A non-trivial family of bundles fixed by the square of Frobenius

C. R. Acad. Sci. Paris, t. 333, Série I, p. 651–656, 2001 Géométrie algébrique/Algebraic Geometry A non-trivial family of bundles fixed by the square of Frobenius Yves LASZLO Institut de mathématiques de Jussieu, Analyse algébrique, Université Paris-6, 4, place Jussieu, 75005 Paris, France E-mail: [email protected] (Reçu le 15 juin 2001, accepté le 25 juin 2001) Abstract. One shows the existence of a smooth projective curve over F2 and of representations of the arithmetic fundamental group of X ⊗ k with values in SL2 (k[[t]]), with k suitable finite field of characteristic 2, such that the image of the geometric fundamental group is infinite. This gives a negative answer to a question of A.J. de Jong.  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Une famille non triviale de fibrés fixés par le carré du Frobenius Résumé. On montre l’existence d’une courbe projective lisse X sur F2 et de représentations continues du groupe fondamental arithmétique de X ⊗ k à valeurs dans SL2 (k[[t]]), avec k corps fini convenable de caractéristique 2, telles que l’image du groupe fondamental géométrique soit infinie. Cela répond par la négative à une question de A.J. de Jong.  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Version française abrégée On montre dans cette Note l’existence d’une courbe projective X, lisse de genre 2 définie sur le corps à 2 éléments et d’une famille paramétrée par une courbe lisse S de fibrés vectoriels de rang 2 sur X, à déterminant trivialisé, qui est isomorphe à son image inverse par le carré du Frobenius (corollaire 3.2). On associe alors à tout point fermé s de S une famille de représentations continues du groupe fondamental arithmétique π1 (X ⊗ k) dans SL2 (k[[t]]) (muni de la topologie t-adique), où k est une extension finie convenable du corps résiduel k(s), telles que l’image du groupe fondamental géométrique de X ne soit pas finie. Ceci répond négativement à une question posée par de Jong. Dans [2], il conjecture que tout Fℓ [[t]]faisceau lisse sur X, avec ℓ un nombre premier différent de p, a une monodromie géométrique finie. Ce résultat assure en particulier que l’hypothèse ℓ = p est indispensable, même si X est projective. On utilise la description du Verschiebung agissant sur l’espace des modules de fibrés de rang 2 et de déterminant trivial sur une courbe ordinaire de genre 2 obtenue par C. Pauly et l’auteur dans [4]. Note présentée par Pierre D ELIGNE. S0764-4442(01)02109-7/FLA  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés 651 Y. Laszlo 1. Introduction In this Note, we show that there exists a projective genus 2 curve X defined over F2 and a family of rank 2 bundles over X, with trivialized determinant, parameterized by a smooth curve S which is isomorphic to its pull-back by the square of Frobenius (Corollary 3.2). One associates to each closed point of S a family of continuous representations of the arithmetic fundamental group π1 (X ⊗ k) in SL2 (k[[t]]) (with the t-adic topology) , where k is a suitable finite extension of the residual field k(s), such that the image of the geometric fundamental group of X is not finite (Lemma 3.3 and Remark 1). This gives a negative answer to a question of de Jong. In [2], he conjectures that any smooth Fℓ [[t]]-sheaf over X, where ℓ = p is prime has finite geometric monodromy. In particular, this shows that hypothesis ℓ = p in cannot be dropped, even if X is projective. 2. Review on theta-constants We recall the basic results of [4]. For X a scheme over a field k of characteristic 2 and n
0, we write n X(n) for the scheme deduced from X by the extensions of scalars t → t2 . For k perfect, one can take n < 0 as well. Let k be a perfect field of characteristic 2 and X an ordinary curve over k of genus 2. By a slight abuse of notations, we denote by F the relative Frobenius F : X(n − 1) → X(n). 2.1. Theta-characteristics and polarizations For each integer n, the canonical bundle of the curve X(n) has a canonical square root B(n) (a thetacharacteristic in classical terminology). For instance, it can be defined as the bundle B(n) = coker(OX(n) → F∗ OX(n−1) ) of locally exact forms in F∗ ΩX(n−1) [7]. More geometrically, one can describe them as follows [5]. Take a rational function f on X(n − 1) which is not a square, meaning that df = 0. One checks easily that the divisor of df ⊗ 1 on X(n) is twice a divisor D, well defined up to linear equivalence. The associated line bundle O(D) is B(n). The Jacobian variety Pic0 (X(n)) of X(n) is canonically isomorphic to the twist (Pic0 (X))(n). We will shorten Pic0 (X(n)) = (Pic0 (X))(n) to JX(n). To B(n) is associated as usual the symmetric divisor Θ(n) on the Jacobian variety JX(n). As a set, one has   
Θ(n) = L ∈ JX(n) such that h0 X(n), L ⊗ B(n)
1 . The line bundle M (n) = O(Θ(n)) defines a principal polarization and we denote by L(n) its square. Notice that the isomorphism class of L(n) does not depend on the particular choice of the symmetric line bundle you have chosen to represent the principal polarization of the jacobian, but of course the divisor 2Θ(n) ∈ |L(n)| does. We will simply denote X(0), B(0), Θ(0), M (0), L(0) by X, B, Θ, M, L. 2.2. Frobenius and Verschiebung We still denote by F : JX → JX(1) the relative Frobenius of the Jacobian (obtained from the Frobenius of the curve by Albanese functoriality). It is a purely inseparable isogeny of degree 2g whose kernel is  In other words, F is the quotient morphism JX → JX/G.  One has F ∗ M (1) = L and denoted by G.  acts on L. therefore G Let V : JX(1) → JX be the Verschiebung (obtained from the Frobenius of the curve by Picard functoriality). One has V ◦ F = [2], where [2] is the multiplication by 2 on JX. Let G be the reduced part 652 A non-trivial family of bundles fixed by the square of Frobenius of Ker([2]) = K(L). It is a reduced finite group scheme of length 2g which maps therefore isomorphically onto the kernel of V . We’ll identify them together.  and the Weil pairing eL is a duality. Both G and G  are isotropic. The addition identifies K(L) and G × G The important observation is that the central extension of K(L) by µ2 defined by the Heisenberg group G(L) has a unique splitting over G. For the convenience of the reader, let me recall the reason. The point is that if g ∈ G(k̄), there exists g̃ ∈ G(L)(k̄) over g, unique because µ2 (k̄) = 1. Therefore, by uniqueness, g ∈ G(L)(k). Because G is isotropic, again because µ2 (k̄) = 1, the map g → g̃ is a morphism. In particular, G acts on L (and also on L(1)) with the previous identification. 2.3. Canonical basis of H0 (L) and H0 (L(1)) We assume from now on that G(k) = G(k̄). We will identify therefore the reduced group G and G(k). Let θ1 be a section of M (1) vanishing along Θ(1). One defines first a basis of H0 (L) by X g = g F ∗ θ1 for all g ∈ G. We denote the dual basis in H0 (L)∗ of Xg , g ∈ G by xg , g ∈ G. One checks that each Xg2 ∈ H0 (L2 ) =  = Ker(F ). Therefore, one gets for each g ∈ G a section Yg ∈ H0 (L(1)) H0 (F ∗ L(1)) is invariant under G ∗ 2 such that F Yg = Xg . This is our basis of H0 (L(1)). One defines then polynomials Pg ∈ Sym2 H0 (L)∗ by:  2   Pg = h∈G xh   Pg = if g = 0, xh xg+h otherwise. h∈G/g 2.4. Theta-constants By the formula (3.6) of [4], there exists λg ∈ k such that  ∗ 2   h∈G Yh = λ0 V X0   Yh Yg+h = λg V ∗ Xg if g = 0, otherwise. h∈G/g Moreover, λ0 = 0 and λg = 0 if and only if g is a singular point of Θ1 . In particular, if the genus of X is 2, then by the Riemann singularity theorem and the Clifford lemma, all the λg ’s are non-zero. 2.5. Rank 2 vector bundles, genus 2 case The main result of [4] goes as follows. The map which associates to a vector bundle E on X(n) its determinant defines a morphism from the coarse moduli space of rank 2 semi-stable vector bundles on X(n) to PicX(n). Let NX(n) be the fiber over the class of trivial line bundle OX(n) in Pic(X(n)). The morphism ιX (n) : NX (n) → |L(n)| which associates to E its theta divisor which as a set is   
ΘE = L ∈ JX(n) such that h0 L ⊗ E ⊗ B(n)
1 is an isomorphism by a theorem of Narasimhan and Ramanan [6] (see the details for the generalization in characteristic 2 in [4]). The pull-back of bundles by F defines a rational morphism, the Verschiebung V : NX(1)  NX . 653 Y. Laszlo Then, the diagram NX(1) V ≀ ιX ιX (1) ≀ |L(1)| NX x→(λg Pg (x)) |L| is commutative. 3. The example The quadratic extensions k̄(x) ֒→ k̄(X) of genus 2 are known for a long time, the equation depending on the number (which is 1, 2 or 3 if the 2-rank is 0, 1 or 2) of Weierstraß points [1]. Let X be the smooth curve (easy check) over the prime field k = F2 of affine equation y 2 + x(x + 1)y = x5 + x2 + x. The projection (x, y) → x is a (2 : 1) covering, ramified over 0, 1, ∞. Near ∞, we take the coordinates x− = x−1 and y− = yx−3 giving the equation 2 y− + x− (x− + 1)y− = x5− + x4− + x− . In the (non-rational) affine model of [1], this curve is given by the equation y 2 + (x2 + x)y = x5 + jx3 + x where j ∈ F4
F2 (change y in y + jx). Over 0 is the point (0, 0), over 1 the point (1, 1) (in the coordinates x, y) and over ∞ the point with coordinates x− = 0 and y− = 0. All three are k-rational. The differences between these points define three divisors whose associated line bundles are the three points of order 2 of G, all being k-rational. 3.1. The line ∆ By the discussion above, all the theta-constants λg , g ∈ G, are in k = F2 and are non-zero: they are all equal to 1. In this case, one has JX = JX(1) and because F ∗ Xg = Xg2 , one has Xg = Yg and the two identifications of |L| = |L(1)| with P3 given by the their canonical basis are the same. We denote the coordinates xg on |L| by a = x0 and b, c, d for the others. Therefore, the rational map V :    P3  P3 , (a, b, c, d) → (a2 + b2 + c2 + d2 : ab + cd : ac + bd : ad + bc), above is conjugate to the Verschiebung V : NX  NX . Let ∆ be the line passing to the unique base point E = F∗ (B −1 ) of V and the trivial bundle T = 2O (notice that a positive dimensional subvariety of the fixed point locus has to contain E in its closure). ∼ L EMMA 3.1. – The restriction of V to ∆ −→ P1 is an involution. Proof. – The theta divisor of T is 2Θ = X0 and therefore its image by ιX in P3 is (1 : 0 : 0 : 0). The image of E is (1 : 1 : 1 : 1), the unique base point of our rational map. Therefore this line has equation b = c = d. The restriction of V to ∆
E can be lifted to ∆ (which is complete!). Explicitly, if the homogeneous 654 A non-trivial family of bundles fixed by the square of Frobenius coordinates on ∆ are (a : b), then V is given by V :    ∆  ∆, (a, b) → (a + b, b), which is a linear involution. ✷ Therefore, the restriction of V ◦ V to ∆ is the identity. Let me explain the geometric situation. The group G acts on P3 by even permutations of the xg ’s. The degree of V is 4 and the orbit of ∆ through G is a union of 4 lines meeting at E (simply because no line fixed by G is ∆). Therefore, the image of ∆ is a line. In this exceptional case all the λg ’s are equal to 1, the image of ∆ is the line ∆ itself. Using the construction of NX as a GIT quotient, one obtains as usual: C OROLLARY 3.2. – There exists a smooth quasi-projective curve S over some finite extension of k, a finite covering S → ∆, a family of semi-stable bundles E on X × S of rank 2 and trivialized determinant, an isomorphism ∼ ι : (F × IdS )2∗ (E) −→ E such that the modular morphism S → NX factors through ∆. One can presumably show that ∆ is a Hecke line (after a suitable field extension F2 ֒→ F2n ), namely the family of elementary transforms of some degree 1 rank 2 bundle at a given point. This should imply that one can assume that S = ∆ ⊗ k, but we do not need that. 3.2. Representations of the fundamental group It is well known that rank r-bundles fixed by some powers of the absolute Frobenius come from continuous representations of the fundamental group in GLr (k) [3]. The analogous statement in our situation, which is certainly well known to the experts, goes as follows. We keep the notations of the Corollary 3.2, assuming further that the degree of k over F2 is even. L EMMA 3.3. – Each closed point s ∈ S defines a continuous representation    ∼     S,s −→ SLr k(s)[[t]] , ρs : π1 X ⊗ k(s) → SLr O where the right hand-side has the t-adic topology (which makes it a profinite group too). The author had a cohomological proof, but as explained to him by Deligne, this is a local question. The general statement is as follows. Let F be the Frobenius morphism x → xq of a scheme over Fq . Let G be a connected algebraic group. Then the embedding G(Fq ) = GF ֒→ G induces a functor from
 G(Fq )-torsors over X to
 ∼ G-torsors P over X with an isomorphism P −→ F ∗ P which is an equivalence of category. In fact, starting with a pair (P, α), one constructs the étale sheaf of sets parameterizing local sections σ of P such that F ∗ (σ) = α(σ). It is a formal homogeneous space under G(Fq ) (use the fact that F ∗ P is the push-out of P by the Frobenius of G) which is a torsor (use the fact that the Lang isogeny of G is étale and surjective). We apply this statement to the G = SLr (OS,ns )-torsor of local sections of E|X×ns respecting the given trivialization where s is a point of S and ns is the subscheme of length n of S supported by s. 655 Y. Laszlo Remark 1. – Notice that the image of the geometric fundamental group of X by ρs is certainly not finite because in this case, after a finite extension of the base field, the whole family would be constant. But by construction it is not the case. This shows that in de Jong’s conjecture [2] about representations of the fundamental group in GLr (F[[t]]), where F is a finite field of characteristic ℓ = p, the hypothesis ℓ = p cannot be dropped, as hoped by this author before. Acknowledgements. The fact that there exists such family has been discovered by the author by computation using the Macaulay software. Then, it was necessary to “clean” the fixed point scheme removing the isolated points to recognize the positive dimensional part. I thank Daniel Perrin which has written the corresponding Macaulay program giving the equations of the nice line ∆ of Section 3. I would like to thank Dominique Lebrigand which gave me a simple equation for a genus 2 curve over Z/2 with 3 Weierstraß points! Especially, I would like to thank A.J. de Jong for his interest and to have introduced me to this question and P. Deligne for all his remarks and to have pointed out an inaccuracy in the first version. Finally, I would like to thank Harvard University, where this Note was worked out, for its hospitality. References [1] Ancochea G., Corps hyperelliptiques abstraits de caractéristique 2, Portugal Math. 4 (3) (1943) 119–128. [2] Jong A.J. de, A conjecture on arithmetic fundamental groups, Preprint MIT, 1998. [3] Lange H., Stuhler U., Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Z. 156 (1) (1977) 73–83. [4] Laszlo Y., Pauly C., The action of the Frobenius map on rank 2 vector bundles in characteristic 2, J. Algebraic Geom. (2000) (à paraître). [5] Mumford D., Theta characteristics on an algebraic curves, Ann. Sci. École Norm. Sup., 4e série 4 (1971) 181–192. [6] Narasimhan M.S., Ramanan S., Moduli of vector bundles on a compact Riemann surface, Ann. of Math. 89 (2) (1969) 14–51. [7] Raynaud M., Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France 110 (1) (1982) 103–125. 656