These are my notes for my talk at the Juvitop Seminar I co-organized in Spring 2021. I gave the introductory talk, which means giving an overview of the paper, Galois Action on Symplectic K-Theory by Feng-Galatius-Venkatesh, and explaining how the subsequent talks fit together. You can find notes for subsequent talks at the seminar website here.

If you were present at the talk and would benefit from seeing the "blackboards" (notability) from the talk instead, here is the PDF

Juvitalk.pdf

While you're here here, enjoy some music. →

https://open.spotify.com/playlist/3lAuKRq7kX19WMWmoY8fP9?si=drpIO5C3QXGBdVfX5yQcdw


It's common in number theory to look for Galois representations by looking at Galois action on (co)homology of arithmetic groups. In this paper, the homotopy and K-theory of arithmetic groups are shown to provide another source of interesting Galois representations.

<aside> <img src="https://s3.us-west-2.amazonaws.com/secure.notion-static.com/a7b705be-8482-4165-9952-4505fc6ac789/Screen_Shot_2020-12-20_at_00.23.32.png?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAT73L2G45O3KS52Y5%2F20201224%2Fus-west-2%2Fs3%2Faws4_request&X-Amz-Date=20201224T153003Z&X-Amz-Expires=86400&X-Amz-Signature=670df25e4e58fee03030132ea4ad6189d79d87135f2a321c50aa1c452f11d55a&X-Amz-SignedHeaders=host&response-content-disposition=filename %3D"Screen%2520Shot%25202020-12-20%2520at%252000.23.32.png"" alt="https://s3.us-west-2.amazonaws.com/secure.notion-static.com/a7b705be-8482-4165-9952-4505fc6ac789/Screen_Shot_2020-12-20_at_00.23.32.png?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAT73L2G45O3KS52Y5%2F20201224%2Fus-west-2%2Fs3%2Faws4_request&X-Amz-Date=20201224T153003Z&X-Amz-Expires=86400&X-Amz-Signature=670df25e4e58fee03030132ea4ad6189d79d87135f2a321c50aa1c452f11d55a&X-Amz-SignedHeaders=host&response-content-disposition=filename %3D"Screen%2520Shot%25202020-12-20%2520at%252000.23.32.png"" width="40px" /> Results of the paper. Symplectic $K$-theory has a Galois group action, and as a Galois module it gives the universal extension of Tate twists $\mathbb{Z}_p(2k-1)$ by a trivial rep.

</aside>

Throughout we take $q=p^n$ to be a power of an odd prime.

[Algebraic] $K$-theory. $K$-theory comes in various different flavors, both algebraic and topological. The algebraic flavors of $K$-theory take as imput a ring and can be obtained by tweaking a bit the classifying spaces of infinite matrix groups. This will all be explained more clearly in Lucy's talk on $K$-theory, but roughly:

Take some classical matrix group $G_n(R)$ on the ring $R$ and stabilize (take colimit) to $G_\infty(R)$. Take the classifying space $BG_\infty(R )$ and "plus construct" (attach some cells to abelianize $\pi_1$, preserving homology) $BG_\infty(R )^+$.

The homotopy groups of this space will be the [adjective] K theory of $R$

$B\text{GL}_\infty(R )^+$ algebraic K-theory

$B\text{Sp}_\infty(R )^+$ symplectic K-theory

$BO^\epsilon_\infty (R )^+$ hermitian K-theory, $O^\epsilon$ are matrices preserving an $\epsilon$-symmetric form (this includes symplectic $K$-theory for $\epsilon = -1$.

One flavor of topological $K$-theory will also be relevant in the paper, the (connective) $K$-theory of complex vector bundles, which comes from the classifying space of unitary groups

$BU$ connective complex topological K-theory $ku$

I was lying a little bit when I said the homotopy groups of these spaces give us [adjective] $K$-theory. It turns out we actually have to fix $\pi_0$ of all of these spaces such that we'll get back the Grothendieck group of $R$-modules with whatever-structure-$G$-preserves, which we do by simply taking the product of $BG_\infty^+$ with the correct $\pi_0$ as a discrete space. So for instance, the right space for connective complex topological $K$-theory is

$$ \mathbb{Z}\times BU $$

Turns out we can also get the K theory groups as homotopy groups of geometric realizations of certain symmetric monoidal module categories, the relevant ones to this paper being