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Andreas Strömbergsson: Rational points on horospheres, and small solutions to linear congruences

Time: Wed 2022-05-11 13.15 - 14.15

Location: Zoom, meeting ID: 694 6016 6420 (password required)

Participating: Andreas Strömbergsson (Uppsala)

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I will discuss a result about the asymptotic distribution of certain point sets in the homogeneous space \(\text{SL}(d,\mathbb{Z})\backslash \text{SL}(d,\mathbb{R})\), i.e. the space of lattices in the Euclidean space \(\mathbb{R}^d\). This result was first proved by Einsiedler, Mozes, Shah and Shapira (2016). I will discuss a new proof, which leads to an explicit rate of convergence. This proof makes use of tools from algebraic geometry (recent work of Erdelyi and Toth giving bounds on matrix Kloosterman sums) and from geometry of numbers (an integration formula by Rogers from 1955). As an application, we also obtain a refinement of a result by Strömbergsson and Venkatesh (2005) on counting small solutions to a system of linear congruences.

Joint work with Daniel El-Baz and Min Lee.

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