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University of Groningen Twists of Genus Three Curves and Their University of Groningen Twists of genus three curves and their Jacobians Meagher, Stephen IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2008 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Meagher, S. (2008). Twists of genus three curves and their Jacobians. s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license. More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne- amendment. Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 28-09-2021 RIJKSUNIVERSITEIT GRONINGEN Twists of genus three curves and their Jacobians Proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op vrijdag 26 september 2008 om 14.45 uur door Stephen James Meagher geboren op 24 september 1980 te Sydney, Australi¨e Promotor Prof. dr. J. Top Beoordelingscommissie Prof. dr. C-L. Chai Prof. dr. G. B. M. van der Geer Prof. dr. D. Kohel Contents Introduction 1 Twists of genus three Jacobians 3 1. Introduction 3 2. Definitions and conventions 5 3. Mumford’s theta group, existence of a principal line bundle and χ18 13 4. Existence and properties of the ∆i 20 5. Theorem 1e 25 6. Theorem 1f 27 Twists of genus three curves over finite fields 29 1. Forms and automorphisms 29 2. The points on a form 34 3. The zoo of symmetric quartics 35 4. Points on Fermat 52 5. Points on Klein 54 6. Moduli theoretic interpretation of twists of C and X 54 Bibliography 57 Samenvatting 61 Acknowledgements 63 iii Introduction The questions treated in this thesis are motivated by the following simple question: 1 let C be a genus g curve over a finite field Fq. What are the possible values of #C(Fq)? One fundamental fact is the Hasse-Weil-Serre upper and lower bounds q +1 g[2√q] #C(F ) q +1+ g[2√q]. − ≤ q ≤ So our question can be rephrased as: which numbers in the Hasse-Weil interval HW(g, q) := N [q +1 g[2√q], q +1+ g[2√q]] ∩ − can occur as the number of rational points on a genus g curve C over Fq? This question can be approached from at least two points of view: fix the genus and let q vary or fix q and let the genus of the curve vary. The second perspective has been taken up by many other authors, and we do not consider it here. The first perspective goes like this: treat genus 0 first, then genus 1, then genus 2, then genus 3, etc. Genus 0 is trivial2 as HW(0, q) = q +1 . Genus 1 was completely understood by the work of Hasse and Deuring:{ the following} numbers can occur subject to the given conditions([58]) N HW(1, q) if p ∤ N q 1 ∈ − − N = q +1 √q if 3 ∤ p 1 and √q Z, ± − ∈ N = q +1 2√q if √q Z, ± ∈ N = q +1 √pq if p =2 or p = 3 and √pq Z, N = q +1± if4 ∤ p 1. ∈ − Genus 2 was treated by Jean-Pierre Serre [48, 49, 50] in the 1980’s. Then we have genus three, which is the first genus where things do not go so smoothly. One question which brings focus to this problem is the following: Does there exist a constant c so that for every integral prime power q there is a genus three curve C so that #C(F ) > q +1+3[2√q] c? q − In words this question says: is the Hasse-Weil upperbound good within a constant not depending on the field Fq? 1Always assumed to be smooth, projective and connected. 2In fact this shows that any twist of P1 has a rational point, hence a very ample divisor of Fq degree 1 and is therefore isomorphic to P1 . Fq 1 2 INTRODUCTION Jean-Pierre Serre conjectured that this question has an affirmative3 answer! Indeed, he more or less sketched a strategy to answer this question, which runs like this: i) For each q construct an indecomposable principally polarised Abelian 3-fold (A, a) over Fq so that tr(Fr ) > 3[2√q] c. − A − ii) Show that (A, a) as in i) can be chosen to be the Jacobian of a genus three curve C. Then the Weil trace formula implies that #C(F )= q +1 tr(Fr ) > q +1+3[2√q] c. q − A − The first part of this strategy is considerably easier than the second. For example one can easily choose c = 24 and take A to be (2, 2, 2) isogenous to the third power of an ordinary elliptic curve. If one wants a better constant, this requires some more work. In fact Auer-Top [1] obtain the constant c = 21 and Lauter-Serre [24] obtain the constant c = 3. The second part is more difficult. However, the Torelli theorem for genus three curves tells us that if (A, a) is not a Jacobian then its [ 1] twist is a Jacobian, and this implies that in any case there is a genus three curve− C so that either #C(F )= q +1 tr(Fr ) q − A or #C(Fq)= q + 1 + tr(FrA). We therefore conclude, following Auer-Top [1] and Lauter-Serre [24], that for each q either the Hasse-Weil upperbound is good within a constant or the Hasse-Weil lower bound is good within a constant. This annoying ambiguity was noticed by Serre and in a letter to Jaap Top [47] he proposed a method for dealing with it. His suggestion is that (A, a) is a Jacobian if and only if the product of its 36 even theta nulls is a square. It is this suggestion which we take up in Chapter 1. During the last 4 years the author made several stabs at the above conjecture by considering special families of smooth quartics. This philosophy, that exotic or un- usual objects should yield interesting results is the motivation of Chapter 2. In Chapter 2 we study all the smooth quartics with many automorphisms. In particular we establish how to explicitly compute the number of points of a twist of a smooth quartic in terms of a 1-cocycle and the Frobenius of the original untwisted quartic. Highlights of this chapter are treatment of the Klein and Fermat quartics. 3He conjectured the analogous questions for genus g ≤ 8 also have affirmative answers [50] pp18-19, pp71-72. The question itself is probably due to Serre, but we are uncertain about this point. Twists of genus three Jacobians 1. Introduction 1.0.1. Let k be a field not of characteristic 2. Let (A, a) be a principally polarised Abelian variety of dimension three over k. 1.0.2. If a is indecomposable and k is algebraically closed there is a smooth genus three curve C and a k isomorphism of principally polarised Abelian varieties1 (Jac(C), λΘ) ∼= (A, a). However, if k admits quadratic extensions, then such a C need not exist (see [45] and [46]). 1.0.3. All of the notions in Theorem 1 will be explained within the first 3 sections of this paper, with the exception of parts e) and f) which will be explained in sections 5 and 6 respectively. Let be the fine moduli scheme of principally polarised Abelian varieties with A3,1,4 a symplectic level 4 structure; it is a smooth scheme over Z[1/2, √ 1]. Let 3′ ,1,4 be the open subscheme corresponding to those principally polarised− AbelianA varieties whose polarisation is indecomposable. We prove Theorem 1. 1. Let (A/ 3,1,4,a) be the universal principally polarised Abelian scheme with a level 4 structure.A There is a symmetric effective degree 1 polar divisor Θ on A and therefore (Θ) has a a global section s which is unique up OA Θ to multiplication by a unit of Z[1/2, √ 1]. Let χ18 be the product of the 36 even theta nulls on , i.e. − A3,1,4 χ18 := p∗sΘ, even p O where the product runs over the even 2 torsion points p : 3,1,4 A. Then χ18 36 A −→ is unique up to multiplication by an element of Z[1/2]∗ . Moreover, χ18 has the following properties: a) χ18 is a modular form, i.e. there is an isomorphism 18 p∗ A(Θ) = ωA/ 3,1,4 O ∼ A even p O which is compatible with the classical isomorphism between analytic theta nulls and analytic modular forms (see 3.3.1 and 3.4.4, cf.
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