Walter Neumann

I recently learnt the sad news that Walter Neumann just passed away. Although I don’t think I have seen him in person for over 25 years, Walter was a pretty significant influence in my mathematical life. Here are some of my recollections. (See also this celebration of Walter on his retirement from people who knew him much better than me!)

Walter was lured to Australia in 1993 by Melbourne University. Although Australia was unable to keep him (he moved back to the US in 1999), he tenure included the entire time I was an undergraduate.

The first thing Walter did for me happened before I even met him. During the summer of 1993-94, there was a (in US parlance) “REU” program run over the summer at Melbourne University in hyperbolic geometry, run in part by Walter and Craig Hodgson. Somehow they offered me a position even though I had only just finished high school and had neither formally met them nor applied. I guess my brother must have said something? I was paid the princely sum of $250 a week for six weeks to learn about volumes of hyperbolic 3-manifolds, invariant trace fields, snappea, and so much more. To top it off, there were generous breaks on Lygon street for lunch and gelato. It was an idyllic summer on campus and I can think of no better introduction to what life as a mathematician could be like at its best.

Then, a few years later, Walter happily agreed to be the advisor for my senior undergraduate thesis. I wanted to generalize a problem of Fermat to quadratic fields. Fermat famously proved that there do not exist four distinct rational squares in arithmetic progression, and the problem I had in mind was understanding (to the extent possible) for what quadratic fields this was still true. If \(a^2, b^2, c^2, d^2\) are in arithmetic progression, then (with \(x = (b/a)^2 -1\)) one obtains a point on the curve

\[y^2 = (1+x)(1+2x)(1+3x)\]

which is isogenous to \(E = X_0(24)\), and so the problem reduces to computing the ranks of the quadratic twists \(E_d\). The first part of my thesis consisted of a \(2\)-descent on the quadratic twists \(E_p\) for \(p\) prime. If I remember correctly, the rank can be computed (conditional on the finiteness of the Tate-Shafarevich group) for any prime \(p\) unless \(p \equiv 1 \bmod 24\). The rank is unequivocally zero if \(p \equiv 5,7,19 \bmod 24\), is conditionally one (on the parity conjecture) if \(p \equiv 11,13,23 \bmod 24\), and is conditionally \(0\) or \(2\) if \(p \equiv 1 \bmod 24\). The first appearance of \(2\)-torsion in the Tate-Shafarevich group occurs for \(p=97\). (Exercise for the reader in their head: when \(p=73\) the rank is positive!) This is not exactly Walter’s main area of interest(!), but he was very happy to spend time doing everything from answering my questions in algebraic geometry to watching me doing explicit \(2\)-descents on his whiteboard. I had also hoped to do something similar to what Tunnell did when he analyzed the congruent number problem, but I wasn’t quite able to compute all the relevant spaces of weight \(3/2\) modular forms. Looking at my thesis again for the first time in very many years, it seems I proved the following:

Theorem: Let
\[g = q \prod_{n=1}^{\infty} (1 – q^{12 n})^2
\cdot \sum q^{6 n^2} = q + 2 q^7 + \ldots = \sum a_d q^d \]
Then for \(d > 0\) odd and \(2d\) squarefree, assuming BSD, one has
\[\mathrm{Rank}(E/\mathbf{Q}(\sqrt{2d})) = 0
\leftrightarrow a_d = 0.\]

I’m happy that this seems to still be true, at least for \(d < 1000\), which includes \(17\) curves of rank \(2\), the first corresponding to \(\mathbf{Q}(\sqrt{134})\).

Perhaps one of the most amazing things Walter did for me was to help arrange for me to visit the Max Planck institute (Gottfried-Claren-Straße! Cash payments of 500DM bills!) for a month when I was still an undergraduate. The idea was that it would be useful for me to spend some time with (his collaborator) Don Zagier. In retrospect this obviously went above and beyond what most faculty might ever do for an undergraduate! It was certainly a transformative act for me.

One piece of mathematics that Walter is very much associated with is is work on the Bloch group and hyperbolic volumes, something which is very much dear to my heart as well. I was a bit too wet behind the ears as an undergraduate to learn about it directly from him. I had considered writing something more extensive about Walter’s work on the Bloch group (for example this paper) but then decided I probably couldn’t do better than the article of Stavros and Don in the link I cited above. Another paper of Walter close to my heart is his work with Alan Reid on Arithmetic of hyperbolic manifolds. There is clearly a direct link between my exposure to this subject in Melbourne and my later work on the Taylor-Wiles method.

Due to our respective differences in position at the time, our social interaction was naturally limited, but not entirely so. Walter was also my “masters thesis” advisor in the interregnum between my finishing my undergraduate degree and starting my PhD at Berkeley (recall the differences in timing of the academic year). My housemate (and still best friend in Australia) Toby and I invited Walter and his wife Anne over for dinner, which Toby generously cooked for (was it some variation of Beef Wellington?). We may have drunk a Chateau Tahbilk 1978 as well, but quite possibly that was another night. (My parents were not great wine collectors, but the family lore is that they did was lay down one bottle each of Penfolds Grange Hermitage of the vintage for each of our birth years. The 1972 we drunk one Boxing day in the ’90s when Danny wasn’t there; it was not a great vintage. The 1975 had disappeared long before I knew about it and had been replaced by a the ’78 Tabilk at probably around 10% of the price.) Apart from the general bonhomie, what I most remember was Anne deciding that she needed to fix Toby up with one of her friends. (After quizzing Toby on his opinions, interests, and tastes, Anne’s response was either of the form “wonderful, the two of you are in complete agreement” or, if that were not the case, “excellent, the two of you will have something to argue about!”.) Relatively soon after that, Toby and I found ourselves invited to a dinner party at the Neumann’s house in Ivanhoe. (I say Ivanhoe with apparent confidence, but I confess that many of the details in this post stretch so far back into the past that even the things I know are true still seem unlikely.) I believe they served sherry as an aperitif, which seemed wonderfully sophisticated. And it turned out that Anne was cunningly playing the role of matchmaker and had also invited her friend in question to dinner.

I have learned a lot of mathematics from a lot of people. But looking back, I can see how my mentors have done so much more — some as simple as introducing us to other mathematicians, but other things behind the scenes which take significantly more effort and time but that don’t always get noticed at the time, but which play a key role in enriching our careers. When Anne was working to set up my friend Toby, Walter was working to set me up with his friend Don Zagier in Bonn! Thanks, Walter, for everything you did for me.

(Oh, and for those curious about how successful Anne’s matchmaking was: reader, she married him.)