A curve $C$ defined over $\mathbb Q$ is modular of level $N$ if there exists a non-constant morphism from $X_1(N)$ onto $C$ defined over $\mathbb Q$ for some positive integer $N$. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve $C$ of genus $3$ and level $N$ such that $\mathrm{Jac}{(C)}$ is $\mathbb Q$-isogenous to a given three dimensional $\mathbb Q$-quotient of $J_1 (N)$. Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus $3$. Let $C$ be a modular curve of level $N$, we say that $C$ is new if the corresponding morphism between $J_1(N)$ and $\mathrm{Jac}{(C)}$ factors through the new part of $J_1(N)$.
Let N be a positive integer and X 1 (N ) (resp. X 0 (N )) be the classical modular curve corresponding to the modular group Γ 1 (N ) (resp. Γ 0 (N )). Many papers have already been devoted to the problem of finding Q-rational models for these modular curves and their quotients [22,12,17,14,15,10]. In this work we are interested in modular curves defined over Q which are dominated over Q by X 1 (N ).
In [13,2] the concept of new modular curve is introduced. These are curves dominated by X 1 (N ) such that the corresponding morphism on their jacobians factors through the new part of the jacobian of X 1 (N ). For the genus 1 case, the concept of new modular curve and modular curve are equivalent. Shimura [19] proved that any elliptic curve with complex multiplication is modular, and provided in this way the first infinite set of new modular curves of genus 1. Furthermore, in a series of papers, Wiles et al. [24,23,4] proved that every elliptic curve defined over Q is modular and thus new modular. Conversely, all new modular curves of genus 1 are elliptic curves defined over Q. In contrast to the new modular elliptic curve case, for a fixed genus g ≥ 2 the set of new modular curves of genus g (up to Q-isomorphism) is finite and computable [2]. In the genus 2 case, [2,13] provide a complete list of new modular curves. More precisely, in their proof [2] of computability of new modular curves of fixed genus g > 1, the authors develop a deterministic method that provides a finite (but enormous) list containing, amongst others, all the new modular curves of genus g. The large amount of curves appearing in this list makes the computation of all new modular curves of genus g > 2 impossible nowadays. For example, for g = 3, 4, 5, 6 there are respectively 10 105 , 10 239 , 10 455 , 10 844 possibilities. Moreover, [2] provides a sufficient and necessary condition to verify if a new modular abelian variety is Q-isogenous to the jacobian of a modular hyperelliptic curve. That is, for each level N , they provide a method to compute all the new modular hyperelliptic curves defined over Q of level N . For the non-hyperelliptic new modular case, they provide a necessary, but not sufficient condition based on the canonical embedding (cf. Remark 1).
The aim of this paper is to study the simplest case of non-hyperelliptic new modular curves, i.e. the case of genus 3 (smooth plane quartics). We first provide a necessary and sufficient condition for a non-hyperelliptic curve to be modular of level N with the additional requirement that its holomorphic differentials correspond to the holomorphic differentials of a given modular abelian 3-fold defined over Q. We then restrict our attention to the computation of all non-hyperelliptic new modular curves of genus 3 up to a fixed level (see Appendix).
This paper is organized as follows: In Sections 2 and 3 we review the necessary technical background about modular curves and non-hyperelliptic genus 3 curves respectively. In Section 4 we present a method that allows us to recognize if a modular abelian 3-fold corresponds to the jacobian of a non-hyperelliptic modular curve of genus 3. We apply this method to compute all the new modular nonhyperelliptic curves of genus 3 up to certain levels. In Section 5 we present some examples that show the ambiguity of the non-new modular case. We conclude this paper with an appendix that gives equations of 44 non-hyperelliptic new modular curves of genus 3, and we expect these to be the complete list of this kind of curves.
Remark: A different approach to the one studied in this paper is the one researched at [18]. There the author developes an algorithm to recognize if a modular abelian 3-fold is the jacobian of a non-hyperelliptic curve of genus 3. Note that in our problem, if a curve is modular then its jacobian is modular. Nevertheless, the converse is not true in general.
Notation: All curves and varieties in this paper are smooth and projective, and all the fields will be of characteristic zero. If X is a variety over a field K, Ω 1 = Ω 1 X/K denotes the sheaf of holomorphic 1-forms. If A and B are two abelian varieties defined over a field K, the notation A K ∼ B means that A and B are K-isogenous. Let k, N ∈ N, we denote by S k (N ) the vector space of cuspidal forms of weight k for the modular subgroup Γ 1 (N ). Throughout the paper all the modular curves and abelian varieties are defined over Q and we will use the labelling of modular forms and abelian varieties as it was introduced in [2, Appendix]. For the sake of completeness, we remind this labelling at the Appendix. Along the paper we will use the canonical identification between the spaces H 0 (C, Ω 1 ) and H 0 (Jac(C), Ω 1 ) (cf. [21, §2.9, Prop. 8]).
This section is dedicated to the basic notions about modularity that will be used in the rest of the paper. [2, §3.1] is a good reference where all the necessary background is included. Definition 1. An abelian variety A over Q is
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