Joint Mathematics Colloquium
Galois representations associated to Drinfeld modules
Speaker: Richard Pink (ETH, Zurich)Date: Friday, 13 June 2025 - 12:30Place: Aula Azul, ICMAT
Online: https://conectaha.csic.es/b/inf-hxq-reh-wzp
Abstract:
The most important tool for algebra in positive characteristic \(p\) is the Frobenius map \(x\mapsto x^p\). This can be thought of as an analogue of differentiation. Thus for constants \(a_0,\ldots,a_r\) in a field \(K\) of characteristic \(p\) with \(a_r\not=0\), the map \(x\mapsto \Phi(x) := a_0x+a_1x^p+a_2x^{p^2}+\ldots+a_rx^{p^r}\) can be thought of as an analogue of an ordinary linear differential operator of order \(r\). In the case \(a_0\not=0\) this is a separable polynomial, and its zeros in an algebraic closure of \(K\) form a vector space of dimension \(r\) over the field \({\mathbb F}_p\) with \(p\) elements. The Galois group of this polynomial is then a subgroup of the general linear group \(\mathop{\rm GL}_r({\mathbb F}_p)\). If the field \(K\) and the coefficients of \(\Phi\) are arbitrary, any subgroup of \(\mathop{\rm GL}_r({\mathbb F}_p)\) can occur. But things become interesting when one studies the Galois groups of all operators that are obtained from a fixed \(\Phi\) by taking composites and \({\mathbb F}_p\)-linear combinations. I plan to review what is known about these in the case that \(K\) is a global or local field.
Prerequisites: Galois theory, basics on rings and modules as in an honors algebra course.
More information:
