalent to h_T'∘ g ∘ h_T'^-1∈_Y/X(T') = G(T') for all g ∈ G(T'), which is precisely the condition that h ∈ N__Y(G).
  
  
   
  Contrary to the situation for abstract groups, Proposition <ref> typically fails if Y is a non-proper variety or the action of G is not free. Indeed, consider any infinitesimal subgroup scheme G ⊆ PGL_2 of length p. The k-linear Frobenius F: Y := ℙ^1 →ℙ^1 =: X is the geometric quotient for the action of G on ℙ^1 and _Y/X =   PGL_2[F] is the kernel of Frobenius on PGL_2. Moreover, we have _F =  PGL_2. Thus, _Y/X and _F are strictly bigger than G and N_ PGL_2(G) even though F is a G-torsor over an open subscheme of X. 
  
  
  Even though Example <ref> shows that Proposition <ref> fails for non-free actions on curves, we can at least describe the k-rational points in Sequence (<ref>) if the quotient is smooth.
  
   
  Let G be a finite group scheme acting faithfully on a proper integral curve D with geometric quotient φ: D → D':=D/G. Assume that D' is smooth. Then, we have
  
      _D/D'(k) = G(k)  and  _φ(k) = N_(D)(G(k)).
  
  
  
  
  We can consider the four groups as subgroups of the group _k(k(D)) of k-linear field automorphisms of k(D) via the injective restriction map (D) ↪_k(k(D)). We have a tower of field extensions k(D') ⊆ k(D)^G(k)⊆ k(D), where k(D') ⊆ k(D)^G(k) is purely inseparable and  k(D)^G(k)⊆ k(D) is a Galois extension with Galois group G(k). An elementary calculation shows that N__k(k(D))(G(k)) is the subgroup of _k(k(D)) of automorphisms preserving k(D)^G(k). Since D' is a curve, we have k(D') = (k(D)^G(k))^p^n for some n ≥ 0, so an automorphism of k(D) preserves k(D)^G(k) if and only if it preserves k(D'). Hence, N__k(k(D))(G(k)) is also the group of automorphisms of k(D) preserving k(D'). On the other hand, since D' is smooth and proper, _φ(k) consists precisely of those automorphisms of D which, when restricted to k(D), preserve k(D'). Hence, we have
  N_(D)(G(k)) = N__k(k(D))(G(k)) ∩(D) = _φ(k),
  and _D/D'(k) = _k(D')(k(D)) ∩(D) = G(k), which is what we had to show.
  
  
  
  
  § PROOF OF THEOREM <REF>
  
  
  Throughout this section, E and C are integral curves of arithmetic genus 1, and we assume that E is smooth and C is either smooth or has a single cusp as singularity. We choose a point O ∈ E and consider E as an elliptic curve with identity element O. We fix a finite subgroup scheme G ⊆ E, and a monomorphism α: G →_C such that α(G) is not contained in the group of translations of C if C is smooth, and not contained in the stabilizer of the cusp if C is singular.
  In particular, the actions of G on E (via translations) and C (via α) give rise to a product action of G on E × C and we set X := (E × C)/G with quotient map π: E × C → X.
  We have the following commutative diagram with two cartesian squares:
  
  
  
      E × C [rd]^π[d]^π_C[r]^π_E    X ×_E' E [r] [d] @[rd] | □    E [d] 
      
      X ×_C' C [r] [d] @[rd] | □    X [r]_f_E[d]^f_C    E/G =: E' 
      
      C [r]     C/α(G) =: C'.
  
  
  Since G acts freely on E, the map π_E induces isomorphisms on the fibers of E × C → E and X ×_E' E → E and thus, as both maps are flat, the morphism π_E is an isomorphism. The following lemma shows that the automorphism group scheme of X is controlled by the fibrations f_E and f_C.
  
   
  There is a unique action of _X on C' and on E' such that both f_E: X → E' and f_C: X → C' are _X-equivariant. In particular, there are exact sequences
  
      1 →_X/E'→_X (f_E)_*→_E'
  
  and
  
      1 →_X/C'→_X (f_C)_*→_C'.
  
  
  
  The _X^∘-action on X descends to both E' and C' by Blanchard's Lemma <cit.>. Since f_E and f_C are the only fibrations of X and E' ≇C' ≅ℙ^1, it is also clear that the action of the abstract group (X) descends to E' and C'. By <cit.>, this is enough to prove that the whole _X-action descends uniquely to the two curves E' and C'.
  
  With respect to the _X-actions of the previous paragraph, we have _X = _f_C = _f_E, hence the short exact sequences in the statement of the lemma are special cases of Sequence (<ref>).
  
  
  The idea for the proof of Theorem <ref> is to use the isomorphism π_E to lift group scheme actions from X to E × C. By Proposition <ref>, the automorphisms of X that come from E × C are induced by the normalizer N__E × C(G). Therefore, before proving Theorem <ref>, we study N__E × C(G). 
  For the following lemma, note that there is a natural inclusion _E ×_C ↪_E × C given by letting _E and _C act on the first and second factor, respectively. In particular, we can consider C__E(G) × C__C(α(G)) and N__E(G) × N__C(α(G)) as subgroup schemes of _E × C.
  
  
  
  The normalizer N__E × C(G) of G in _E × C satisfies the following properties:
  
  
    *  N__E × C(G) ⊇ C__E(G) × C__C(α(G)).
  
    *   N__E × C^∘(G) = C__E^∘(G) × C__C^∘(α(G)).
  
    *   where ad_h_E and ad_h_C denote conjugation by h_E and h_C, respectively.
  
  
    *  The quotient maps C → C' and E → E' are N__E × C(G)-equivariant.
  
  
  
  Claim (<ref>) is clear.
  
  For Claim (<ref>), the inclusion N__E × C^∘(G) ⊇ C__E^∘(G) × C__C^∘(α(G)) follows from Claim (<ref>) and we have to show the other inclusion.
  By <cit.>, we have _E × C^∘ = _E^∘×_C^∘. In particular, being connected, N__E × C^∘(G) is contained in ^∘_E×^∘_C. 
  Hence, it suffices to show that N__E × C^∘(G) centralizes G on the first factor of E × C. Since G ⊆_E^∘ is a subgroup scheme of the connected commutative group scheme _E^∘, we have _E^∘⊆ C^∘__E(G) ⊆ N^∘__E(G) ⊆_E^∘, so N__E^∘(G) centralizes G. Therefore, N__E × C^∘(G) centralizes G as well.
  
  Claim (<ref>) holds for N__E × C^∘(G) by Claim (<ref>), so it suffices to prove the statement for T =  k. Let h ∈ N__E × C(G)(k). Since h normalizes G, it descends to X by Proposition <ref>. The induced automorphism of X preserves both f_C and f_E, because they are the only fibrations of X and E' has genus 1, while C' ≅ℙ^1.
   Since the projections E × C → E and E × C → C coincide with the Stein factorizations of f_E ∘π and f_C ∘π, respectively, both projections are preserved by h. Hence, h ∈(E) ×(C). An automorphism of this form normalizes the G-action on E × C if and only if it normalizes the G-action on both factors and the automorphisms of G induced by the two conjugations are identified via α. This proves Claim (<ref>).
   
   
  Claim (<ref>) follows from the N__E × C(G)-equivariance of π,f_E, and f_C, since the two projections E × C → E and E × C → C are faithfully flat.
  
  
  Recall that, by Proposition <ref>, the action of N__E × C(G) on E × C descends to X and we denote the corresponding homomorphism by π_*: N__E × C(G) →_X. 
  After these preparations, we are ready to prove the following refined version of Theorem <ref>.
  
  [cf. Theorem <ref>] 
  Let X = (E × C)/G be a bielliptic or quasi-bielliptic surface. Then:
  
  
    *  _X/C' = π_*(C__E(G) × (C__C(α(G)) ∩_C/C')).
  
    *  If G is étale, then _X/C'≅ C__E(G).
  
    *   _X/E' = π_*((C__E(G) ∩_E/E') × C__C(α(G))).
  
    *  _X/E'≅ C__C(α(G)).
  
    *  There is a short exact sequence of group schemes
  
      1 → (C__E(G) × C__C(α(G)))/G π_*→_X → M → 1,
  
  where G is embedded via id×α, M is finite and étale, and M(k) is a subquotient of the groups _E'(k)/((f_E)_* C__E(G)(k)) and N_(C)(α(G)(k))/(C__C(α(G))(k)).
  
    *  If every element of M(k) can be represented by an automorphism of X that lifts to E × C, then
  
      M(k) ≅{(h_E,h_C) ∈ N__E(G) × N__C(α(G)) |α∘ ad_h_E =  ad_h_C∘α}/ C__E(G)(k) × C__C(α(G))(k).
  
  This always holds if X is bielliptic.
  
  
  
  
  For Claim (<ref>), we first show that the _X/C'-action lifts to E × C. For this, choose a general point c ∈ C and let c' ∈ C' be its image in C', so that π restricted to E ×{c} yields an identification of E with the fiber F of f_C over c'. Via this identification, the morphism (f_E)|_F: F → E' is identified with the quotient map E → E/G = E'.
  By Lemma <ref>, the action of _X/C' on X descends to an action on E', and we can use the restriction homomorphism _X/C'→_F and the identification of F with E to get a compatible action of _X/C' on E. Using the isomorphism π_E: E × C → X ×_E' E, we thus obtain an action of _X/C' on E × C that lifts the action of _X/C' on X. Hence, _X/C' is in the image of π_* and it remains to describe its preimage.
  
  By Lemma <ref> (<ref>), a subgroup scheme H ⊆ N__E × C(G) ⊆_E ×_C maps to _X/C' via π_* if and only if it maps to _C/C' under the second projection. To prove Claim (<ref>), we have to show that such an H in fact centralizes G. By Lemma <ref> (<ref>) this holds for H^∘, so we have to prove that H(k) centralizes α(G). 
  Observe that H(k) is mapped to _C/C'(k) under the second projection and _C/C'(k) = α(G)(k) by Proposition <ref>. This, and the fact that G is abelian, implies that H(k) centralizes α(G). Now, Lemma <ref> (<ref>), shows that H(k) centralizes the G-action on E × C.
  
  For Claim (<ref>), it suffices to show that C__C(α(G)) ∩_C/C' = α(G), since there is an isomorphism (C__E(G) ×α(G))/G ≅ C__E(G). This holds if G is étale, for then _C/C' is the constant group scheme associated to α(G) by Proposition <ref>. 
  
  For Claim (<ref>), we only have to show that the _X/E'-action lifts to E × C, because the description of the preimage of _X/E' under π_* works as in the proof of Claim (<ref>). Since _X/E' acts trivially on E', we can use the trivial action of _X/E' on E to define an action of _X/E' on X ×_E' E lifting the action of _X/E' on X. Using the isomorphism π_E: E × C → X ×_E' E, we thus obtain the desired lifting.
  
  For Claim (<ref>), we use that the G-action on E is free. By Proposition <ref> this implies that _E/E' = G. Hence, Claim (<ref>) shows that _X/E' = π_*(G × C__C(α(G)) ≅ C__C(α(G)). 
   
  Next, let us prove Claim (<ref>). By Proposition <ref>, the image of C__E(G) × C__C(α(G)) under π_* is isomorphic to (C__E(G) × C__C(α(G)))/G. By Claim (<ref>) and Claim (<ref>), this image coincides with the subgroup scheme of _X generated by the two normal subgroup schemes _X/C' and _X/E', hence it is itself normal. In particular, the quotient M and the exact sequence in Claim (<ref>) exist. It remains to describe M.
  
  First, consider the exact sequence
  
      1 → C__C(α(G)) →_X (f_E)_*→_E'
  
  from Lemma <ref>, where we used Claim (<ref>) to describe the kernel of (f_E)_*. 
  The homomorphism (f_E)_* identifies the group scheme M with a subgroup scheme of _E'/((f_E)_* C__E(G)). 
  We can choose the image O' ∈ E' of O ∈ E as the neutral element of a group law on E'. Then, we have _E'≅ E' ⋊_E',O' for the finite and étale stabilizer _E',O' of O'. 
  Using the translation action, we can consider E as a subgroup scheme of _E×_C. In fact, we have E ⊆ C__E(G), since G ⊆ E and E is commutative. By Lemma <ref> (<ref>), the induced action on E' coincides with the translation action of E' on itself. Hence, E' ⊆ (f_E)_* C__E(G). In particular, M is a subquotient of _E',O' and hence it is finite and étale.
  
  Let H := (f_E)_*^-1 (_E',O') ⊆_X and let F be the fiber of f_E over O'. Then, the restriction of π to {O}× C gives an identification of C with F such that the quotient map φ: C → C/α(G) = C' is identified with (f_C)|_F: F → C'. In the following, we use this identification to write C instead of F and φ instead of (f_C)|_F. Since _E',O' fixes O', the action of H on X preserves C and the morphism φ is H-equivariant, since f_C is _X-equivariant by Lemma <ref>.
  In other words, the H-action on C factors through _φ. By Claim (<ref>), the kernel of this action is contained in (C__E(G) × C__C(α(G)))/G, hence M is a subquotient of _φ/C__C(α(G)). Now, it suffices to observe that _φ(k) = N_(C)(α(G)(k)), which follows from Proposition <ref>.
  
  Finally, for Claim (<ref>), the description of M follows immediately from Lemma <ref> (<ref>) and Proposition <ref>. By the previous paragraph, we can lift every element of M(k) to an automorphism g ∈(X) mapping under (f_C)_* to the image of _φ(k) →_C'(k). In particular, g lifts to an automorphism h' of X ×_C' C. Since π_C: E × C → E ×_C' C is birational, we obtain a birational automorphism h of E × C. Now, if X is bielliptic, then E × C is smooth, minimal, and non-ruled hence h extends to a biregular automorphism of E × C lifting g.
  
  
  
   
  We remark that if G is not étale, then the group N_(C)(α(G)(k)) will usually be bigger than N__C(α(G))(k). Only later it will turn out that M is in fact a subquotient of the smaller group N__C(α(G))(k)/C__C(α(G))(k) in every case.
  
  
   
  In the case-by-case analysis of quasi-bielliptic surfaces in Section <ref>, we will show that the assumptions of Theorem <ref> (<ref>) are also satisfied for all quasi-bielliptic surfaces, hence the description of M also holds for these surfaces.
  
  
   
  It will follow from the calculations of Section <ref> that Theorem <ref> (<ref>) holds for all bielliptic surfaces. Indeed, the situation where X is bielliptic and G is not étale only occurs if p = 2 and G = μ_2 ×/2 and in this case explicit calculations show that C__C(α(G)) ∩_C/C' = α(G), hence the existence of an isomorphism _X/C'≅ C__E(G) follows from Theorem <ref> (<ref>).
  In particular, for bielliptic surfaces, we always have _X/C'∩_X/E' = π_*(G ×α(G)) ≅ G.
  
  If X is quasi-bielliptic, then it is not true in general that _X/C'≅ C__E(G). Indeed, for example if p = 3 and G = α_3, then C → C' is purely inseparable of degree 3, hence _C/C' = _C[F]. Calculations (see Section <ref>, Case (d)) show that C__C(α_3)^∘≅α_3^2 and C__E(G) ≅ E ⋊/3. Hence, by Theorem <ref> (<ref>), _X/C' is non-reduced while C__E(G) is reduced, so they cannot be isomorphic.
  In particular, for quasi-bielliptic surfaces, _X/C'∩_X/E' can be larger than G.
  
  
  We end this section with a description of (_X^∘)_. We are thankful to the editors for sharing an observation that allowed us to avoid forward references to Section <ref> in the proof of the following proposition.
   
  We have E ≅ (_X^∘)_ and (_X^∘)_ is normal in _X.
  
  
  Since X is not birationally ruled, <cit.> implies that (_X^∘)_ does not contain a connected linear algebraic group, hence, by <cit.>, (_X^∘)_ is an Abelian variety. 
  Then, by <cit.>, the stabilizers of the (_X^∘)_-action on X are finite. Since X is not an Abelian surface, the (_X^∘)_-action on X cannot be transitive, hence (_X^∘)_ is either trivial or an elliptic curve. Now, by Theorem <ref>, the action of E on the first factor of E × C descends to a faithful action of E on X. This yields a monomorphism, and hence an isomorphism, of elliptic curves E → (_X^∘)_.
  To see that (_X^∘)_ is normal in _X, let (_X^∘)_ ant be the largest anti-affine subgroup scheme of _X^∘ (see <cit.>). By <cit.>, (_X^∘)_ ant is smooth and connected, and it contains (_X^∘)_, since the latter is anti-affine. Hence, (_X^∘)_ = (_X^∘)_ ant. By <cit.>, (_X^∘)_ ant = (_X^∘)_ is normal in _X^∘, hence also normal in _X. This finishes the proof.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § COMPUTING CENTRALIZERS AND NORMALIZERS
  
  
  First, recall that if D is an integral curve of arithmetic genus p_a(D) = 1 with smooth locus D^sm and with a chosen point O ∈ D^sm, then there is a decomposition _D = D^sm⋊_D,O, where the group scheme _D,O of automorphisms fixing O acts on the group scheme D^sm of translations t_s by points s ∈ D^sm via g ∘ t_s ∘ g^-1 = t_g(s). This is because _D,O acts on D^sm via group scheme automorphisms, see <cit.> and <cit.>. We use the letter D here, since, using the terminology of Section <ref>, the following Lemma <ref> applies to both D = E and D = C.
  
  
  
  
  Let D be an integral curve with p_a(D) = 1 and with a chosen point O ∈ D^sm. Let G_1 ⊆_D,O and G_2 ⊆ D^sm be subgroup schemes.
  Then, the following are equivalent:
  
  
    *  G_2 normalizes G_1.
  
    *  G_1 and G_2 commute.
  
    *  G_2 ⊆ D^G_1.
  
  
  
  
  Note that if T is a k-scheme, g ∈ G_1(T), and t_s ∈ G_2(T), then we have
  
      t_s  ∘ g ∘ t_-s = t_s - g(s)∘ g.
  
  In particular, if s = g(s) for all t_s ∈ G_2(T), then G_1 and G_2 commute, hence (<ref>)⇒(<ref>). The implication (<ref>)⇒(<ref>) is clear, hence it remains to prove (<ref>)⇒(<ref>):
  if G_2 normalizes G_1, then Equation (<ref>) shows that t_s - g(s)(O_T') = O_T' for all T-schemes T' and t_s ∈ G_2(T). This is only possible if s = g(s), hence G_2(T) ⊆ D^G_1(T).
  
  
  
  
   §.§ Bielliptic surfaces
  
  We use the notation of Section <ref> and Lemma <ref>, but assume that D is smooth. In each of the cases p ≠ 2,3, p = 3 and p = 2, we will recall the structure of the subgroup scheme _D,O⊆_D. Moreover, for every commutative subgroup H ⊆_D,O, we list the fixed locus D^H and, if _D,O is non-commutative, also the centralizer and normalizer of H in Lemma <ref>, Lemma <ref>, and Lemma <ref>. All of this is well-known and elementary to check, and we refer the reader to <cit.> for details. 
  Together with Lemma <ref>, it will be straightforward to calculate the groups C__E(G) and C__C(α(G)) of Theorem <ref> and produce Table <ref>. We will leave the details to the reader, but we will explain how the calculations work in Example <ref>. Using Theorem <ref> (<ref>), we calculate M in every case. The results of the calculations of this section are summarized in Table <ref>.
  To simplify notation, we define 
      N := N_(C)(α(G)(k))/(C__C(α(G))(k)).
  
  
  
  
    §.§.§ Characteristic p ≠ 2,3
  
  By Bombieri and Mumford <cit.>, the group schemes G leading to bielliptic surfaces X = (E × C)/G are the seven groups 
  
      /2,/3,/4,/6,(/2)^2,(/3)^3,/4×/2.
   The translation subgroup of α(G) is trivial in the first four of these cases, and isomorphic to the group /2,/3, or /2 in the other three cases, respectively.
  
  
  
  The non-trivial commutative subgroup schemes H of _D,O and their fixed loci D^H are as in Table <ref>.
  
  
  
  
  
  We explain how to calculate the centralizers in the case where the group is G = /2.
  
  For the calculation of C__E(G), recall that translations in E always commute with G. Next, by Lemma <ref>, an automorphism h_E ∈_E,O commutes with G precisely if G ⊆ E^h_E. Now, we apply Lemma <ref>:
  if j(E) ≠ 1728, or j(E) = 1728 and G does not coincide with the fixed locus of an automorphism h_E of order 4 in _E,O, then C__E(G)/E ≅/2. This is Case a) in the first row of Table <ref>. 
  If j(E) = 1728 and G = E^h_E, then C__E(G)/E ≅/4. This is Case b) in the first row of Table <ref> and it seems to be missing from <cit.>, see also Remark <ref>.
  
  For the calculation of C__C(α(G)), we apply Lemma <ref> to find the subgroup of translations of C that commute with α(G). By Lemma <ref>, this group is isomorphic to (/2)^2. Next, by Lemma <ref>, the group α(G) is in the center of _C,O, so C__C(α(G)) ≅ (/2)^2 ⋊_C,O. Now, if j(E) ≠ 0,1728, then C__C(α(G))/α(G) ≅ (/2)^2, if j(E) = 1728, then C__C(α(G))/α(G) ≅ D_8, and if j(E) = 0, then C__C(α(G))/α(G) ≅ A_4. These are the Cases i), ii), and iii) in the first row of Table <ref>.
  
  
  Similarly, one can calculate the centralizers of G and α(G) for all seven possibilities of G. They are listed in Table <ref>. As for the group N, we have the following:
  
   
   The group N is as in Table <ref>.
  
  
  
  
  
  
  If α(G) does not contain translations, then N__C(α(G)) = C__C(α(G)) by Lemma <ref> and because _C,O is abelian. Hence, N is trivial in these cases.
  
  If G = (/2)^2, then conjugation by N__C(α(G)) fixes the unique non-trivial 2-torsion point c in α(G). By Lemma <ref> and Lemma <ref>, this implies |N| | 2. The non-trivial element of N is induced by a 4-torsion point c' of C with 2c' = c.
  
  If G = (/3)^2, then conjugation by N__C(α(G)) preserves the subgroup ⟨ c ⟩⊆α(G) generated by a non-trivial 3-torsion point c in α(G). Thus, the action of N__C(α(G)) descends to C” := C/⟨ c ⟩. There, it maps to the normalizer in _C” of a subgroup G”⊆_C”,O” of order 3, where O” is the image of O.  By Lemma <ref> and Table <ref>, the normalizer of G” is isomorphic to /3⋊/6, where G” sits inside the second factor. Thus, N is isomorphic to a subgroup of S_3. One can check that the involution in _C,O and a 3-torsion point not contained in ⟨ c ⟩ induce non-trivial elements of N, hence N ≅ S_3.
  
  Finally, if G = /4×/2, then, again, conjugation by N__C(α(G)) fixes the unique non-trivial 2-torsion point c in α(G). In this case, however, the involution in α(G) ∩_C,O is the unique element in α(G) which is divisible by 2, hence it is also fixed by N__C(α(G)). Thus, by Lemma <ref>, a translation can be in N__C(α(G)) only if it is a translation by a 2-torsion point. The non-trivial 2-torsion point that commutes with α(G) is already contained in α(G), hence N ≅/2 is generated by one of the other two non-trivial 2-torsion points.
  
  
  
  The cases where M is non-trivial are precisely the following:
  
  
    *  G = (/2)^2, j(E) = 1728, and the fixed points G^h_E of the automorphism h_E of order 4 in _E,O act as translations on C. In this case, M = /2.
  
    *  G = (/3)^2, j(E) = 0, and the fixed points G^h_E of the automorphism h_E of order 3 in _E,O act as translations on C. In this case, M = /3.
  
  
  
  
  
  Assume that M is non-trivial.
  By Theorem <ref> (<ref>) and Table <ref>, this can only happen if G ∈{ (/2)^2,(/3)^2, /4×/2}.
  
  Assume G = (/2)^2. By Theorem <ref> (<ref>) and Table <ref>, we have |M| | 2. If j(E) ≠ 1728, then _E'/((f_E)_* C__E(G)) has odd order, hence M = {1} by Theorem <ref> (<ref>). 
  If j(E) = 1728, we use Theorem <ref> (<ref>):
  by our description of the centralizers and normalizers, both N__E(G)/C__E(G) and N__C(α(G))/C__C(α(G)) are isomorphic to /2 and every non-trivial element of M(k) can be represented by h = (h_E,h_C), where h_E ∈_E,O is of order 4 and h_C is translation by a non-trivial 4-torsion point such that h_C^2 ∈α(G). By Lemma <ref> and Table <ref>, we have α∘ ad_h_E =  ad_h_C∘α if and only if the fixed point of h_E maps via α to the unique translation in α(G). This is Case (<ref>).
  
  Next, assume G = (/3)^2. Let h = (h_E,h_C) be an automorphism of E × C lifting a non-trivial element of M(k). By our description of C__E(G) and N, we may assume that h_C is either the involution in _C,O or translation by a 3-torsion point c' ∉α(G), and that h_E ∈_E,O.
  If h_E is an involution, then ad_h_E fixes only the identity in G, while ad_h_C has more fixed points on α(G). Hence, by Theorem <ref> (<ref>), h does not normalize the G-action on E × C in this case, a contradiction to Proposition <ref>.
  Thus, we may further assume that j(E) = 0 and h_E has order 3. Then, we may assume that h_C is translation by c'. By Lemma <ref> and Table <ref>, we have α∘ ad_h_E =  ad_h_C∘α if and only if the fixed points of h_E on E map to translations in α(G). This is Case (<ref>).
  
  Finally, assume G = /4×/2. Assume M is non-trivial and, using Theorem <ref> (<ref>), let h = (h_E,h_C) be an automorphism mapping to a non-trivial element in M(k). We may assume that h_E ∈_E,O is the involution and h_C is a translation by one of the 2-torsion points not contained in α(G). Observe that ad_h_E maps elements of order 4 in G to their inverses while ad_h_C maps the automorphism σ of order 4 in α(G) ∩_C,O to σ∘ t_c, where c is the non-trivial 2-torsion point in α(G). Hence, we have α∘ ad_h_E≠ ad_h_C∘α. This contradiction shows that M = {1} in this case. 
  
   
  
   
  
  
    §.§.§ Characteristic p = 3
  
  By Bombieri and Mumford <cit.>, the groups G leading to bielliptic surfaces X = (E × C)/G are the six groups 
  
      /2,/3,/4,/6, (/2)^2,/4×/2.
   The translation subgroup of α(G) is trivial in the first four of these cases, and isomorphic to /2 in the other two cases.
  
  
  The non-trivial commutative subgroup schemes H of _D,O, their fixed loci D^H, centralizers C__D,O(H) and normalizers N__D,O(H) are as in Table <ref>.
  
  
  
  As in characteristic ≠ 2,3, it is straightforward to calculate the centralizers of G and α(G) and they are listed in Table <ref>.  
  
  
   The group N is as in Table <ref>.
  
  
  
  
  
  If α(G) does not contain translations, then a translation in _C normalizes α(G) if and only if it centralizes α(G) by Lemma <ref>. Thus, in these cases, N can be read off from the last two columns of Table <ref>. The proof of the two remaining cases is the same as for Lemma <ref>.
  
  
  
  
  The cases where M is non-trivial are precisely the following:
  
  
    * G = (/2)^2 and j(E) = 0. In this case, M ≅/2.
  
    * G ∈{/3, /6}. In these cases, M ≅/2.
  
  
  
  
  
  By Theorem <ref> (<ref>) and Table <ref>, we may assume G ∈{/3,/6 , (/2)^2, /4×/2}.
  For G ∈{(/2)^2,/4×/2}, the proof is essentially the same as in Proposition <ref>. The only difference is that every non-trivial 2-torsion point of E is fixed by some automorphism of order 4 in _E,O, so we do not have an extra condition as in Proposition <ref>.
  
  For G ∈{/3, /6}, it suffices to find a non-trivial element in M. 
  By Lemma <ref>, there is an element h_C ∈ N__C,O(α(G)) of order 4 such that ad_h_C swaps the two generators of α(G). The inversion h_E on E induces the same action on G. By Theorem <ref> (<ref>), this shows M ≅/2.
  
  
  
  
    §.§.§ Characteristic p = 2
  
  By Bombieri and Mumford <cit.>, the group schemes G leading to bielliptic surfaces X = (E × C)/G are the six group schemes 
  
      /2,/3,/4,/6, μ_2 ×/2,(/3)^2.
   The translation subgroup scheme of α(G) is trivial in the first four of these cases, and isomorphic to μ_2 and /3, respectively, in the other two cases.
  
  
  
  The non-trivial commutative subgroup schemes H of _D,O, their fixed loci D^H, centralizers C__D,O(H) and normalizers N__D,O(H) are as in Table <ref>.
  
  
  
  As before, it is straightforward to calculate the centralizers of G and α(G) and they are listed in Table <ref>.  
  
  
   The group N is as in Table <ref>.
  
  
  
  
  
  If α(G) does not contain translations, then a translation in _C normalizes α(G) if and only if it centralizes α(G) by Lemma <ref>. Thus, in these cases, N can be read off from the last two columns of Table <ref>. 
  For G = (/3)^2, the proof is the same as for Lemma <ref>.
  Finally, if G = μ_2 ×/2, then N_(C)(α(G)(k)) is generated by α(G)(k) and the unique non-trivial 2-torsion point in C(k) by the same argument as in the proof of Lemma <ref>. Translation by this 2-torsion point commutes with α(G), hence N is trivial.
  
  
  
  
  The cases where M is non-trivial are precisely the following:
  
  
    * G = (/3)^2 and j(E) = 0. In this case, M = /3.
  
    * G = /4. In this case, M = /2.
  
  
  
  
  By Theorem <ref> (<ref>) and Table <ref>, we may assume G ∈{(/3)^2,/4}.
  The proof for G =(/3)^2 is the same as in Proposition <ref> with the only difference that every non-trivial 3-torsion point in E is fixed by some automorphism of order 3, so we do not have an extra condition as in Proposition <ref>.
  
  If G = /4, consider the automorphism h = (h_E,h_C) of E × C where h_C ∈ N__C,O(α(G)) is of order 4 and not contained in α(G) and h_E is the inversion involution on E. 
  By Lemma <ref> (<ref>), h normalizes the G-action on E × C and, by Proposition <ref>, induces a non-trivial element of M. Hence, we have M ≅/2.
  
   
  
  
  
  
  
  
  
   §.§ Quasi-bielliptic surfaces
  
  
  In the case of quasi-bielliptic surfaces, E is still smooth, so the group C__E(G)/E can be calculated using the results of the previous section. We will thus focus on the calculation of C__C(α(G))/α(G) and M. We identify the smooth locus of C with 𝔸^1 =  k[t] and use the description of automorphisms of 𝔸^1 coming from C given in <cit.>.
  
  
  
    §.§.§ Characteristic p = 3
  
  
  By <cit.> the T-valued automorphisms of 𝔸^1 coming from C are of the form
  
      t ↦ bt + c + dt^3
  
  with b ∈_m(T), c,d ∈_a(T) and d^3 = 0. By <cit.>, the subgroup schemes α(G) leading to quasi-bielliptic surfaces are the following:
  
  
    * μ_3: t ↦ at + (1-a)t^3 with a^3 = 1
  
    * μ_3 ×/2: μ_3 as in (a) and t ↦± t.
  
    * μ_3 ×/3: μ_3 as in (a) and t ↦ t + i with i^3 = i
  
    * α_3: t ↦ t + at^3 with a^3 = 0
  
    * α_3 ×/2: α_3 as in (d) and t ↦± t
  
  
   
  As noted in <cit.>, Case (f) of <cit.> does not exist, because the group scheme given there is isomorphic to α_9 and thus not a subscheme of an elliptic curve.
  
   
  Now, let us calculate C__C(α(G)) and M for the surfaces in Case (a),...,(e). To this end, we take a k-scheme T and arbitrary elements g ∈α(G)(T) as in the above list and h ∈_C(T) as in (<ref>). One can check that the inverse of h is given by
  
      t ↦ b^-1t + b^-4(c^3d - b^3c) - b^-4dt^3
  
  
  
  
    * We calculate
  
      h ∘ g ∘ h^-1: t ↦ at + (1-a)b^-1(c^3-c) + (1-a)(b^2 - b^-1d)t^3.
  
  Thus, h normalizes α(G) if and only if it centralizes α(G) if and only if c^3 = c and b^3 = d + b. Taking the cube of the second equation, we obtain b^6 = 1. Thus, the centralizer of α(G) is the group scheme of maps
  
      t ↦ bt + i + (b^3 - b)t^3  with  b^6 = 1  and  i^3 = i.
  
  This group scheme is isomorphic to μ_3 × S_3. Therefore, we have C__C(α(G))/α(G) ≅ S_3. 
  
  To calculate M, first note that |M| | 2, since E and E' are ordinary, M is a subquotient of _E'/((f_E)_* C__E(G)) by Theorem <ref> (<ref>), and _E'^∘⊆ ((f_E)_* C__E(G)). If M is non-trivial, then it can be represented by an automorphism g ∈(X) that induces the inversion involution on E'. This involution can be lifted to E, hence g lifts to an automorphism of E × C. However, by the above calculations there is no element of (C) that acts as an inversion on α(G). So, Theorem <ref> (<ref>) shows that M is trivial.
  
  
  
  
    * Since μ_3 is the identity component of μ_3 ×/2, the normalizer of μ_3 ×/2 in _C is contained in the normalizer of μ_3 in _C. By Case (a), the latter is isomorphic to μ_3 × S_3. Thus, N__C(α(G)) equals the normalizer of μ_3 ×/2 in μ_3 × S_3, hence N__C(α(G)) = C__C(α(G)) = α(G). By the same argument as in (a), we also have M = {1}.
  
  
  
  
    * Similar to case (b), we obtain that C__C(α(G))/α(G) = {1} and M = {1}.
  
  
  
  
    * We calculate
  
      h ∘ g ∘ h^-1: t ↦ t + ab^-1c^3 + ab^2t^3.
  
  Thus, h normalizes α(G) if and only if c^3 = 0, and it centralizes α(G) if and only if additionally  holds. Thus, C__C(α(G)) is a semi-direct product α_3^2 ⋊/2 and N__C(α(G)) is a semi-direct product (α_3)^2 ⋊_m.
  In particular, we have C__C(α(G))/α(G) ≅α_3 ⋊/2.
  
  Next, we calculate M. Using  Lemma <ref> and Lemma <ref>, one can check that C__E(G)/E ≅/3.
  Thus, there is an isomorphism _E'/((f_E)_* C__E(G)) ≅_E',O/(/3) ≅/4, where we use the structure of _E',O recalled in Lemma <ref>. So, by Theorem <ref> (<ref>), M is a subquotient of /4.
  
  Choose any automorphism h_E ∈_E,O of order 4. Since α_3 ⊆ E is the kernel of Frobenius, it is preserved by h_E. Moreover, by Lemma <ref> and Lemma <ref>, the centralizer of α_3 in _E,O has order 3, so conjugation by h_E induces an automorphism of α_3 of order 4. By the calculations of the first paragraph, we have a surjection N__C(α(G)) →_α(G)≅𝔾_m, hence we can find an h_C ∈ N__C(α(G))(k) such that 
  h = (h_E,h_C) ∈ N__E ×_C(G)(k) by Lemma <ref> (<ref>). By Proposition <ref>, h descends to an automorphism of X that induces an element of order 4 in M. Therefore, we have M ≅/4.
  
  
  
  
    * Let g: t ↦ - t. Then,
  
      h ∘ g ∘ h^-1: t ↦ -t + b^-1c - b^-4c^3d.
  
  Since α_3 is the identity component of α_3 ×/2, we can use the results of (d) to deduce that h normalizes α(G) if and only if c = 0 and it centralizes α(G) if and only if additionally b^2 = 1. Thus, we get C__C(α(G)) ≅α_3 ×/2 and the normalizer of α(G) is N__C(α(G)) ≅α_3 ⋊_m. In particular, C__C(α(G))/α(G) = {1}.
  
  Since the automorphism g generates the group α(G)(k), the calculation of the previous paragraph also shows that N_(C)(α(G)(k)) = _m(k). Thus, M is isomorphic to a subquotient of _m(k) by Theorem <ref> (<ref>) and, in particular, the order of M is prime to 3. By the same theorem, M is also a subquotient of _E'/((f_E)_* C__E(G)), which is isomorphic to /3⋊/4 since C__E(G) ≅ E in the current case. Hence, M is a subquotient of /4. Using the same construction as in (d), one can show that M ≅/4.
  
  
  
  
  
  
    §.§.§ Characteristic p = 2
   
  By <cit.> the T-valued automorphisms of 𝔸^1 coming from C are of the form
  
      t ↦ bt + c + dt^2 + et^4
  
  with b ∈_m(T), c,d,e ∈_a(T) and d^4 = e^2 = 0. The subgroup schemes α(G) leading to quasi-bielliptic surfaces are the following, where λ∈ k:
  
  
    * μ_2: t ↦ at + λ(a+1)t^2 + (a+1)t^4 with a^2 = 1.
  
    * μ_2 ×/3: μ_2 as in (a) with λ = 0 and t ↦ω t, where ω^3 = 1.
  
    * μ_2 ×/2: μ_2 as in (a) and t ↦ t + ζ, where ζ is a fixed root of x^3 + λ x + 1.
  
    * μ_4: t ↦ at + (a+a^2)t^2 + (1+a^2)t^4 with a^4 = 1
  
    * μ_4 ×/2: μ_4 as in (d) and t ↦ t + 1.
  
    * α_2: t ↦ t + λ a t^2 + at^4 with a^2 = 0, and with λ∈{0,1}.
  
    * α_2 ×/3: α_2 as in (f) with λ = 0 and /3 as in (b)
  
    * M_2: t ↦ t + a + λ a^2 t^2 + a^2 t^4 with a^4  =0, and with λ≠ 0.
  
  
  
   
  In <cit.>, Bombieri and Mumford do not give restrictions on the parameter λ∈ k in Case (f). However, all the α_2-actions with λ≠ 0 described by them are conjugate, so we may assume λ∈{0,1}. For more details, we refer the reader to the discussion of Case (f) below.
  
  
  
  To see that the group scheme in Case (h) is indeed M_2, denote the transformation in Case (h) associated to z_i with z_i^4 = 0 by t_z_i. Observe that t_z_1∘ t_z_2 = t_z_1 + z_2 + λ z_1^2 z_2^2. So, if G =  k[z]/z^4 is the group scheme in Case (h), then its co-multiplication is given by 
      z ↦ z_1 ⊗ 1 + 1 ⊗ z_2 + λ z_1^2 ⊗ z_2^2.
  
  Consider the supersingular elliptic curve E with affine Weierstrass equation y^2 + λ y = x^3 and set z = x/y,w = 1/y, so that the equation becomes z^3 = w + λ w^2. Then, the 2-torsion subscheme M_2 of E is the subscheme given by z^4 = w^2 = 0, and thus w = z^3. By <cit.> the co-multiplication on k[z]/z^4 induced by the group structure on E is precisely the one described above. Hence, we have G = M_2.
  
  For later use, we note that by <cit.>, the group of automorphisms of E preserving w = z = 0 is given by the substitutions x ↦ b^2x + c^2, y ↦ y + b^2cx +d with b^3 = 1, c^4 + λ c = 0 and d^2 + λ d + c^6 = 0. In particular, they act on k[z]/z^4 as
  
      z ↦b^2x + c^2/ y + b^2cx +d = b^2z + c^2w/1+ b^2cz + dw = (b^2z + c^2z^3)(1 + b^2cz + dz^3)^3 = b^2z + bcz^2.
  
  
  In particular, if we think of the substitutions in Case (h) above as defining a homomorphism of group schemes E ⊇ M_2 α→_C, then precomposing α with ad_h_E where h_E ∈_E,O is as described in the previous paragraph, then α∘ ad_h_E corresponds to M_2 acting on C as 
  
      t ↦ t + (b^2a + bca^2) + b λ a^2t^2 + ba^2t^4.
  
  
  
  
   
  Now, we are prepared to calculate C__C(α_C) and M in Cases (a),...,(h). As in characteristic 3, we take a k-scheme T and arbitrary elements g ∈α(G)(T) as in the above list and h ∈_C(T) as in (<ref>). One can check that the inverse of h is given by
  
      t ↦ b^-1t + b^-7(b^6c + b^2c^4e + b^4c^2d + c^4d^3) + b^-3 d t^2 + b^-7(d^3 + b^2e)t^4.
  
  
  
  
    * We calculate
  
      h ∘ g ∘ h^-1:     t ↦    at + (a+1)b^-1(c + λ c^2 + c^4)  + (a+1)(b^-1d + λ b)t^2 
             + (a+1)(b^-1e + λ b^-1 d^2 + b^3)t^4.
  
  Thus, h normalizes α(G) if and only if it centralizes α(G) if and only if
  
      c^4 + λ c^2 + c    =    0, 
      
      d    =   λ (b^2 + b),  and
      
      e    =    b^4 + b + λ^3 (b^4 + b^2) .
  
  
  If λ≠ 0, the fourth power of (<ref>) yields b^4 = 1, while the square of (<ref>) yields b^6 = 1, so we have b^2 = 1. Hence, in this case C__C(α(G)) is the group scheme of maps
  
      t ↦ bt + c + λ(1 + b)t^2 + (1+b)t^4  with  b^2 = 1  and  c^4 + λ c^2 + c = 0 ,
  
  which is isomorphic to (/2)^2 ×μ_2 since c^4 + λ c^2 + c has 4 distinct roots. Therefore, we have C__C(α(G))/α(G) ≅ (/2)^2.
  
  If λ = 0, then d = 0, and the square of (<ref>) yields b^6 = 1. Thus, the centralizer of α(G) is the group scheme of maps
  
      t ↦ bt + c + (b + b^4)t^4  with   b^6 = 1  and  c^4 = c,
  
  which is isomorphic to A_4 ×μ_2. We deduce that C__C(α(G))/α(G) ≅ A_4.
  
  In both cases λ≠ 0 and λ = 0, note that _E' = (f_E)_* C__E(G), so M = {1} follows immediately from Theorem <ref> (<ref>).
  
  
  
  
  
    * Since μ_2 is the identity component of the group scheme /3×μ_2, it suffices to calculate the normalizer of /3×μ_2 in A_4 ×μ_2, which is equal to its centralizer and both are equal to /3×μ_2. In particular, C__C(α(G))/α(G) = {1}. To see that M = {1}, one can use the same arguments as in Case (a) in characteristic 3 to show that the action of M lifts to E × C. Since N__C(α(G)) = C__C(α(G)), Theorem <ref> (<ref>) shows that M is trivial.
  
  
  
  
  
    * We take the centralizer of /2×μ_2 in (/2)^2 ×μ_2 if λ≠ 0 and in A_4 ×μ_2 if λ = 0. Both are equal to the normalizer and also equal to (/2)^2 ×μ_2. Thus, C__C(α(G))/α(G) ≅/2. As in Case (a), we have M = {1}.
  
  
  
  
  
    * We calculate
  
      h ∘ g ∘ h^-1: t    ⟼     at + (a+1)b^-1(c + ac^2 + (a+1)(b^-2c^2d + b^-2c^4d + c^4))  + (a+a^2)(b^-1d + b)t^2 
            + (a+1)(b^-1e + a b^-1d^2 + (a+1)(b^-3d^3 + bd + b^3))t^4.
  
  Thus, h normalizes α(G) if and only if it centralizes α(G). For h to centralize the subgroup scheme where a^2 = 1, we obtain the conditions
  
      c + c^2    =    0, 
      
      d    =    b^2 + b,  and
      
      e    =    b^4 + b^2.
  
  Since d^4  = 0, this implies b^4  = 1. Plugging these conditions back into the equation for h ∘ g ∘ h^-1, it turns out that the subgroup scheme of transformations satisfying these conditions centralizes all of α(G).
  Therefore, the centralizer C__C(α(G)) is given by the group scheme of maps
  
      t ↦ bt + c + (b + b^2)t^2 + (1 + b^2)t^4  with  b^4 = 1  and  c ∈{0,1},
  
  which is isomorphic to μ_4 ×/2. Therefore, C__C(α(G))/α(G) ≅/2. By the same argument as in Case (b), we have M = {1}.
  
  
  
  
  
    * Since μ_4 is the identity component of μ_4 ×/2, we can use the computations of (d) to immediately conclude that centralizer and normalizer of α(G) are both equal to μ_4 ×/2 and thus C__C(α(G))/α(G)= {1}. Also, M = {1} follows by the same argument as in Case (b).
  
  
  
  
  
  
    * We calculate
  
      h ∘ g ∘ h^-1: t ↦ t + ab^-1(λ c^2 + c^4) + λ ab t^2 + a (λ b^-1d^2 + b^3)t^4.
  
  This shows that all the α_2-actions with λ≠ 0 are conjugate to the one with λ = 1 by conjugating with the map t ↦√(λ)t. Hence, we may assume λ∈{0,1}.
  
  Suppose λ = 1. Then, h normalizes α(G) if and only if it satisfies the conditions
  
      c^2 + c^4    =    0,  and
      
      d^2    =    b^4 + b^2 .
  
  Squaring (<ref>), we get b^4 = 1. We also note that h centralizes α(G) if and only if additionally b = 1. Therefore, the normalizer of α(G) is the group scheme of maps
  
      t ↦ bt + c + dt^2 + et^4  with  b^4 = 1, c^4 = c^2, d^2 = b^4 + b^2,  and  e^2 = 0,
  
  which is isomorphic to a semi-direct product (α_2^3 ⋊/2) ⋊μ_4 and the centralizer of α(G) is isomorphic to a semi-direct product α_2^3 ⋊/2. Hence, we have C__C(α(G))/α(G) ≅α_2^2 ⋊/2. To calculate M, note that _E'/((f_E)_* C__E(G)) ≅/3, so |M| | 3 by Theorem <ref> (<ref>). Since E → E' is purely inseparable we can lift the M-action from E' to E and hence to E × C, where it normalizes G. Since N__C(α(G))/C__C(α(G))(k) ≅μ_4(k) is trivial, this shows that M = {1}.
  
  If λ = 0, then h normalizes α(G) if and only if c^4 = 0 and it centralizes α(G) if and only if additionally b^3 = 1. Thus, the normalizer of α(G) is the group scheme of maps
  
      t ↦ bt + c + dt^2 + et^4  with  c^4 = d^4 = e^2 = 0,
  
  which is isomorphic to (α_4 ⋊ A) ⋊_m and the centralizer is isomorphic to (α_4 ⋊ A) ⋊/3, where A is a non-split extension of α_4 by α_2 = α(G). Thus, we have C__C(α(G))/α(G) ≅ (α_4 ⋊α_4) ⋊/3.
  
  Finally, let us explain how to compute M in the case λ = 0. As in the case λ = 1, we have |M| | 3. Choose an element h_E ∈_E,O of order 3. Since α(G) = α_2 is the kernel of Frobenius on E, it is preserved by h_E. By Lemma <ref> and Lemma <ref>, conjugation by h_E induces an automorphism of α_2 of order 3. On the other hand, the conjugation action of N__C(α(G)) on α_2 factors through N__C(α(G))/C__C(α(G)). By the calculations of the previous paragraph and since _α_2≅_m, we can find an automorphism h_C ∈ N__C(α(G)) of order 9 such
  that α∘ ad_h_E =  ad_h_C∘α. By Lemma <ref> (<ref>), h = (h_E,h_C) normalizes the G-action on E × C. By Proposition <ref>, h descends to X and induces a non-trivial element of M. Hence, we have M ≅/3. 
  
  
  
  
  
    * Let g: t ↦ω t, where ω^2 + ω = 1. Then,
  
      h ∘ g ∘ h^-1: t ↦ω t + ω^2b^-1(c + b^-4c^4 e + ω^2 b^-2c^2d + b^-6c^4 d^3) + b^-1dt^2 +  b^-3d^3t^4.
  
  Thus, h normalizes /3 if and only if it centralizes /3 if and only if
  
      d    =    0,  and
      
      c^4e + b^4 c    =    0.
  
  Putting  this together with the conditions obtained in (f), we deduce that the normalizer of α(G) is the group scheme of maps
  
      t ↦ bt + et^4  with  e^2 = 0,
  
  which is isomorphic to α_2 ⋊_m. Moreover, we see that C__C(α(G)) = α(G). 
  
  Since the automorphism g generates the group α(G)(k), the calculation of the previous paragraph also shows that N_(C)(α(G)(k)) = _m(k). Thus, M is a subquotient of _m(k) by Theorem <ref> (<ref>) and, in particular, the order of M is prime to 2. By the same theorem, M is also a subquotient of _E'/((f_E)_* C__E(G)), which is isomorphic to Q_8 ⋊/3 since C__E(G) ≅ E in the current case. Hence, M is a subquotient of /3. Using the same construction as in (f), one can show that M = /3.
  
  
  
  
  
    * We compute
  
      h ∘ g ∘ h^-1: t ↦ t + ab^-1(1 + a(λ c^2 + c^4 + b^-2d)) + λ a^2 b t^2 + a^2(λ b^-1d^2 + b^3)t^4.
  
  This means that h normalizes α(G) if and only if it satisfies
  
      b^3    =    1,  and 
      λ d^2    =    b + b^2.
  
  In fact, since d^4 = 0, we can square (<ref>) to deduce b = 1, and since λ≠ 0, we get d^2 = 0. Now, h  centralizes α(G) if and only if additionally
  
      d    =    c^4 + λ c^2.
  
  Squaring (<ref>), we obtain c^8 + λ^2 c^4 = 0. Hence, the centralizer C__C(α(G)) of α(G) is the group scheme of maps
  
      t ↦ t + c + (c^4 + λ c^2)t^2 + et^4  with  e^2 = 0  and  c^8 + λ^2 c^4 = 0,
  
  which is isomorphic to (M_2 ×α_2) ⋊/2, and the normalizer of α(G) is the group scheme of maps
  
      t ↦ t + c + dt^2 + et^4  with  d^2 = e^2 = 0,
  
  which is isomorphic to _a ⋊α_2^2. In particular, we have C__C(α(G))/α(G) ≅α_2 ×/2.
  
  To calculate M, note first that M is a subquotient of _E'/((f_E)_*C__E(G)) ≅ A_4 by Theorem <ref> (<ref>). Since E → E' is purely inseparable, we can lift the action of _X to E × C, where it normalizes the G-action. By the previous paragraph, we have N__C(α(G))/C__C(α(G)) ≅_a and therefore M is isomorphic to a subquotient of (/2)^2, again by Theorem <ref> (<ref>). We may assume that E is given by the equation y^2 + λ y = x^3.
  Choose c,d ∈ k such that c^3 = λ and d^2 + λ d + λ^2 = 0 and let h_E,c,d be the corresponding automorphism of E as in Remark <ref>. Then, by the calculations of the previous paragraph and by Remark <ref>, α_T ∘ ad_h_E,c,d =  ad_h_C,c'∘α_T, where h_C,c' is a substitution 
  h_C,c': t ↦ t + c'
  with c'^4 + λ c'^2 = c. Therefore the automorphisms (h_E,c,d,h_C,c') of E × C descend to X. The three different values of c yield three distinct non-trivial elements of M, so M ≅ (/2)^2.
  
  
  
  
  
  
  This finishes the calculation of the groups C__E(G)/E, C__C(α(G))/α(G),M, and thus also of the full automorphism group schemes for all bielliptic and quasi-bielliptic surfaces in all characteristics.
  The results are summarized in Table <ref>, Table <ref> and Table <ref>.
  
  
  
  
  
  
  
  ABD+66+++
  
  [ABD+66]SGA3
  M. Artin, J.-E. Bertin, M. Demazure, A. Grothendieck, P. Gabriel, M. Raynaud,
    and J.-P.  Serre.
  Schémas en groupes,
  Séminaire de Géométrie Algébrique du Bois Marie (SGA3),
  Institut des Hautes
    Études Scientifiques, Bures-sur-Yvette,
    1963–1966.
  
  [BdF10]Bagnera
  G. Bagnera and M. de Franchis,
  Le nombre ρ de Picard pour les surfaces hyperelliptiques,
  Palermo Rend. 30 (1910), 185–238.
  
  [BM90]BennettMiranda
  C. Bennett and R. Miranda,
  The automorphism groups of the hyperelliptic surfaces,
  Rocky Mountain J. Math. 20 (1990), no. 1, 31–37.
  
  
  [BM76]BombieriMumford3
  E. Bombieri and D. Mumford,
  Enriques' classification of surfaces in char. p. III,
  Invent. Math. 35 (1976), 197–232.
  
  [BM77]BombieriMumford2
  ,
  Enriques' classification of surfaces in char. p. II,
  in: Complex Analysis and Algebraic Geometry (A Collection of Papers Dedicated to K. Kodaira), pp.  23–42, Cambridge University Press, Cambridge, 1977.
  
  [Bri11]BrionTorsor
  M. Brion,
  On automorphism groups of fiber bundles,
  Publ. Mat. Urug. 12 (2011), 39–66.
  
  [Bri18]BrionNotes
  ,
  Notes on automorphism groups of projective varieties,
  Lecture notes for the School and Workshop on Varieties and Group Actions, Warsaw, 2018.
  Available at <http://www-fourier.univ-grenoble-alpes.fr/ mbrion/autos_final.pdf>
  
  [BSU13]Brion
  M. Brion, P. Samuel, and V. Uma,
  Lectures on the structure of algebraic groups and geometric applications,
  CMI Lecture Series in Mathematics, vol. 1, Hindustan Book Agency, New Delhi; Chennai Mathematical Institute (CMI), Chennai, 2013.
  
  [Lan79]Langquell
  W. E. Lang,
  Quasi-elliptic surfaces in characteristic three,
  Ann. Sci. Éc. Norm. Supér. 12 (1979), no. 4, 473–500.
  
  [MO68]MatsumuraOort
  H. Matsumura and F. Oort,
  Representability of group functors, and automorphisms of algebraic
    schemes,
  Invent. Math. 4 (1967/1968), 1–25.
  
  [MFK94]MumfordGIT
  D. Mumford, J. Fogarty, and F. Kirwan,
  Geometric invariant theory, Ergebnisse der
    Mathematik und ihrer Grenzgebiete (2), vol. 34,
  Springer-Verlag, Berlin, 1994.
  
  [Pop16]Popov
  V. L. Popov,
  Birational splitting and algebraic group actions,
  European Journal of Mathematics 2, 283–290 (2016).
  
  [Sil09]Silverman
  J. H. Silverman,
  The arithmetic of elliptic curves, 
  Graduate Texts in Mathematics, vol. 106,
  Springer, Dordrecht, 2009.
  
  
   
  