An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams E 6 , E 7 , E 8 . These curves are non-hyperelliptic of genus 3 or 4. We prove that a positive proportion of each family consists of curves with integral points everywhere locally but no integral points globally.

Let k be a finite extension of Q p , let G be an absolutely simple split reductive group over k, and let K be a maximal unramified extension of k. To each point in the Bruhat-Tits building of G K , Moy and Prasad have attached a filtration of G(K) by bounded subgroups. In this paper we give necessary and sufficient conditions for the dual of the first Moy-Prasad filtration quotient to contain stable vectors for the action of the reductive quotient.Our work extends earlier results by Reeder and Yu, who gave a classification in the case when p is sufficiently large. By passing to a finite unramified extension of k if necessary, we obtain new supercuspidal representations of G(k).

We prove that the average size of the 3-Selmer group of a genus-2 curve with a marked Weierstrass point is 4. We accomplish this by studying rational and integral orbits in the representation associated to a stably Z/3Z-graded simple Lie algebra of type E8. We give new techniques to construct integral orbits, inspired by the proof of the fundamental lemma and by the twisted vertex operator realisation of affine Kac-Moody algebras. Contents

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