# Open conjectures and expected applications of homotopy theory to arithmetics

I hope this question is not too broad to be asked here; if it is, please feel free to close the question.

I'm currently near the end of my masters studies and subsequently search for a particular field of maths I want to focus on in the near future.

Recently, the idea of mingling homotopy theory with arithmetic geometry has been very inspiring to me; be it through the anabelian conjectures of Grothendieck, their natural generalization to étale topoi or étale homotopy theory...

What I feel I'm still missing though is a more general 'overview' of what maths is part of 'arithmetic homotopy theory/homotopical arithmetic geometry', what expectations people have in those fields et cetera.

1. What (other) parts of mathematics are part of the circle of ideas trying to inject homotopy theoretic machinery into arithmetic questions?

2. What kind of applications might one hope to achieve by these methods? I.e., what are some interesting (open) conjectures people are or have been working on within the area?

Given that $$\infty$$-categories (in the sense of lurie) are kind of made to build homotopy theory into the very foundation mathematics, it seems very plausible to me that they should come up in research of homotopical arithmetic geometry; hence I was wondering:

1. Are $$\infty$$-categorical methods entering the field recently and what would be good examples of research in this area using $$\infty$$-categories.

Interesting examples I'm aware of are the Exodromy paper by Barwick, Glasman and Haine and Schmidt & Stix's paper on Anabelian geometry with etale homotopy types.

I'd be really interested in any thoughts on these questions: (other) related papers, descriptions of conjectures, points to the literature, ...

• I'm far from an expert about them, but the applications of THH to p-adic Hodge theory come to mind. May 1, 2020 at 14:14
• Does the algebraic K-theory of $\mathbb Z$ count? May 1, 2020 at 17:06
• I was always curious about this kind of questions. Given $L$ a finite extension of rational field $\mathbb{Q}$ and a scheme $X$ over $\mathbb{Q}$, let $X(L)$ be the set of its rational points. I was always wondering if there are homotopical methods to answer the question of (non)emptiness of $X(L)$.
– GSM
May 1, 2020 at 20:00
• @DanielD. I essentially just studied everything related to Algebraic Geometry and Algebraic Number Theory my university offered and at one point just started reading into this direction by myself to be honest
– lush
May 4, 2020 at 5:27
• This falls under the umbrella of what @DenisNardin mentions, but here some examples: Bhatt-Morrow-Scholze (arxiv.org/abs/1802.03261), Clausen-Mathew-Morrow (arxiv.org/abs/1803.10897), and Antieau-Mathew-Morrow-Nikolaus (arxiv.org/abs/2003.12541) are worth pointing out. There's currently lots of activity involving trace methods (K-theory, THH, TC) and arithmetic geometry being carried out by the authors I've listed above and many others whom I've neglected to mention. May 5, 2020 at 17:54