## View abstract

### 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. document.getElementById('cloak3b00e471b962a271bca19565263480d2').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy3b00e471b962a271bca19565263480d2 = '&#97;rr&#101;ch&#101;' + '&#64;'; addy3b00e471b962a271bca19565263480d2 = addy3b00e471b962a271bca19565263480d2 + '&#117;td&#97;ll&#97;s' + '&#46;' + '&#101;d&#117;'; var addy_text3b00e471b962a271bca19565263480d2 = '&#97;rr&#101;ch&#101;' + '&#64;' + '&#117;td&#97;ll&#97;s' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak3b00e471b962a271bca19565263480d2').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy3b00e471b962a271bca19565263480d2 + '\'>'+addy_text3b00e471b962a271bca19565263480d2+'<\/a>';

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).