The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
By Manjul Bhargava, B. Gross. 2012
111 citations recorded by the source corpus.
We prove that when all hyperelliptic curves of genus $n\geq 1$ having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational points on most of these hyperelliptic curves; for example, we show that a majority of hyperelliptic curves of genus $n\geq 3$ with a rational Weierstrass point have fewer than 20 rational points.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Information Systems
View Paper: https://www.semanticscholar.org/paper/6aea4c78e366a61756a6966d15c743949f45cf6c
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Computational Theory and Mathematics
- David Zureick-Brown
- Bjorn Poonen
- Michael Stoll
- Matthew Satriano
- Xiaoheng Wang
- Jordan S. Ellenberg
- Manjul Bhargava
- Samir Siksek
In Geometry and Topology
- Bjorn Poonen
- David Zureick-Brown
- Michael Stoll
- Xiaoheng Wang
- Jordan S. Ellenberg
- Matthew Satriano
- Manjul Bhargava
- Samir Siksek
In Information Systems
- David Zureick-Brown
- Bjorn Poonen
- Michael Stoll
- Matthew Satriano
- Xiaoheng Wang
- Jordan S. Ellenberg
- Manjul Bhargava
- Samir Siksek
Citations in the corpus
- Most odd degree hyperelliptic curves have only one rational point (2013, methodology): “For each g ≥ 1 and prime p, we may now formulate the following equidistribution conjecture, compatible with both the heuristics in [PR12] and the theorems for p = 2 in [BG13].”
- Maximal linear spaces contained in the base loci of pencils of quadrics (2013, methodology): “For any field k′ containing k, J [2](k′) acts on LT{d1,.”
- Heights on stacks and a generalized Batyrev–Manin–Malle conjecture (2021, methodology): “26 In particular, if Y is a hyperelliptic curve over Q with Weierstrass equation (3.9), where the ai are chosen to be integers so that there is no prime p with p2i|ai, the height of Y is log max |ai|1/2i, which again is equivalent to the notion of height typically used for hyperelliptic curves with ”