title: Free dihedral actions on Abelian varieties
authors: Bruno Aguiló Vidal
text: 
  Complete classification of cosmological teleparallel geometries
      Manuel Hohmann
      Received XXX 2019 / Accepted XXX 2020
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  We give a simple construction for hyperelliptic varieties defined as the quotient of a complex torus by the action of a dihedral group that contains no translations and fixes no points. This generalizes a construction given by Catanese and Demleitner for D_4 in dimension three.
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  MSC codes: 14K99, 14L30.
  
  Key words: Abelian varieties, dihedral group, free action.
      
  
  
  
  
  
  
  
      Introduction
  
  
  
  A Generalized Hyperelliptic Manifold X is defined to be a quotient X = T/G of a complex torus T by the free action of a finite group G which contains no translations. We say that X is a Generalized Hyperelliptic Variety if moreover the torus T is projective, i.e., it is an Abelian variety A.
  
  Uchida and Yoshihara showed that the only non Abelian group that gives such an action in dimension three is the dihedral group D_4 of order 8 <cit.>. Later, Catanese and Demleitner gave a simple and explicit construction for that action <cit.> and completed the characterization of three-dimensional hyperelliptic manifolds <cit.>.
  
  The purpose of this note is to generalize Catanese and Demleitner's construction to bigger dihedral groups acting in higher dimension. Specifically, for every n∈ℕ we give a Generalized Hyperelliptic Variety of dimension 2n+1 defined by the action of the dihedral group D_4n of order 8n acting on a family of Abelian varieties, from which the construction by Catanese and Demleitner remains as the particular case for n=1. 
  
  We end with a simple corollary that explains how this allows us to create Generalized Hyperellyptic Varieties using any dihedral group.
  
  
  
  
      The construction
  
  
  Let E, E' be any two elliptic curves,
  
  
      E=ℂ/( ℤ + ℤτ), E'=ℂ/( ℤ + ℤτ').
  
  
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  Now, for n∈ℕ set A':= E^2n× E', A:= A'/⟨ w ⟩, where w:= (1/2,  1/2,  ...,  1/2,  0).
  
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  Theorem. The Abelian Variety A admits a free action with no translations of the dihedral group D_4n of order 8n.
  
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   First, let us recall that for k∈ℕ, the dihedral group of order 2k is defined as 
  
  
      D_k := ⟨ r,s |  r^k = 1, s^2 = 1,  (rs)^2 = 1⟩.
  
  
  Now, set, for z:= (z_1,  z_2,  ...,  z_2n,  z_2n+1)∈ A':
  
  
      r(z)     :=    (-z_2n,  z_1,  z_2,  ...,  z_2n-1,  z_2n+1 + 1/4n)
          =     R(z) + (0, ..., 0, 1/4n), 
      
          s(z)     :=    (-z_2n + b_1,  -z_2n-1 + b_2, ...,  -z_1 + b_2n,  -z_2n+1)
          =     S(z) + (b_1, b_2, ..., b_2n, 0),
  
  
      where, for i=1, ..., n, b_2i-1 := 1/2 + τ/2 and b_2i := τ/2.
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  Step 1. It is easy to verify that r and R have order exactly 4n on A', and that R(w) = w, so that r descends to an automorphism of A of order exactly 4n. Moreover, any power r^j, 0< j<4n, acts freely on A since the (2n+1)-th coordinate of r^j (z) equals z_2n+1 + j/4n, and clearly none of this powers is a translation.
  
  
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  Step 2. s^2 (z) = z+w, since for i = 1, ...,  2n, b_i - b_2n + 1-i= 1/2; moreover, S(w)=w, hence s descends to an automorphism of A of order exactly 2.
  
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  Step 3. We have
  
      rs(z)     =     r(-z_2n + b_1,  -z_2n-1 + b_2, ...,  -z_1 + b_2n, -z_2n+1)
          =    (z_1 - b_2n,  -z_2n + b_1, ...,  -z_2 + b_2n-1, -z_2n+1 + 1/4n).
  
  
  hence
  
  
      (rs)^2 (z)     =    (z_1 -2b_2n, z_2 + b_1 - b_2n-1, ...,  z_i + b_i-1 - b_2n - (i-1), ..., z_2n + b_2n-1 - b_1,  z_2n+1)
          =     z,
  
  
  and we have an action of D_4n on A, since the orders of r, s and rs are precisely 4n, 2 and 2.
  
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  Step 4. We claim that also the symmetries in D_4n act freely on A and are not translations. Since there are exactly two conjugacy classes of symmetries, those of s and rs, it suffices to observe that these two transformations are not translations. In the next step we show that they both act freely.
  
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  Step 5. It is rather immediate that rs acts freely, since rs(z) = z in A is equivalent to 
  
  
      ( -b_2n,  -z_2n - z_2 + b_1, ...,  -z_2 - z_2n + b_2n-1,  -2z_2n+1 + 1/4n)
  
  
  being a multiple of w in A', but this is absurd since 2w = 0 and -b_2n = τ/2 ≠ 0, 1/2.
  
  
  On the other hand, s acts freely in A because s(z) = z is equivalent to
  
  
      ( -z_2n - z_1 + b_1, -z_2n-1-z_2 + b_2, ...,  -z_1 - z_2n + b_2n, -2z_2n+1)
  
  
  being a multiple of w in A', but the first and 2n-th coordinate of multiples of w are equal, while here the difference between them is 1/2≠ 0.
  
  
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      Using any dihedral group
  
  
  
  Notice that, although the previous  construction is somewhat restrictive because it works with very specific dihedral groups, since it is true that D_n⊆ D_nk for all n, k∈ℕ, we have the following corollary:
  
  
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  Corollary. For all n∈ℕ, there exists a free action of the dihedral group D_n of order 2n on some Abelian variety of dimension mcm(4,n)/2 + 1 that contains no translations.
  
  
  
  Acknowledgements: I would like to thank professors Robert Auffarth and Giancarlo Lucchini Arteche for introducing me to this topic and for dedicatedly guiding me into getting this result as an undergraduate student.
  
  
  
  
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  B. Aguiló Vidal, Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Ñuñoa, Santiago, Chile.
  
  
  E-mail address: bruno.aguilo@ug.uchile.cl
  