Coregular spaces and genus one curves

By Manjul Bhargava, Wei Ho. 2013

59 citations recorded by the source corpus.

A coregular space is a representation of an algebraic group for which the ring of polynomial invariants is free. In this paper, we show that the orbits of many coregular irreducible representations where the number of invariants is at least two, over a (not necessarily algebraically closed) field k, correspond to genus one curves over k together with line bundles, vector bundles, and/or points on their Jacobians. In forthcoming work, we use these orbit parametrizations to determine the average sizes of Selmer groups for various families of elliptic curves.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/05a8a57d42138970147fb04eb215470d55323caa

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Citations in the corpus

The geometric sieve and the density of squarefree values of invariant polynomials (2014, background): “ cut out by a bidegree (2,2)-form on P1 ×P1; and a genus one curve in P2 ×P2 is similarly cut out by three bidegree (3,3)-forms on P2 ×P2. These cases will be carried out in more detail in [10]. (See [9] also for other examples of such spaces of genus one models.) For all these genus one models over”
Most hyperelliptic curves over Q have no rational points (2013, methodology): “to v. If the elliptic curve Ev is expressed in Weierstrass form as y3 = x3 + A vx+ Bv, then Av and Bv give the two independent invariants of this representation of SL2(Q) ×SL4(Q) on V(Q). See [19] or [8] for details on these parametrizations. When only SL± 4 (Q) (or SL4(Q)) is acting on our represen”