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Abel’s 1826 Paris Memoir: The Birth of Algebraic Curves and an Under-Appreciated Formula

Abel’s 1826 Paris Memoir: The Birth of Algebraic Curves and an Under-Appreciated Formula

I will discuss two aspects of the amazing Memoir Abel submitted to the Paris Academy in 1826. First, I will begin with some topics touched on in the Memoir, including linear systems and two definitions of genus, one related to Riemann-Roch and the other related to holomorphic differentials. Second, I will explore one of the simpler versions of Abel’s Theorem that appears in the memoir, where a sum of Abelian integrals equals a logarithmic and rational function of the parameters of the linear system. In one sense, this is an elementary result; what makes it fun is the explicit formula Abel gives for the sum, which involves residues and the Global Residue Theorem for P^1. What’s amazing is that all this was done in 1826!

Refreshments at 3:45PM.

## Department of Mathematics and Statistics