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a-Genus 2 curves covering elliptic curves



I was introduced to this area by H. Volklein, who was my advisor at the University of Florida. First papers I read on the subject were by Frey, Frey/Kani, and a preprint of M. Fried. My first paper on the topic was written while I was visiting University of Erlangen on a DFG fellowship in 2000.

The Jacobian of a genus 2 curve is (n, n)-split if there is a degree n cover from C to an elliptic curve. For a fixed n, the space of all genus 2 curves with (n, n)-split Jacobians, say Ln, is an irreducible 2-dimensional variety if n is odd.  In my thesis  were computed for the first time the loci Ln, for n=2,3,5.

  • Curves of genus two covering elliptic curves. Thesis (Ph.D.)–University of Florida. 2001. 72 pp. ISBN: 978-0493-20012-5, ProQuest LLC

Such results appeared in the literature in the following sequence

  • Shaska, T.; Voelklein, H, Elliptic subfields and automorphisms of genus 2 function fields. Algebra, arithmetic and geometry with  applications (West Lafayette, IN, 2000),703--723, Springer, Berlin, 2004.

This paper basically deals with the case n=2 and considers Hurwitz spaces and generating tuples. It discovers the dihedral invariants u and v and a nice embedding of M1 to M2. It was submitted in September 2000.

  • Shaska, T. Genus 2 fields with degree 3 elliptic subfields, Forum Math. 16 (2004), no. 2, 263--280.

Starting from a paper of Kuhn, which gives a parametric equation for the genus 2 curve, we study the function field of L3 and show that k(L3)=k(r1, r2) for some parameters r1, r2. The parameters r1, r2, are cleverly chosen as the invariants of a pair of cubics. Using such invariants, we were able to compute the equation of L3. This was the first time that the equation of Ln as a sublocus of M2 was computed for any n>2. The reader can find the maple equations in the following files

equation of L3           eq. r1, r2   

  • Genus 2 curves with (3,3)-split Jacobian and large automorphism group. Algorithmic number theory (Sydney, 2002),  205--218, Lecture Notes in Comput. Sci., 2369, Springer, Berlin, 2002.

In this paper we investigate the rational points on some genus 2 curves. We prove that the intersection L2 and L3 in M2 has only finitely many rational points. We compute such points explicitly as triples (i1, i2, i3) of absolute invariants. These points correspond to the genus 2 curves which have a degree 2 and degree 3 elliptic subcovers.

  • Magaard, KayShaska, TanushVölklein, Helmut; Genus 2 curves that admit a degree 5 map to an elliptic curve. Forum Math. 21 (2009), no. 3, 547–566.

Paper from Arxiv All references below refer to the version of the Arxiv paper.
Expressions of a and b in terms of u, v, z, as in Eq. (16)
Computation of L5 locus as in page 9
Maple file for Case 1
Maple file for Case 2
Maple file for Case 3

Erratum: There is a wonderful paper posted on Arxiv by A. Kumar, where it is pointed out that we have made a mistake in determining the function field k(L5) and have determined a degree 2 extension of k(L5) instead.  

Later with some of my students I focused on the degree even coverings. The following papers are on degree even:

In this short note is given a quick review of the general situation for degree even coverings C--> E. All the ramification structures are determined for the general cases and all its degenerate cases. 

Abstract: We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to $\mathbb C$-isomorphism) defined over $\mathbb Q$, which have degree 2 and 3 elliptic subcovers also defined over $\mathbb Q$.






Notes and remarks:

There have been many other papers published on this topic after some of my work. 




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  61k v. 2 Dec 20, 2014, 7:24 PM Tony Shaska
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