Two problems in arithmetic geometry inspired by homotopy theory: enriched curve counting for the Yau-Zaslow formula, and a homotopic approach to the valuative section conjecture

This thesis comprises work the author has done on two separate problems in arithmetic geometry whose origins lie in homotopy theory. The work in these problems are separate, but linked by the underlying techniques and philosophies. The first part of this thesis is based around an enriched version of the Yau–Zaslow formula for counting rational curves on K3 surfaces, and is joint work with Ambrus P´al. In order to produce this arithmetic enrichment, we first need an invariant that allows us to count. We first construct this invariant, then relate this invariant to more classical invariants. We then prove some basic properties about this invariant, such as its compatibility with transfer maps and Galois twists. We then focus on the question of whether this invariant is compatible with natural power structures, and answer the question affirmitively in dimension 0. Following this, we discuss a “Fubini theorem” for this invariant proving a version of a Fubini theorem, as well as showing that the na¨ıve version of the Fubini theorem does not hold in this context. Finally, we pull this all together to provide an arithmetic refinement of the Yau–Zaslow formula, closely following Beauville’s original strategy. The second part of this thesis is based around a variation of a result of Pop and Stix, relating the section conjecture to valuations on our variety. We first show that we can rephrase this result in terms of ´etale homotopy types and Berkovich spaces. We then show that we can relate the ´etale homotopy type of a variety to its Berkovich space, before using this to give an alternative proof of this theorem. Our alternative proof extends the result to hold over more general fields, and we can use this flexibility to extend the theorem to a larger class of varieties as well.Open Acces