# Papers and preprints

• Rational points and unipotent fundamental groups (Ph.D. thesis, June 2018).

We investigate rational points on higher genus curves over number fields using Kim's non-abelian Chabauty method. We provide an exposition of this method, including a brief survey of the literature in the area. In joint work with Ellenberg, we then study the Selmer varieties of smooth projective curves of genus at least two defined over Q which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that the non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov–Tschinkel and Poonen on unramified correspondences, we deduce that any cover of P1 with solvable Galois group, and in particular any superelliptic curve over Q, has only finitely many rational points over Q.

We also present a strategy for generalizing the non-abelian Chabauty method to real number fields: A conjecture on certain transcendence properties of the unipotent Albanese map is formulated in the final two chapters of this thesis, together with a proof that this conjecture allows a generalization of several major results in the non-abelian Chabauty method to curves over a real number field.

• Jordan S. Ellenberg and Daniel Rayor Hast, Rational points on solvable curves over Q via non-abelian Chabauty (submitted).

We study the Selmer varieties of smooth projective curves of genus at least two defined over Q which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and Kim to show that Kim's non-abelian Chabauty method applies to such a curve. By combining this with results of Bogomolov–Tschinkel and Poonen on unramified correspondences, we deduce that any cover of P1 with solvable Galois group, and in particular any superelliptic curve over Q, has only finitely many rational points over Q.

• Daniel Rayor Hast and Vlad Matei, Higher moments of arithmetic functions in short intervals: a geometric perspective, in International Mathematics Research Notices (2018).
• We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius function, in short intervals of polynomials over a finite field Fq. Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the l-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree n in the limit as q → ∞. The results of this paper can be viewed as a geometric explanation for asymptotic results that can be proved using analytic number theory over function fields.
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