What Gauss Told Riemann About Abel's Theorem



presented in the Florida Mathematics History Seminar, Spring 2002, as part of John Thompson's 70th birthday celebration



Interpreting Abel’s Problem Abel’s Theorem Implications from an odd σ with the log-differential property Compact surfaces from cuts and a puzzle Modular curve generalizations Riemann’s formulation of the generalization Competition between the algebraic and analytic approaches

Using Riemann to vary algebraic equations algebraically The impact of Riemann’s Theorem Final anecdote

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Galois' using his unsolvability result to show parameters for Y 0 (p) (p>3) are not "solvable" in the classical j parameter. Riemann's partial success in finding algebraic parameters for Riemann surface families by "dragging," by its branch points, a function on one of them.

Theta Functions.

§4.2 – Source of the cuts and modular curves

§6.1.1 – Complex spaces, topologically a subspace of Riemann's sphere



throughout §7.

Complex Variables

Riemann-von Mangoldt formula

Inverse Galois Problem