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

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

Yes, the well-over 60-year-old Gauss actually did talk to the just 20-year-old Riemann. Historian Otto Neuenschwanden studied Riemann’s library record in Göttingen. He also consulted job position letters in Germany in the 3rd quarter of the 19th century. These show Riemann relied on personal discussions with Gauss (in the late 1840’s) about harmonic functions. Mathematicians rejected that early approach after Riemann’s death (in 1866) until near the end of the 20th century’s 1st quarter.

Using Riemann’s theta functions required generalizing Abel’s most famous work in two distinct ways. Abel compared functions on the universal cover of a complex torus; everything was in one place. Riemann had to compare functions from two types of universal covers. I explain why we still struggle to understand the Gauss-Riemann approach. Topics:

- 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

- 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.

- §4.2 – Source of the cuts and modular curves
- §6.1.1 – Complex spaces, topologically a subspace of
Riemann's sphere

- throughout §7.

To see that these topics are still serious, consider the comment in §7.3 on references (sic) from Alfhor's classic Complex Variables. Many well-trained in complex variables learned it from that book. (I've taught from it several times, hard as it is for most first year graduate students.) So tell me, those who hold it also in high regard, if you aren't shocked by that observation?

Little can set an admirer of the Abel-Galois-Riemann triumverate back more than realizing most of the world at large hasn't a clue about Riemann. Even stranger, most of the mathematical community knows only his famous hypothesis, a one-shot paper. Albeit, this uses his tour-de-fource command of harmonic functions (the Riemann-von Mangoldt formula), but still … §10 has an anecdote from a recent conversation with a German humanities scholar that puts my personal exasperation in a context.

Both Abel and Galois owe much to Crelle (and his journal) for his understanding of the revolutionary nature of their work, and his energy in making that work accessible. Still, if it weren't for the genius of Jordan, who knows if Galois' work would have survived? As we emphasize, Riemann's resurrection makes an even more complicated story. It feels incomplete – to me – even to this day.

The Norwegian Academy of Sciences web site has a section "Articles about Abel and his mathematics." It includes reference to this html file and the attached paper. I think it is well-known from Erik Temple Bell's book that Jacobi competed heavily with Abel. That competition showing markedly after Abel's death from consumption at 27. The Academy site references

Relevant to my comment on Abel in item #1, the site includes