Department of Mathematical Sciences

Mathematics

In 1996, Shreeram Abhyankar constructed a certain family of equations, and conjectured that their Galois groups (that is, the set of symmetries of the equation) were of the form GL(n,q), the group of  invertible nxn matrices with coefficients in the finite field with q elements, where q is a prime power depending on the equation. It turns out that these equations are related to Drinfeld modules, which in turn are analogous to elliptic curves. The elliptic curve version of the conjecture is actually a theorem, and applying these insights to Drinfeld modules has allowed Florian Breuer to prove the conjecture.
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