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Session III.6 - Symbolic Analysis

Tuesday, June 20, 15:30 ~ 16:00

Galois groups for linear integrable systems of differential and difference equations over elliptic curves

Carlos Arreche

UT Dallas, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $C$ be an algebraically closed field of characteristic zero, let $E$ be an elliptic curve defined over $C$, and let $K$ be the field of rational functions on $E$. Let $\delta$ be an invariant derivation on $K$ (unique up to a constant multiple), and $\sigma$ denote the automorphism on $K$ induced by addition by a fixed non-torsion $C$-point of $E$ under the elliptic group law. We consider a linear system \[\delta(Y)=AY; \qquad \sigma(Y)=BY;\] where $A\in\mathfrak{gl}_n(K)$ and $B\in\mathrm{GL}_n(K)$ satisfy the integrability condition \[\delta(B)=\sigma(A)B-BA.\] There are several (five!), a priori different and seemingly incomparable, Galois groups that one can attach to such a system. We explain why some of them must be abelian, and why (conjecturally) all of them must be solvable.

Joint work with Matthew Babbitt (UT Dallas).

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