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
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
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
- Interpreting Abel’s Problem
- Abel’s Theorem
- Implications from an odd σ
the log-differential property
- Compact surfaces from cuts
and a puzzle
- Modular curve
- Riemann’s formulation of
- Competition between the
algebraic and analytic approaches
- Using Riemann to vary
- The impact of Riemann’s
- Final anecdote
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
- Galois' using his unsolvability result to show parameters for Y0(p)
(p>3) are not "solvable" in the classical j
- Riemann's partial success in finding algebraic parameters for
Riemann surface families by "dragging," by its branch points, a function on
one of them.
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
and Riemann. We discuss its modern implications.
- §4.2 – Source of the cuts and modular curves
- §6.1.1 – Complex spaces, topologically a subspace of
- 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
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.
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.