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Arithmetic statistics and Vinberg representations https://doi.org/10.17863/CAM.131736 Oller Riera, Martí Thorne, Jack 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 $p^2$ 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. University of Cambridge 2026-03-19 Thesis Doctoral Doctor of Philosophy (PhD) eng https://www.repository.cam.ac.uk/handle/1810/405562 https://www.repository.cam.ac.uk/bitstreams/bc496a9e-0f1f-4a85-9187-50d17032ba44/download fe1be77fba4628e0d9e3b6001af62174 https://www.repository.cam.ac.uk/bitstreams/fbbc4273-e94b-46bf-aa53-2f5e920b2c35/download 87eda9de84448d1f82354d60eee3eb5f http://purl.org/NET/rdflicense/allrightsreserved number theory arithmetic statistics Lie algebras graded Lie algebras Selmer groups geometry-of-numbers squarefree sieve arithmetic geometry