Arithmetic statistics and Vinberg representations

In this thesis, we obtain results in arithmetic statistics through Vinberg representations. Very often, the rational and integral orbits of Vinberg representations encode information about certain arithmetic objects of interest. In view of that, we can use Bhargava's techniques in geometry-of-numbers to count these orbits and prove results on the distribution of the aforementioned arithmetic objects. There are three main chapters in this thesis. In the first one, we obtain a result on the density of squarefree values of certain discriminant polynomials. It is conjectured that such a density should be equal to the product of local densities, but this result is only known assuming the $abc$ conjecture. We build on a result of Bhargava--Shankar--Wang, which is concerned with the density of squarefree values of polynomial discriminants. Using Vinberg theory, we prove a more general result on the discriminant polynomials arising from the invariant theory of Lie algebras. The proof of our result follows the same steps as Bhargava--Shankar--Wang's proof, reinterpreted in further generality within the general context of Vinberg theory. We adapt their $Q$-invariant method to turn mod p² sieves into mod $p$ sieves. We also need to develop new methods in geometry-of-numbers. This includes a construction of ``box-shaped'' fundamental domains that allow us to count orbits in the cusp of the representation. In the second main chapter, we extend the results of the first one into number fields. This requires several technical improvements, particularly in the construction of fundamental domains to be able to use geometry-of-numbers methods in the cusp. In the last chapter, we use new parametrisations within Vinberg theory to prove statistical results on the average size of some isogeny Selmer groups. The parametrisations arise from the $B$ and $C$ Dynkin diagrams, and are new in the literature. We also observe that some of our average sizes diverge: this is related to the behaviour of a product of Tamagawa ratios arising from a formula of Greenberg--Wiles.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2026

Author dc:creator

• Oller Riera, Martí

Advisor dc:contributor.advisor

• Thorne, Jack

• number theory
• arithmetic statistics
• Lie algebras
• graded Lie algebras
• Selmer groups
• geometry-of-numbers
• squarefree sieve
• arithmetic geometry

Licence

• https://www.repository.cam.ac.uk/bitstreams/fbbc4273-e94b-46bf-aa53-2f5e920b2c35/download
• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.131736

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/405562