Arithmetic Statistics - outline and references
CMI-HIMR summer school Bristol 24-28 June 2019
John Cremona
1. Introduction and first examples
2. Local and global densities
- p-adic densities
- real densities
3. Quadratics and quadrics
4. Plane curves
- conics
- cubics
5. The Ekedahl sieve
- an infinite Chinese Remainder Theorem
- application to Weierstrass densities
References:
1. (JC with Manjul Bhargava, Tom Fisher, Jon Keating, Nick Jones):
"What is the probability that a random integral quadratic form in n
variables has an integral zero?". Int. Math. Res., 2016:12 (2016).
DOI:10.1093/imrn/rnv251.
2. (JC with Manjul Bhargava and Tom Fisher): "The proportion of plane
cubic curves over Q that everywhere locally have a point".
International Journal of Number Theory, Vol.12, No.4} (2016). DOI:
10.1142/S1793042116500664.
3. (JC with Mohammad Sadek) "Densities for elliptic curves", in
preparation.
4. Bjorn Poonen and J.-F. Voloch, "The Cassels-Tate pairing on
polarized abelian varieties", Ann. of Math. (2), vol.150 no.3
(1999), pp.1109--1149.
5. Bjorn Poonen and Michael Stoll, "A local-global principle for
densities", in "Topics in number theory (University Park, PA,
1997)", Kluwer 1999. Math. Appl. vol 467, pp.241-244.
6. Torsten Ekedahl, "An infinite version of the Chinese remainder
theorem", Comment. Math. Univ. St. Paul., vol.40 no.1 (1991),
pp.53--59.