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.