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Weil's Galois Descent Theorem from a Computational Point of View

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Weil's Galois Descent Theorem from a Computational Point of View WEIL’S GALOIS DESCENT THEOREM FROM A COMPUTATIONAL POINT OF VIEW RUBEN´ A. HIDALGO AND SEBASTIAN´ REYES-CAROCCA Abstract. Let L/K be a finite Galois extension and let X be an affine alge- braic variety defined over L. Weil’s Galois descent theorem provides necessary and sufficient conditions for X to be definable over K, that is, for the existence of an algebraic variety Y defined over K together with a birational isomor- phism R : X → Y defined over L. Weil’s proof does not provide a method to construct the birational isomorphism R. The aim of this paper is to give an explicit construction of R. 1. Introduction Let K be a perfect field and let C be an algebraic closure of it. An affine algebraic variety X ⊂Cn is said to be defined over a subfield R of C if its corresponding ideal of polynomials I(X) can be generated by a finite collection of polynomials with coefficients in R. Let us assume that X is defined over a subfield L of C which is a finite Galois extension of K. Under this assumption, we will say that X is definable over K, with respect to the Galois extension L/K, if there is an algebraic variety Y defined over K and a birational isomorphism R : X → Y defined over L. To decide whether or not X is definable over K is, in general, a difficult task. For instance, if L is the field of complex numbers and K is the field of real numbers, then there are known explicit examples of complex algebraic curves which are not definable over the reals. These examples were provided by Shimura [21] and Earle [5, 6] and later by Huggins [12] (in the hyperelliptic case) and by the first author [9] and Kontogeorgis [15] (in the non-hyperelliptic situation). The natural action of Gal(L/K) on the ring of polynomials with coefficients in L induces a well-defined action (σ, X) → Xσ on the set of birational isomorphism classes of algebraic varieties. A collection of birational isomorphisms arXiv:1203.6294v6 [math.AG] 2 Jan 2020 σ {fσ : X → X : σ ∈ Gal(L/K)} defined over L satisfying the so-called Weil’s co-cycle condition τ fτσ = fσ ◦ fτ for each σ, τ ∈ Gal(L/K) is called a Galois descent datum for X with respect to L/K. Assume that X is definable over K. Namely, suppose the existence of a birational isomorphism R : X → Y defined over L where Y is an affine algebraic variety defined over K. Then Y = Y σ for each σ ∈ Gal(L/K), and the collection {(Rσ)−1 ◦ R : X → Xσ : σ ∈ Gal(L/K)} 2010 Mathematics Subject Classification. 14E99, 14A10, 12F10. Key words and phrases. Algebraic varieties, Galois extensions, Fields of definition. The first author was partially supported by Fondecyt Grant 1150003. The second author was partially supported by Fondecyt Grant 11180024, 1190991 and Redes Grant 2017-170071. 1 2 RUBEN´ A. HIDALGO AND SEBASTIAN´ REYES-CAROCCA is a Galois descent datum for X with respect to L/K. In other words, the existence of such a Galois descent datum for X is a necessary condition for X to be definable over K. Conversely, Weil in [22] proved that the existence of such a Galois descent datum is also sufficient condition. More precisely: Weil’s Galois descent theorem. Let K be a perfect field, let C be an algebraic closure of K and let L be a subfield of C that is a finite Galois extension of K. Set Γ = Gal(L/K) and assume that X is an affine algebraic variety defined over L. (a) If X admits a Galois descent datum {fσ}σ∈Γ with respect to L/K, then there exists an algebraic variety Y , defined over K, and there exists a bi- σ rational isomorphism R : X → Y , defined over L, such that R = R ◦ fσ for every σ ∈ Γ. Moreover, if all the isomorphisms fσ are biregular then R can be chosen to be biregular. (b) If there is another birational isomorphism Rˆ : X → Yˆ , defined over L, σ where Yˆ is defined over K, such that Rˆ = Rˆ ◦ fσ for every σ ∈ Γ, then there exists a birational isomorphism J : Y → Yˆ , defined over K, such that Rˆ = J ◦ R. Weil’s proof does not provide an algorithm to construct the isomorphism R explicitly. However, in the proof of [22, Proposition 1], it was observed that if {fσ} is a Galois descent datum, if each fσ is biregular and if an explicitly birational map R as before is known, then there is an explicit method to obtain a new biregular isomorphism X → Z, defined over L, with Z still defined over K. Such an explicit method is given by considering the map F : X → Y n defined by x 7→ (R(x), Rσ2 (x),...,Rσn (x)), where Γ = {σ1 = e, σ2,...,σn} and then, as there is a natural permutation action of Γ, to consider the classical invariant theory to construct a regular map Ψ : Y n → Z, defined over L, so that Z is defined over K and Ψ ◦ F : X → Z is a biregular isomorphism. In this article, we follow similar ideas as above to construct explicitly a rational map R : X → Cm, defined over L, such that Y = R(X) is defined over K and R : X → Y is a birational isomorphism. This explicit construction is done in terms of equations for X and of a Galois descent datum for X with respect to L/K. Since R is explicitly given, the algorithm can be used to compute explicit equations for Y over K. Indeed, for the sake of completeness, in the last section we will work out an example where X is a complex algebraic curve of genus five defined over Q(i). 4 This curve admits a group of conformal automorphisms isomorphic to Z2 and it is also endowed with an anticonformal involution; so, it is definable over Q. 2. Preliminaries Let K be a perfect field, let C be an algebraic closure of it, and let L be a subfield of C which is a finite Galois extension of K. We denote by Γˆ = Gal(C/K) and Γ = Gal(L/K) the Galois group associated to the extensions C/K and L/K respectively. Each η ∈ Γˆ induces a natural bijection n n ηˆ : C →C given by (y1,...,yn) 7→ (η(y1),...,η(yn)), WEIL’SGALOISDESCENTTHEOREM 3 η and if P ∈ C[z1,...,zn] then we denote by P the polynomial obtained after ap- plying η to the coefficients of P . In other words, the following diagram commutes. P Cn C ηˆ η P η Cn C Let n X = {(y1,...,yn) ∈C : Pj (y1,...,yn)=0, 1 ≤ j ≤ r} ˆ η be an affine algebraic variety where each Pj ∈ L[z1,...,zn]. If η ∈ Γ then Pj ∈ L[z1,...,zn] and n η ηˆ(X)= {(y1,...,yn) ∈C : Pj (y1,...,yn)=0, 1 ≤ j ≤ r}. Let us denote by ρ : Γˆ → Γ the canonical epimorphism defined by restriction. Note that: η σ (1) if ρ(η)= σ then Pj = Pj , and (2) if ρ(η1)= ρ(η2) thenη ˆ1(X)=η ˆ2(X). Then, if ρ(η)= σ then we denoteη ˆ(X) by Xσ. Let {e1,...,em} be a basis of L as a K−vector space. Then the matrix e1 e2 ··· em σ2(e1) σ2(e2) ··· σ2(em) A = . ∈ M(m × m, L) . .. σ (e ) σ (e ) ··· σ (e ) m 1 m 2 m m is non-singular (see, for example, [13]). The trace map m Tr : L → K given by a 7→ Σj=1σj (a) extends naturally to polynomial rings m σj Tr : L[x1,...,xn] → K[x1,...,xn] given by P 7→ Σj=1P . Lemma 1. Under the above notations, we have the following. (a) If P ∈ L[x1,...,xn] then P ∈ SpanL(Tr(e1P ),..., Tr(emP )). σ (b) If I < L[x1,...,xn] is an ideal so that P ∈ I for every σ ∈ Γ and every P ∈ I, then I can be generated as ideal by polynomials in I ∩ K[x1,...,xn]. Proof. For each j =1,...,m we define Qj := Tr(ejP ) ∈ L[x1,...,xn], σ and notice that Qj ∈ K[x1,...,xn] because Qj = Qj for each σ ∈ Γ. As the matrix A is non-singular, there are values λ1,...,λm ∈ L so that Aλ = E, t t where λ = [λ1 λ2 ... λm] and E = [1 0 ... 0]. In other words, we have m m Σj=1λj ej =1 and Σj=1λj σk(ej )=0, for each k =2,...,m; hence P can be written as m m σk m λ1Q1 + ··· + λmQm = Σj=1λj ej P +Σk=2 P Σj=1λj σk(ej ) and the first statement is proved. The second statement follows from the first one σ together with the fact that, if P ∈ I for every σ ∈ Γ then Qj ∈ I. 4 RUBEN´ A. HIDALGO AND SEBASTIAN´ REYES-CAROCCA Lemma 2. Let Y ⊂ Cn be an affine algebraic variety defined over L. If Y σ = Y for every σ ∈ Γ then Y is defined over K. More precisely, if Y is defined by the polynomials P1,...,Pr ∈ L[x1, ..., xn] then Y is also defined by the polynomials Tr(ej Pi) ∈ K[x1,...,xn], where i ∈{1,...,r} and j ∈{1,...,m}. Proof. As Y is defined over L, its associated ideal of polynomials I < C[x1,...,xn] is generated by a finite collection of polynomials with coefficients in L.
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