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

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:
1. Interpreting Abel’s Problem
2. Abel’s Theorem
3. Implications from an odd σ with the log-differential property
4. Compact surfaces from cuts and a puzzle
5. Modular curve generalizations
6. Riemann’s formulation of the generalization
7. Competition between the algebraic and analytic approaches
8. Using Riemann to vary algebraic equations algebraically
9. The impact of Riemann’s Theorem
10. Final anecdote
Abel's explicit production of all analytic functions on a single complex torus is well-known. Less known is his development of parameters for all functions of a special type: Those from one complex torus mapping through a prime (p) degree cover to another complex torus. Those parameters describe what we today call the modular curve Y0(p). Even less known, are early uses of this:
1. Galois' using his unsolvability result to show parameters for Y0(p) (p>3) are not "solvable" in the classical j parameter.
2. Riemann's partial success in finding algebraic parameters for Riemann surface families by "dragging," by its branch points, a function on one of them.
The 2nd came from Riemann's conversations with Gauss about the complex torus case. This presented more problems for modern topics than did generalizing the first of Abel's famous theorems. I've packed the paper with historical asides, based on such literature as Fay's book on Theta Functions. Though I'm not a professional historian, I've drawn conclusions extending those of Neuenschwanden. These appear in
• §4.2 – Source of the cuts and modular curves
• §6.1.1 – Complex spaces, topologically a subspace of Riemann's sphere
• throughout §7.
Here is the puzzle in §4. There is no branch of log description of most (analytic) functions on a complex torus: Galois’ discovery. So, how can we picture such a function? That was the issue between Gauss and Riemann. We discuss its modern implications.

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 C. G. J. Jacobi Considerationes generales de transcendentibus Abelianis, Journal für die reine und angewandte Mathematik (1832?), 394-403. This competition has some aspects of that between Riemann and Weierstrass, as reported by Neuenschwanden. My impression, however,  was that Jacobi was fairer than Weierstrass in this respect – possibly influenced by his knowledge that Gauss had met, and thought well of, Abel? (Again, I rely on Bell for this.)

Relevant to my comment on Abel in item #1, the site includes W. R. Hamilton On the Argument of Abel, respecting the Impossibility of expressing a Root of any General Equation above the Fourth Degree, by any finite Combination of Radicals and Rational Functions, Transactions of the Royal Irish Academy 18(2) (1839) 171-259. Read 22 May 1837. (Reprinted in Hamilton Papers, vol. 3, 517-569). My source is L.T. Rigatelli, Evariste Galois: 1811-1832, Vol. 11, translated from the Italian by John Denton, Vita Mathematica, Birkhäuser, 1996. I discuss this from a modern perspective in reviewing the Inverse Galois Problem book by Matzat and Malle  matmal04-19-01.pdf.