250133 VO Algebra meets Analysis and Number Theory (2022W)

Around 1866, Lazarus Fuchs asked himself whether it is possible characterize those ordinary linear differential equations

a_n(x)y*(n) + a_n−1(x)y*(n−1) + · · · + a_1(x)y′ + a_0(x)y = 0

with holomorphic coefficients a_i, defined in a neighborhood of 0 in C, which admit a basis of solutions which are moderate in the sense that they are either holomorphic at 0 or converge to ∞ at most polynomially in any sector in C with vertex 0. The second case can only occur when 0 is a singularity of the equation, say, when a_n(0) = 0.

This was the starting point of an exciting and multi-faceted story, opening up a new field which one calls nowadays Differential Algebra. It was truly a vein of gold Fuchs had discovered: First, he was able to establish a purely algebraic criterion to characterize the before mentioned equations (now called Fuchsian equations). Further, he (and also Frobenius and Thome ́) then described explicitly all solutions of these equations. Powers of logarithms appear in combinations with holomorphic functions. Using then analytic continuation of the solutions along a closed curve around 0 one constructs the associated Monodromy Group of the equation. Using the language of differential fields (a field like C(x), equipped with a derivation) one is able to interpret this group as the (differential) Galois group of the related field extension. This group gives precise information about the solutions: for instance, it is finite if and only if there is a basis of algebraic solutions, i.e., functions which satisfy a polynomial equation.

One can also ask when the solutions of the equation have integer coefficients (assuming that the equation is defined over Z), thus entering number theory. The question is still not settled, with many mysterious examples and phenomena. Not to speak of the unsolved Grothendieck-Katz p-curvature conjecture, which predicts the algebraicity of the solutions by looking at the reduction of the equation modulo primes p.

These are just a few of the numerous aspects which pop up and which are fascinating to discuss and describe. In the course, we start with a systematic introduction to Fuchsian equations and then go on to some of the most striking phenomena and puzzles.