Workshop on curves of low genus

Given an elliptic curve $E$ over a number field $K$ and a point $A$ in $E(K)$, one may ask, following Lang and Trotter, for how primes $p$ of $K$ the point $A$ is "primitive" at $p$, i.e., generates the point group of the reduced curve ($E$ mod $p$). We present a characterization, in terms of the Galois representation of $E$, of the phenomenon of ``never-primitive points".

The inverse Galois problem is one of the greatest open problems in group theory and also one of the easiest to state: is every finite group a Galois group? My interest around this problem is connected to the realization of linear and symplectic groups as Galois groups over $\mathbf Q$ and over number fields. In this talk I will describe uniform realizations using elliptic curves, genus 2 and 3 curves. After this introduction, I~will explain how to extend these results using Jacobians of higher genus (joint work with Vladimir Dokchitser). Here, Goldbach's conjecture will play a central role.

Let $C:y^2 = f(x)$ be a hyperelliptic curve over a local field $K$ of odd residue characteristic. We show how several arithmetic invariants of the curve and its Jacobian, including its potential stable reduction, Galois representation and (in the semistable case) Tamagawa numbers, can be simply extracted from combinatorial data coming from the roots of $f(x)$.

Let $k$ be a field of characteristic $p\geq0$, and fix an algebraic closure $\overline{k}$ of $k$. By a smooth $\overline{k}$-plane curve $C$ of genus $g=\frac{1}{2}(d-1)(d-2)$, we mean a smooth curve $C$ that admits a non-singular plane model $F_C(X,Y,Z)=0$ of degree $d$ in $\mathbb{P}^{2}_{\overline{k}}$. The existence of universal families for a moduli space helps to recover the information on its points and allows to write down the attached objects to a point of this space. It becomes difficult to deal with a moduli space when a universal family does not exist. % However, universal families do not exist for the moduli space $\mathcal{M}_g$. %For instance, looking at curves with extra structure, and then not having automorphisms, we could look for those universal families. R. Lercier, C. Ritzenthaler, F. Rovetta and J. Sijsling in \cite[\S2]{LR} introduced three good substitutes for the notion of universal family: complete, finite and representative families. For genus $3$, we get plane quartic curves, and different arithmetics properties have been investigated by many people around. We mention, for example, a classification up to isomorphism with good properties (complete and representative families) can be found in \cite{LR,Loth}. For genus $6$, the dimension of the (coarse) moduli space $\mathcal{M}_6$ of smooth curves of genus $g=6$ over $\overline{k}$ is equal to $3g-3=15$. The stratum $\mathcal{M}_6^{Pl}$ of smooth $\overline{k}$-plane curves of genus $6$ has dimension equal to $21-9=12$, since there are $21$ monic monomial of degree $5$ in $3$ variables and all the isomorphisms are given by projective matrices of size $3\times 3$. In particular, this dimension is larger than the dimension of the hyperelliptic locus, which is $2g-1=11$. The aim of this talk is to present a so-called representative classification for the strata by automorphism group of smooth $\overline{k}$-plane curves of genus $6$, where $\overline{k}$ is a fixed algebraic closure of a perfect field $k$ of characteristic $p=0$ or $p>13$. We start with a classification already obtained in \cite{BB3} (jointly with F. Bars), in which we give the determination of the exact list of finite groups $G$ where there exists a smooth $\overline{k}$-plane curve $C$ of genus $6$ whose full automorphism group $\text{Aut}(C)$ is isomorphic to $G$. For each group $G$ in the list, we construct a so-called geometrically complete family over $k$ for each stratum. After, we mimic the techniques in \cite{LR} and \cite{Loth}. Interestingly, in the way to get these families for the different strata (jointly with E. Lorenzo Garc\'ia, see \cite{BL}), we find two remarkable phenomenons that did not appear before (the genus $3$ case). One is the existence of a non $0$-dimensional \textit{final stratum} of plane curves. At a first sight it may sound odd, but we will see that this is a normal situation for higher degrees and we will give a explanation for it. We explicitly describe representative families for all strata, except for the stratum with automorphism group $/5\mathbf Z$. Here we find the second difference with the lower genus cases where the previous techniques do not fully work. Fortunately, we are still able to prove the existence of such family by applying a version of L\"{u}roth's theorem in dimension $2$. \begin{thebibliography}{} \bibitem{BB3}E. Badr and F. Bars, \emph{Automorphism groups of non-singular plane curves of degree 5 }, Commun. Algebra \textbf{44} (2016), 327-4340. doi:\,10.1080/00927872.2015.1087547. \bibitem{BL} E. Badr and E. Lorenzo Garc\'ia, \emph{A note on the stratification by automorphisms of smooth plane curves of genus 6} arXiv:1701.06065, (2017). \bibitem{LR} R. Lercier, C. Ritzenthaler, F. Rovetta, J. Sijsling, \emph{Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields}, (LMS Journal of Computation and Mathematics, Volume 17, Special Issue A (ANTS XI), LMS, London, pp. 128--147 (2014). \bibitem{Loth} E. Lorenzo Garc\'ia, \emph{Arithmetic properties of non-hyperelliptic genus 3 curves}, PhD dissertation, Universitat Polit\`ecnica de Catalunya (2015), Barcelona. %\bibitem{Lo2} E. Lorenzo Garc\'ia, \emph{Twists of non-hyperelliptic curves of genus $3$}, arXiv:1604.02410. \end{thebibliography}{}

The character method, developed by Yih-Dar Shieh in his thesis, recognizes a Sato-Tate from an associated Frobenius distribution. Previous methods used moments of prescribed characters --- coefficients of a characteristic polynomial of Frobenius. The correspond to symmetric product characters, which decompose into direct sums of high multiplicity. As a result, the moment sequences converge poorly to large integers. The character method replaces the moments with a precomputed list of irreducible characters, and from the orthogonality relations of characters imply that a Sato-Tate group G is recognized by inner products yielding 0 or 1 (for which the minimal precision to recognize one bit suffices). We make explict the character theory method for the exceptional Lie group $G_2$, and demonstrate its effectiveness with certain character sums associated to families of curves known to give rise $G_2$ as its Sato-Tate group. This is joint work with Yih-Dar Shieh.

We will prove the Hasse-Weil inequality for genus $2$ curves with a rational Weierstrass point using explicit and constructive methods. This is inspired on the genus $1$ proof of the Hasse inequality presented in Russian by Manin in 1956, later translated to English by the AMS Translations, ser. 2, 13 (1960), 1-7.\\ The strategy is to consider the divisor $\Theta$ in the Jacobian $J$ of the curve $H/\mathbb{F}_q$ and a suitable function $\kappa_4\in\mathscr{L}(2\Theta)\subset\mathbb{F}_q(J)$. Let $\Phi_n:=\text{Frob}_{J/\mathbb{F}_q}+[n]\in\text{End}_{\mathbb{F}_q}(J)$ and $\Psi_n$ the composition \[ H\to\Theta\to {\Phi_n}_*\Theta \xdashrightarrow{\kappa_4} \mathbb{P}^1. \] We prove that $\deg\Psi_n=2(2n^2+Tn+2q)$ for a general $n$, where $T$ denotes the trace of Frobenius. The discriminant of this polynomial will lead us to the Hasse-Weil inequality.

It is well known that any finite non-trivial group $G$ acts as the group of automorphisms of an algebraic curve (not necessarily smooth) of some genus. Let $\mathcal{M}_g$ be the (coarse) moduli space of smooth curves of genus $g\geq 2$ over $\overline{k}$, a fixed algebraic closure of a field $k$ of characteristic $0$ for simplicity. The stratum of $\mathcal{M}_g$, representing $\overline{k}$-isomorphism classes of smooth curves of genus $g$ having $G$ as a subgroup of automorphisms, is denoted by $\mathcal{M}_g(G)$. By $\widetilde{M_g}(G)$ we mean the substratum of smooth curves of genus $g$ whose automorphism group is exactly $G$. There is another stratum inside $\mathcal{M}_g$, denoted by $\mathcal{M}_g^{Pl}$, which consists of smooth plane curves of degree $d$ over $\overline{k}$ (in this case, $g=(d-1)(d-2)/2$). In the same way, we define $\mathcal{M}_g^{Pl}(G):=\mathcal{M}_g(G)\cap\mathcal{M}_g^{Pl}$ and $\widetilde{\mathcal{M}_g^{Pl}}(G):=\widetilde{\mathcal{M}_g}(G)\cap\mathcal{M}_g^{Pl}$. Henn (1976) and Komiya-Kuribayashi (1978) studied the strata $\mathcal{M}_3^{Pl}(G)$ and $\widetilde{\mathcal{M}_3^{Pl}}(G)$. In particular, the list of finite groups $G$, up to isomorphism, representing automorphism groups of smooth plane curves of genus $3$ over $\overline{k}$ is given. Moreover, for each such $G$, the description of $\widetilde{\mathcal{M}_3^{Pl}}(G)$, via a set of families of homogenous polynomial equations $\mathcal{F}_G(X,Y,Z)=0$ of degree $4$ in $\mathbb{P}^2_{\overline{k}}$ is provided. The coefficients of $\mathcal{F}_G(X,Y,Z)$ are parameters assuming values in $\overline{k}$, with respect to some algebraic restrictions so that the resulting plane model is non-singular and corresponds to a smooth curve in the stratum $\widetilde{\mathcal{M}_3^{Pl}}(G)$. Conversely, for any element of $\widetilde{\mathcal{M}_3^{Pl}}(G)$, one can assign a non-singular plane model through a certain specialization of the parameters of some of these families. In the language of Lercier, Ritzenthaler, Rovetta and Sisjling in \cite{LeRit}, a geometrically complete family over $\overline{k}$ for every non-empty $\widetilde{\mathcal{M}_3^{Pl}}(G)$ is determined. In the first part of the talk, we aim to make a certain general approach to the strata $\mathcal{M}_g^{Pl}(G)$ and $\widetilde{\mathcal{M}_g^{Pl}}(G)$. Observe that in order to appear $G$ in such loci, we need a faithful representation $\varrho:G\hookrightarrow\operatorname{PGL}_3(\overline{k})$, the substratum $\varrho(\mathcal{M}_g^{Pl}(G))$ is defined to be the set of all elements $\alpha\in\mathcal{M}_g^{Pl}(G)$, such that $\varrho(G)$ acts as a subgroup of automorphisms of a non-singular plane model $F_{\alpha}(X,Y,Z)=0$ in $\mathbb{P}^2_{\overline{k}}$ for $\alpha$. Similarly, $\varrho(\widetilde{\mathcal{M}_g^{Pl}}(G))$ is defined to be the substratum of $\widetilde{\mathcal{M}_g^{Pl}}(G)$ with $\varrho(G)=\operatorname{Aut}(F_{\alpha})$. Easily one obtains the union decomposition: $$\mathcal{M}_g^{Pl}(G)={\bigcup}_{[\varrho]\in A_G} \varrho(\mathcal{M}_g^{Pl}(G));\;\;\; \widetilde{\mathcal{M}_g^{Pl}}(G)=\bigsqcup_{[\varrho]\in A_G}\,\varrho(\widetilde{\mathcal{M}_g^{Pl}}(G)),$$ where $A_G:=\{\varrho:G\hookrightarrow \operatorname{PGL}_3(\overline{k})\}/\sim,$ and $\varrho_1\sim\varrho_2$ if and only if $\exists Q\in \operatorname{PGL}_3(\overline{k})$, such that $\varrho_1(G)=Q^{-1}\varrho_2(G) Q$. Following the result of Dolgachev \cite{Dol}, we can describe (fixing the degree $d$) an algorithm to list the $\rho$ and cyclic groups $\mathbf Z/m$ for which $\rho(M_g^{Pl}(\mathbf Z/n))\neq \emptyset$, and we may list the tables until degree 9, moreover such loci are described by a certain homogenious polynomial $\mathcal{F}_{\rho(\mathbf Z/m)}(X,Y,Z)$ (this phenomena of description by a certain homogenious polynomial also happens for the strata $\varrho(\widetilde{\mathcal{M}_g^{Pl}}(G))$ and $\varrho({\mathcal{M}_g^{Pl}}(G))$). For example, $m$ should divide one of the integers either $d, d-1, d^2 -3d+3, (d-1)^2, d(d-2)$, or $d(d-1)$. Moreover we can describe the loci $M_g^{Pl}(\mathbf Z/m)$ when $m$ is ``very large", i.e. if $m$ is either $d^2-3d+$, $(d-1)^2$ , $d(d-2)$, or $d(d-1)$. Next, one sees in the determination of Henn and Komiya-Kuribayashi the following phenomenon: Given a finite non-trivial group $G$ such that $\widetilde{\mathcal{M}_3^{Pl}}(G)\neq\emptyset$, there exists a single class $[\varrho]$ such that $\widetilde{\mathcal{M}_3^{Pl}}(G)=\varrho(\widetilde{\mathcal{M}_3^{Pl}}(G))$. In this case, we call the stratum $\widetilde{\mathcal{M}_3^{Pl}}(G)$ to be \emph{ES-irreducible}. We construct explicit examples of non ES-Irreducible strata of the form $\widetilde{\mathcal{M}_g^{Pl}}(\mathbf Z/m\mathbf Z)$ for infinitely many genera $g\geq6$ with at least two components. Now, consider a smooth $\overline{k}$-plane curve $C$ over $k$, that is $C$ is a smooth curve over $k$ that admits a non-singular plane model defined over $\overline{k}$. The second part of the talk is devoted to the study of fields of definition of non-singular plane models of $C$, more concretely: Given a smooth $\overline{k}$-plane curve $C$ over a field $k$, is it a smooth plane curve over $k$, i.e. $C$ is $k$-isomorphic to a non-singular plane curve $F(X,Y,Z)=0$ defined over $k$?; and secondly, if the answer is yes, is every twist of $C$ over $k$ also a smooth plane curve over $k$? For both questions the answer is no in general, it is not. We obtain results for the curves for which the above questions always have an affirmative answer, and we show different examples concerning the negative general answer. Interestingly, in the way to get these examples, we need to handle with non-trivial Brauer-Severi surfaces, and we are able to compute explicit equations of a non-trivial one. The first part of the talk is a joint work with Eslam Badr, and resulted into \cite{BB1}\cite{BB2}. The second part of is a joint work with Eslam Badr and Elisa Lorenzo Garc\'ia, see \cite{BBL}. \begin{thebibliography}{} \bibitem{BB1} E. Badr and F. Bars, \emph{On the locus of smooth plane curves with a fixed automorphism group}, Mediterr. J. Math. \textbf{13} (2016), 3605-3627. doi:\,10.1007/s00009-016-0705-9. \bibitem{BB2}E. Badr and F. Bars, \emph{Non-singular plane curves with an element of ``large" order in its automorphism group }, Int. J. Algebra Comput. 26 (2016), 399-434. doi:\, 10.1142/S0218196716500168. \bibitem{BBL} E. Badr, F. Bars and E.Lorenzo, \emph{On twist of smooth plane curves}, arxiv 2016, submitted. \bibitem{Dol} I. V. Dolgachev. \emph{Classical algebraic geometry: a modern view}. Cambridge University Press, 2012. \bibitem{LeRit} R. Lercier, C. Ritzenthaler, F. Rovetta, J. Sijsling, \emph{Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields}, (LMS Journal of Computation and Mathematics, Volume 17, Special Issue A (ANTS XI), LMS, London, pp. 128--147 (2014). \end{thebibliography}{}

The Burkhardt quartic is a 3-dimensional projective hypersurface of degree 4 with many special properties that are classically known. For instance, its singular locus consists of 45 nodal singularities, and it has a particularly large projective automorphism group. It also has a modular interpretation: it parametrizes abelian surfaces with full level 3 structure. It is also known that the surface is birational to $\mathbf P^3$ over the complex numbers. We look at several arithmetic questions concerning the Burkhardt quartic: We show that the threefold is already rational over $\mathbf Q$ (Baker's original parametrization really required cube roots of unity). The exact parametrization also allows us to compute the zeta function of the various reductions (previously, this was only done for primes congruent to 1 mod 3). We also look at the moduli interpretation and find an equation for an (almost) universal genus 2 curve with full level structure on its Jacobian, and find a particularly explicit description of how the level structure arises from the geometry of the Burkhardt quartic.

In a recent work (2015) Bhargava, Cremona and Fisher computed the probability that a cubic plane curve defined over $\mathbb{Q}$ is everywhere locally solvable. The extension from the cubic to the quartic case is highly non-trivial. The fundamental ingredients are all the locally density formulas at each completion of $\mathbb{Q}$. In order to tackle the solubility densities at the non-archimedean completions, we classify and count by reducibility and singularities the quartics over $\mathbb{F}_q$. Through an interpolating method, we show these quantities are polynomials in the cardinality of the base field. Results regarding the real solubility density will be discussed as well. This is a report of joint work in progress with J.Cremona.

We present improvements of Mestre method for reconstructing a genus 2 or 3 hyperelliptic model from its invariants over the rationals, with application to plane quartics with complex multiplication.

We compute the cohomology of various moduli spaces of curves of genus three with level two structure. To reach this end we use various purity arguments - on the one hand via point counts over finite fields and on the other by studying arrangements of hypertori and hyperplanes. The moduli spaces in question all come with natural actions of the symplectic group on a six dimensional vector space over the field of two elements and we determine the cohomology as a representation of this group.

The CM class number one problem for elliptic curves asked to find all CM elliptic curves defined over $\mathbf Q$. For higher genus CM curves it is the problem of determining all CM curves of genus $g$ defined over the reflex field. In this talk, I will give a solution for CM curves of genus 2 and genus 3. We also prove that there are only 37 CM curves of genus 3 defined over $\mathbf Q$.

The Schottky problem wonders about which principally polarized abelian varieties arise as Jacobians of curves. There exists a natural morphism between the moduli space of curves of genus $g$ and the moduli space of principally polarized abelian varieties of dimension $g$, so by counting dimensions one realizes that the Schottky problem has a positive answer for $g \leq 3$. However, we can still consider the \emph{effective} version of the question, which seeks to determine such a curve whenever there is one. There are methods for the construction of such curves in genus $2$ and $3$. Assuming that the curve has complex multiplication (CM) by the maximal order of a certain field, in this talk we will see the current situation for genus 3 and we will focus on the family of Picard curves, giving a conjectural list of models defined over $\mathbb{Q}$ and possible generalizations of the method.

The $j$-invariants of elliptic curves with complex multiplication (CM) are algebraic integers. For invariants of genus $g = 2$ or $3$, this is not the case, though suitably chosen invariants do have smooth denominators in many cases. Bounds on the primes in these denominators have been given for $g=2$ (Goren-Lauter) and some cases of $g=3$. For Picard curves of genus 3, we give a new approach based not on bad reduction of curves but on a very explicit type of good reduction. This approach simultaneously yields much sharper bounds and a simplification of the proof. This is joint work with Pinar Kiliçer and Elisa Lorenzo García.

I will describe some of the geometry and arithmetic of certain hyperelliptic curves arising from rank two hypergeometric motives defined over totally real cyclotomic fields. This is joint work with S. Sikander and G. Tornaria.

We will give an algorithm to compute equations of non-trivial Brauer-Severi varieties over any perfect field of characteristic 0. The case of dimension 1 yields to an algorithm to construct twists of hyperelliptic curves (joint work with Davide Lombardo). Moreover, we will explicitly describe some interesting examples for the dimension 2 case.

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, David Kohel described the structure of isogeny graphs for elliptic curves and showed that one can exploit this structure to compute the endomorphism ring of an elliptic curve defined over a finite field. In dimension 2, the structure of isogeny graphs has been an open problem for some time. In joint work with E. Thome, we described isogeny graphs of genus 2 Jacobians, with maximal real multiplication. In this talk, I will show how to extend these ideas to describe the isogeny graph structure in its full generality. Over finite fields, we also show how to use this structure to derive an algorithm for endomorphism ring computation. This is work in progress with C. Martindale, D. Robert and M. Streng.