Reading the correspondence between Serre and Tate has been as delightful as one could expect. What is very nice to see — although perhaps not so surprising — is the utter delight that both Serre and Tate find in discussing numerical examples. One of the beautiful aspects of number theory is that there is an abundance of examples, each of which exhibit both special cases of a vast general theory and yet each delighting with their own idiosyncracies: \(\mathbf{Q}(\sqrt{-23}),\) X_0(11), 691, 144169, etc. (It is precisely the absence of such examples, or at least any discussion of them, why geometric Langlands tends to leave me completely cold.) Take, for example, the following:
Letter from Tate to Serre, Dec 8, 1958:
Are you aware that the class number of the field of 97th roots of 1 is divisible by 3457 and 118982593? And that 3457 = 36 * 96 + 1 and 118982593 = 1239402 * 96 + 1?
If reading that doesn’t give you just a little thrill, then you have no soul. Does it have any significance mathematically? The class number is large, of course, which relates to the fact (proved by Odlyzko) that there are only finitely many Galois CM fields with bounded class number. (The reason why one can access class numbers of CM fields F/F+ is that the unit group of F and F^+ are the same up to finite index, so the *ratio* of zeta values \(\zeta_{F}(1)/\zeta_{F^+}(1)\) is directly related to the minus part of the class group \(h^{-}\) uncoupled from any regulator term, so one can access this analytically.) Alternatively, one might be interested in the congruences of the primes q dividing the class number. In this case, we see a reflection of the conjectures of Cohen and Lenstra. Namely, we expect that there is a strong preference for the class group to be “more cyclic,” especially for larger primes. The class group also has an action of \((\mathbf{Z}/97\mathbf{Z})^*\) which is cyclic of order 96. Since one expects the plus part \(h^{+}\) to be very small (and indeed in this case it is trivial), this means that complex conjugation should act non-trivially, which means that the group of order 96 should (at least) act through a quotient of order at least 32. So if the class group is actually cyclic, this forces the prime divisors q of h_F to be 1 mod 32, and even 1 mod 96 if the class group of F doesn’t secretly come from the degree 32 subfield of F (which it doesn’t). (Not entirely irrelevant is Rene Schoof’s nice paper on computing class groups of real cyclotomic fields.)
Both Serre and Tate are unfailingly polite to each other. As a running joke, the expression “talking through one’s hat” occurs frequently, as for example the letter of Nov 14, 1961, where the subtle issue of the failure of \(B \otimes_A C \rightarrow B \widehat{\otimes}_A C\) is discussed. (Another amusing snippet from that letter “Even G. himself makes mistakes when he thinks causally.”) The correspondence is also fascinating from the perspective of mathematical history — one sees the progress of many ideas as they are created, including the Honda-Tate theorem and the Tate conjecture over finite fields. The first time the latter appears (as a very special case) it actually turns out to be an argument of Mumford, who shows Tate an argument (using Deuring) why when two elliptic curves have the same zeta function they are isogenous. This elicits the following reaction from Tate:
Letter from Tate to Serre, May 9, 1962:
“Damn! The result is certainly new to me, and it frankly makes me mad that I never noticed it”
We have all been there, although, to be fair, most of us have the excuse of not being Tate!
“It is precisely the absence of such examples, or at least any discussion of them, why geometric Langlands tends to leave me completely cold.” Thanks for summarizing in a nutshell why number theorists and geometers have sometime such difficulties talking and especially listening to each other. Can I quote that sentence and “If reading that doesn’t give you just a little thrill, then you have no soul.”? With proper attribution, of course.
Happy New Year from a (soulless!) geometer.
Since this is a public blog, requests for permission to quote are hardly necessary. That said, I would remark that this blog adopts a certain tone (implicit in the title) which is always worth considering when interpreting posts, and what I say above is best understood in that context. Most of all, however, I’m surprised by your thesis that number theorists and geometers have difficulty communicating (Serre and Grothendieck?) — many users of the blog certainly use deep geometric ideas in their work. I would also argue that the delight that Barry Mazur (say) takes in X_0(11) is not so different from the delight the Joe Harris takes with the 3264 plane conics tangent to five general plane conics. Does that leave you cold also? (surely not!)
Would you happen to have a reference to the proof of the isogeny theorem using Deuring? Perhaps Tate describes it in the letter mentioned in the post?
I’m guessing the argument was roughly as follows, let me just do the ordinary case for convenience. Suppose that A and B have the same zeta function. Then End(A) and End(B) are (possibly different) orders in the ring of integers of an imaginary quadratic field K in which p splits. Deuring proves the existence of lifts X and Y over some number field F with the property that there exists a prime P in F with X mod P = A and Y mod P = B and such that both X and Y still have CM (necessarily by an order in O_K). Then one uses the global theory to obtain an isogeny between X and Y and thus A and B. (Something that is special to this case is that one can find CM lifts without extending the residue field.)
I think this problem (zeta_A(s) = zeta_B(s) ===> A and B isogenous for elliptic curves A and B) is also an exercise in AOEC somewhere, but I don’t have the book handy at home so I can’t find the reference.