The Mumford-Tate conjecture for the product of an abelian surface and a K3 surface

In this paper we prove the Mumford-Tate conjecture in degree 2 for the product of an abelian surface $A$ and a K3 surface $X$ over a finitely generated field $K \subset \mathbb{C}$. The Mumford-Tate conjecture is a precise way of saying that the Hodge structure on singular cohomology conveys the same information as the Galois representation on $\ell$-adic \'{e}tale cohomology. To make this precise, let $G_{\mathrm{B}}$ be the Mumford-Tate group of the Hodge structure $H^{2}_{\text{sing}}(A(\mathbb{C}) \times X(\mathbb{C}), \mathbb{Q})$. Let $G_{\ell}^{\circ}$ be the connected component of the identity of the Zariski closure of the image of the Galois group $\textrm{Gal}(\bar{K}/K)$ in $\mathrm{GL}(H^{2}_{\text{\'{e}t}}(A_{\bar{K}} \times X_{\bar{K}}, \mathbb{Q}_{\ell}))$. The Mumford-Tate conjecture asserts that $G_{\mathrm{B}} \otimes \mathbb{Q}_{\ell} \cong G_{\ell}^{\circ}$. The proof presented in this paper uses input from number theory (Chebotaryov's density theorem), Lie theory, and some facts about K3 surfaces over finite fields.Comment: 19 pages, all comments are welcom

Abstraction boundaries and spec driven development in pure mathematics

In this article we discuss how abstraction boundaries can help tame complexity in mathematical research, with the help of an interactive theorem prover. While many of the ideas we present here have been used implicitly by mathematicians for some time, we argue that the use of an interactive theorem prover introduces additional qualitative benefits in the implementation of these ideas.Comment: To appear in a special volume of the Bull. Amer. Math. Soc.; 14 pg.; feedback welcome

Formalizing the ring of witt vectors

The ring of Witt vectors W R over a base ring R is an important tool in algebraic number theory and lies at the foundations of modern p-adic Hodge theory. W R has the interesting property that it constructs a ring of characteristic 0 out of a ring of characteristic p > 1, and it can be used more specifically to construct from a finite field containing ĝ.,Currency sign/pĝ.,Currency sign the corresponding unramified field extension of the p-adic numbers ĝ.,sp (which is unique up to isomorphism). We formalize the notion of a Witt vector in the Lean proof assistant, along with the corresponding ring operations and other algebraic structure. We prove in Lean that, for prime p, the ring of Witt vectors over ĝ.,Currency sign/pĝ.,Currency sign is isomorphic to the ring of p-adic integers ĝ.,Currency signp. In the process we develop idioms to cleanly handle calculations of identities between operations on the ring of Witt vectors. These calculations are intractable with a naive approach, and require a proof technique that is usually skimmed over in the informal literature. Our proofs resemble the informal arguments while being fully rigorous