A venue for invited and local speakers to present their research on topics surrounding algebraic geometry and number theory, broadly conceived. All meetings start at 14:10 sharp and end at 15:10. Meetings are held in the subterranean room -101. We expect to broadcast most meetings over Zoom at the URL
Refined Chabauty–Kim computations for the thrice-punctured line over $Z[1/6]$.
If $X$ is a curve of genus at least $2$ defined over the rational numbers, we know by Faltings’s Theorem that the set $X(Q)$ of rational points is finite but we don’t know how to systematically compute this set. In 2005, Minhyong Kim proposed a new framework for studying rational (or S-integral) points on curves, called the Chabauty–Kim method. It aims to produce $p$-adic analytic functions on $X(Q_p)$ containing the rational points $X(Q)$ in their zero locus. We apply this method to solve the S-unit equation for S={2,3} and computationally verify Kim’s Conjecture for many choices of the auxiliary prime $p$.
If $X$ is a curve of genus at least $2$ defined over the rational numbers, we know by Faltings’s Theorem that the set $X(Q)$ of rational points is finite but we don’t know how to systematically compute this set. In 2005, Minhyong Kim proposed a new framework for studying rational (or S-integral) points on curves, called the Chabauty–Kim method. It aims to produce $p$-adic analytic functions on $X(Q_p)$ containing the rational points $X(Q)$ in their zero locus. We apply this method to solve the S-unit equation for S={2,3} and computationally verify Kim’s Conjecture for many choices of the auxiliary prime $p$.
The Katz-Litt theorem gives an explicit recipe to describe Vologodsky integration on curves with semi-stable reduction in terms of Coleman integration on on the rigid analytic domains reducing to the smooth components of the reduction. In work with Mueller and Srinivasan we gave an alternative recipe, more closely related to our past work with Zerbes, which was proved to follow from the Katz-Litt theorem by Katz. In this talk I will describe this alternative recipe and prove it directly. This new proof is significantly simpler than the original proof.
The Effective Siegel Problem aims to explicitly construct, in finite computation, a complete list of integral points on a given affine hyperbolic curve. Recent advances on this problem include the groundbreaking methods of Chabauty–Kim and Lawrence–Venkatesh. Both approaches study the variation of Hodge structures on bundles associated with the hyperbolic curve of interest, yet each has distinct strengths and limitations. The Chabauty–Kim method, while conditioned on the Bloch–Kato conjecture, has successfully facilitated effective computation of integral points on various curves. Conversely, the Lawrence–Venkatesh method is unconditional but has not yet been practically applied to compute integral points for any specific curve.
Kantor’s thesis was a promising initial effort toward bridging these two methods, aiming to combine their strengths through the theory of relative completions. In joint work with David Corwin, titled “The Unipotent Chabauty–Kim–Kantor Method for Relative Completions,” we present the first genuine synthesis of these powerful methods.
In this talk, we will briefly review previous developments to highlight our contribution as a natural progression toward a unified framework. We will introduce the concept of relative completions, outline Kantor’s initial approach, and then discuss our variant method, which resolves several key limitations identified in Kantor’s work. Our main result reduces the problem of Diophantine finiteness to a dimension inequality involving a pair of algebraic spaces—one arithmetic and the other geometric. If time permits, we will derive this dimension inequality explicitly for modular curves under the Bloch–Kato conjecture.
The Zilber-Pink conjecture is a far reaching and widely open conjecture in the area of “unlikely intersections” generalizing many previous results in the area, such as the recently established André-Oort conjecture. Recently the ``G-functions method’’ of Y. André has been able to consistently establish the missing arithmetic result needed to establish cases of this conjecture for Shimura varieties. I will discuss how, using properties of the p-adic values of G-functions, we can get new cases of this conjecture in $\mathcal{A}_2$.
n stable homotopy theory, one attempts to understand the category of spectra. This category is morally akin to the derived category of abelian groups. But it lends itself to a form of localization that is not available in classical algebra - chromatic localization.
The study of chromatically localized spectra is central to stable homotopy theory. I will explain a novel approach for such study (pioneered by Barthel-Schlank-Stapleton-Weinstein): the use of the isomorphism of the Lubin-Tate and Drinfeld towers of rigid analytic geometry.
I shall explain a structural result, accounting for much of the rational behavior of chromatically localized spectra, obtained in collaboration with Shay Ben-Moshe.