Bielliptic Curves Research Articles

Bielliptic quotient modular curves of 𝑋₀(𝑁)

Let N ≥ 1 N\geq 1 be a non-square free integer and let W N W_N be a nontrivial subgroup of the group of the Atkin-Lehner involutions of X 0 ( N ) X_0(N) such that the modular curve X 0 ( N ) / W N X_0(N)/W_N has genus at least two. We determine all pairs ( N , W N ) (N,W_N) such that X 0 ( N ) / W N X_0(N)/W_N is a bielliptic curve and the pairs ( N , W N ) (N,W_N) such that X 0 ( N ) / W N X_0(N)/W_N has an infinite number of quadratic points over Q \mathbb {Q} .

Explicit quadratic Chabauty over number fields

We generalize the explicit quadratic Chabauty techniques for integral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary number fields using restriction of scalars. This is achieved by combining equations coming from Siksek’s extension of classical Chabauty with equations defined in terms of p-adic heights attached to independent continuous idele class characters. We give several examples to show the practicality of our methods.

Geometry of Prym varieties for certain bielliptic curves of genus three and five

We construct two pencils of bielliptic curves of genus three and genus five. The first pencil is associated with a general abelian surface with a polarization of type $(1,2)$. The second pencil is related to the first by an unramified double cover, the Prym variety of which is canonically isomorphic to the Jacobian of a very general curve of genus two. Our results are obtained by analyzing suitable elliptic fibrations on the associated Kummer surfaces and rational double covers among them.

Intersections of loci of admissible covers with tautological classes

For a finite group G, let overline{mathcal {H}}_{g,G,xi } be the stack of admissible G-covers Crightarrow D of stable curves with ramification data xi , g(C)=g and g(D)=g'. There are source and target morphisms phi :overline{mathcal {H}}_{g,G,xi }rightarrow overline{mathcal {M}}_{g,r} and delta :overline{mathcal {H}}_{g,G,xi }rightarrow overline{mathcal {M}}_{g',b}, remembering the curves C and D together with the ramification or branch points of the cover respectively. In this paper we study admissible cover cycles, i.e. cycles of the form phi _* [overline{mathcal {H}}_{g,G,xi }]. Examples include the fundamental classes of the loci of hyperelliptic or bielliptic curves C with marked ramification points. The two main results of this paper are as follows: firstly, for the gluing morphism xi _A:overline{mathcal {M}}_Arightarrow overline{mathcal {M}}_{g,r} associated to a stable graph A we give a combinatorial formula for the pullback xi ^*_A phi _*[overline{mathcal {H}}_{g,G,xi }] in terms of spaces of admissible G-covers and psi classes. This allows us to describe the intersection of the cycles phi _*[overline{mathcal {H}}_{g,G,xi }] with tautological classes. Secondly, the pull–push delta _*phi ^* sends tautological classes to tautological classes and we give an explicit combinatorial description of this map. We show how to use the pullbacks to algorithmically compute tautological expressions for cycles of the form phi _* [overline{mathcal {H}}_{g,G,xi }]. In particular, we compute the classes and of the hyperelliptic loci in overline{mathcal {M}}_5 and overline{mathcal {M}}_6 and the class of the bielliptic locus in overline{mathcal {M}}_4.

Quadratic Chabauty for (bi)elliptic curves and Kim’s conjecture

We explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. We remove the assumption of semistability in the description of the quadratic Chabauty sets $\mathcal{X}(\mathbb{Z}_p)_2$ containing the integral points $\mathcal{X}(\mathbb{Z})$ of an elliptic curve of rank at most $1$. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic difference $\mathcal{X}(\mathbb{Z}_p)_2\setminus \mathcal{X}(\mathbb{Z})$. We also consider some algorithmic questions arising from Balakrishnan--Dogra's explicit quadratic Chabauty for the rational points of a genus-two bielliptic curve. As an example, we provide a new solution to a problem of Diophantus which was first solved by Wetherell. Computationally, the main difference from the previous approach to quadratic Chabauty is the use of the $p$-adic sigma function in place of a double Coleman integral.

Bielliptic smooth plane curves and quadratic points

Let [Formula: see text] be a smooth plane curve of degree [Formula: see text] defined over a global field [Formula: see text] of characteristic [Formula: see text] or [Formula: see text] (up to an extra condition on [Formula: see text]). Unless the curve is bielliptic of degree four, we observe that it always admits finitely many quadratic points. We further show that there are only finitely many quadratic extensions [Formula: see text] when [Formula: see text] is a number field, in which we may have more points of [Formula: see text] than these over [Formula: see text]. In particular, we have this asymptotic phenomenon valid for Fermat’s and Klein’s equations. Second, we conjecture that there are two infinite sets [Formula: see text] and [Formula: see text] of isomorphism classes of smooth projective plane quartic curves over [Formula: see text] with a prescribed automorphism group, such that all members of [Formula: see text] (respectively [Formula: see text]) are bielliptic and have finitely (respectively infinitely) many quadratic points over a number field [Formula: see text]. We verify the conjecture over [Formula: see text] for [Formula: see text] and [Formula: see text]. The analog of the conjecture over global fields with [Formula: see text] is also considered.

On the Bielliptic and Bihyperelliptic Loci

We study some particular loci inside the moduli space M g , namely the bielliptic locus (i.e. the locus of curves admitting a 2 : 1 cover over an elliptic curve E ) and the bihyperelliptic locus (i.e. the locus of curves admitting a 2 : 1 cover over a hyperelliptic curve C ' , g ( C ' ) ≥ 2 ). We show that the bielliptic locus is not a totally geodesic subvariety of A g if g ≥ 4 (whereas it is for g = 3 , see [18]) and that the bihyperelliptic locus is not totally geodesic in A g if g ≥ 3 g ' . We also give a lower bound for the rank of the second Gaussian map at the generic point of the bielliptic locus and an upper bound for this rank for every bielliptic curve.

Bielliptic quotient modular curves with N square-free

Let N≥1 be an square free integer and let WN be a non-trivial subgroup of the group of the Atkin-Lehner involutions of X0(N) such that the modular curve X0(N)/WN has genus at least two. We determine all pairs (N,WN) such that X0(N)/WN is a bielliptic curve and the pairs (N,WN) such that X0(N)/WN has an infinite number of quadratic points over Q.

On the Bielliptic and Bihyperelliptic Loci

We study some particular loci inside the moduli space $\mathcal{M}_g$, namely the bielliptic locus (i.e. the locus of curves admitting a $2:1$ cover over an elliptic curve $E$) and the bihyperelliptic locus (i.e. the locus of curves admitting a $2:1$ cover over a hyperelliptic curve $C'$, $g(C') \geq 2$). We show that the bielliptic locus is not a totally geodesic subvariety of $\mathcal{A}_g$ if $g \geq 4$ (while it is for $g=3$, see [16]) and that the bihyperelliptic locus is not totally geodesic in $\mathcal{A}_g$ if $g \geq 3g'$. We also give a lower bound for the rank of the second gaussian map on the generic point of the bielliptic locus and an upper bound for this rank for every bielliptic curve.

Bielliptic curves of genus three and the Torelli problem for certain elliptic surfaces

We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three. We prove that the period map for the second cohomology has one dimensional fibers, and the period map for the total cohomology is of degree twelve, and moreover, by adding the information of the Hodge structure of the canonical divisor, we prove a generic Torelli theorem for these elliptic surfaces. Finally, we give explicit examples of the pair of non-isomorphic elliptic surfaces which have the same Hodge structure on themselves and the same Hodge structure on their canonical divisors.

Bielliptic modular curves $X_0^*(N)$ with square-free levels

Let N=1 be a square-free integer such that the modular curve X*0(N) has genus =2. We prove that X*0(N) is bielliptic exactly for 19 values of N, and we determine the automorphism group of these bielliptic curves. In particular, we obtain the first examples of nontrivial Aut(X*0(N)) when the genus of X*0(N) is =3. Moreover, we prove that the set of all quadratic points over Q for the modular curve X*0(N) with genus =2 and N square-free is not finite exactly for 51 values of N.

A characterization of bielliptic curves via syzygy schemes

We prove that a canonical curve C of genus ≥11 is bielliptic if and only if its second syzygy scheme Syz2(C) is different from C.

Nontautological bielliptic cycles

Let $[\overline{\mathcal{B}}_{2,0,20}]$ and $[\mathcal{B}_{2,0,20}]$ be the classes of the loci of stable resp. smooth bielliptic curves with 20 marked points where the bielliptic involution acts on the marked points as the permutation (1 2)...(19 20). Graber and Pandharipande proved that these classes are nontatoulogical. In this note we show that their result can be extended to prove that $[\overline{\mathcal{B}}_{g}]$ is nontautological for $g\geq 12$ and that $[\mathcal{B}_{12}]$ is nontautological.

Moduli of abelian covers of elliptic curves

For any finite abelian group G, we study the moduli space of abelian G-covers of elliptic curves, in particular identifying the irreducible components of the moduli space. We prove that in the totally ramified case, the moduli space has trivial rational Picard group, and it is birational to the moduli space M1,n, where n is the number of branch points. In the particular case of moduli of bielliptic curves, we also prove that the boundary divisors are a basis of the rational Picard group of the admissible covers compactification of the moduli space. Our methods are entirely algebro-geometric.

THE HEAVIEST WEIERSTRASS N-SEMIGROUPS ON NON-HYPERELLIPTIC CURVES?

We propose the two heaviest Weierstrass n-semigroups for non- hyperelliptic curves (one on trigonal curves with maximal Maroni invariant and with totally ramification points and one on bielliptic curves and with as points some of the ramification points). We compute them when n = 2.

Theta divisors of stable vector bundles may be nonreduced

A generic strictly semistable bundle of degree zero over a curve $$X$$ has a reducible theta divisor, given by the sum of the theta divisors of the stable summands of the associated graded bundle. The converse is not true: Beauville and Raynaud have each constructed stable bundles with reducible theta divisors. For $$X$$ of genus $$g \ge 5$$ , we construct stable vector bundles over $$X$$ of rank $$r$$ for all $$r \ge 5$$ with reducible and nonreduced theta divisors. We also adapt the construction to symplectic bundles. In the “Appendix”, Raynaud’s original example of a stable rank 2 vector bundle with reducible theta divisor over a bi-elliptic curve of genus 3 is generalized to bi-elliptic curves of genus $$g \ge 3$$ .

Bielliptic curves of genus 3 in the hyperelliptic moduli

In this paper we study bielliptic curves of genus 3 defined over an algebraically closed field $k$ and the intersection of the moduli space $\M_3^b$ of such curves with the hyperelliptic moduli $\H_3$. Such intersection $\S$ is an irreducible, 3-dimensional, rational algebraic variety. We determine the equation of this space in terms of the $Gl(2, k)$-invariants of binary octavics as defined in \cite{hyp_mod_3} and find a birational parametrization of $\S$. We also compute all possible subloci of curves for all possible automorphism group $G$. Moreover, for every rational moduli point $\p \in \S$, such that $| \Aut (\p) | > 4$, we give explicitly a rational model of the corresponding curve over its field of moduli in terms of the $Gl(2, k)$-invariants.

Holomorphic triples on bielliptic curves

Abstract Here we study the α-stability for holomorphic triples on bielliptic curves. In particular some existence theorems for α-stable triples are proved using as main tool the study of holomorphic triples on elliptic curves. Elementary transformations of triples are also taken into consideration.

Ample vector bundles with zero loci having a bielliptic curve section of low degree

Let $${\mathcal{E}}$$ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of $${\mathcal{E}}$$ whose zero locus Z is a smooth subvariety of dimension n − r ≥ 3 of X. Let H be an ample line bundle on X such that its restriction H Z to Z is very ample. Triplets $$(X,{\mathcal{E}},H)$$ are classified under the assumption that (Z,H Z ) has a smooth bielliptic curve section of genus ≥ 3 with $$H^{n-r}c_r({\mathcal{E}}) \leq 8$$ .

WDVV Equations for 6d Seiberg–Witten Theory and Bi-Elliptic Curves

We present a generic derivation of the WDVV equations for 6d Seiberg–Witten theory, and extend it to the families of bi-elliptic spectral curves. We find that the elliptization of the naive perturbative and nonperturbative 6d systems roughly “doubles” the number of moduli describing the system.