## Origin of symbol *l* for a prime different from a fixed prime?

I've never seen an authoritative explanation for the choice of the lower case letter $\ell$ or $l$ to denote an arbitrary prime different from a given prime $p$. This now has its own LaTeX command \ell, but has been in use at least since the old work of Taniyama and Weil involving L functions. That use of the upper case letter might have suggested the lower case here, I guess(?) The letter q would seem more natural in elementary number theory. The write-up of Serre's 1967 McGill lectures was published in 1968 by W.A. Benjamin under the title Abelian l-adic representations and elliptic curves. There his convention is to denote prime numbers by $\ell, \ell', p, \dots$, stating: "we mostly use the letter $\ell$ for $\ell$-adic representations and the letter $p$ for the residue characteristic of some valuation".

I've heard this question raised but not answered quite a few times. For instance, after a colloquium talk in Hamburg given by Bhama Srinivasan on Deligne-Lusztig characters, the elderly Ernst Witt asked the non-technical question I've just raised. (He had done impressive work in his youth but became a convert to the Nazi cause without apparently committing any war crimes. Possibly he was the young man reported to have shown up once at Emmy Noether's seminar wearing a pro-Nazi uniform. In old age he had retained some mental acuity but developed phobias about for example the flooring material in the math tower, which required talks like the ones Bhama and I gave to move to a remote building.)

[ADDED] Both Franz and quim point in the direction of how the symbol $l$ became common for prime numbers in Hilbert's development of Kummer's work. There he considers an $l$th root of unity ($l$ an odd prime) instead of $\lambda$ used earlier by Kummer. Later on I guess it became a default option for many people to use $l$ for a prime different from a given prime $p$, especially when $q$ became used commonly for a power of $p$.

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Witt was an SA (not SS) member, but "convert to the Nazi cause" seems a bit over the top (unlike his teacher Hasse). I don't have any "inside" information, but from what I read, it didn't appear any different from people in USSR and Warsaw Pact countries becoming members of communist youth organizations, often because it was advantageous or even necessary for academic career. – Victor Protsak Jul 2 2010 at 7:43
Thanks for the correction. The details about Witt at that time are not well documented. Some people attracted to the right-wing were far more militant than others. In any case, Witt's postwar visit to Stony Brook was coolly received by many people there, according to what I was later told. And Witt's mathematical career like many others in Europe got derailed in the 1930s and 1940s. – Jim Humphreys Jul 2 2010 at 11:12
Hopefully this will be the last remark on the subject - but the SA was most certainly not a youth organisation; it was an extremely violent paramilitary organisation for adults. Witt was an active member of the NSDAP. See the references on his Wikipedia page. – H A Helfgott Jul 2 2010 at 15:05
For those interested in this topic, see the book "Mathematicians under the Nazis" by Sanford L Segal. There is a section on Witt there. – James Borger Jul 2 2010 at 16:31
I have seen $q$ used for the alternative prime in some textbooks on elementary number theory. The trouble with $q$ is that it is often used to denote a power of $p$. – Timothy Chow Jul 2 2010 at 18:37

This elaborates quim's answer. Kummer did indeed use $\lambda$ for denoting primes (in connection with cyclotomic fields); he borrowed the notation from Jacobi's articles on cyclotomy as well as from his notes of the number theory lectures in 1836/37. When Hilbert rewrote Kummer's contributions in his Zahlbericht, he started the chapter on cyclotomic fields with "Let $l$ denote an odd prime number". The reason for switching from the Greek to the Latin alphabet was Hilbert's custom to use Latin letters for rational numbers. Hilbert also used ${\mathfrak l}$ for prime ideals above $2$.

Edit. For what it's worth: Euler used primes $\lambda n + 1$ in art. 92 of his article E449.

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Vielen Dank! You and quim provide probably as rational an explanation as one will get for the notational evolution. Number theory is ancient, but some conventions tend to stabilize when found in an influential source like the Zahlbericht (which takes up about 300 pages in the first volume of Hilbert's collected papers). He favors lower case roman letters for rational numbers but lower case greek letters for algebraic numbers in general. Kummer's work is a focal point of Hilbert's final two parts Der Kreiskorper and Der Kummersche Zahlkorper, where $l$ constantly appears. – Jim Humphreys Jul 6 2010 at 20:20
Ja, vielen Dank! – quim Jul 6 2010 at 21:38
This explains $l$ as notation a prime, but was it not Weil who established the tradition of using that notation for auxiliary primes distinct from the characteristic? – T. Jul 8 2010 at 5:47

For a mathematician $k$ would be more logical (reflection symmetry in the alphabet)? But confusing because $k$ is a field ... so take one step more. Any better explanations?

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@Mariano: Charles' logic is that $p$ is the third letter in the second half of the alphabet whereas $k$ is the third to last letter in the first half of the alphabet. This is a quite impressively silly explanation (seriously). But I have no idea why number theorists use $\ell$ for primes. Moreover, we use $\ell$ and $p$ for different purposes: $\ell$-adic cohomology is not the same as $p$-adic cohomology! That's also pretty impressive, I think. – Pete L. Clark Jun 30 2010 at 19:21
l and p are the first and last letters in the rythm segment l-m-n-o-p in the abc song. – Adam Gal Jun 30 2010 at 19:34
If we're going to make symmetry-based arguments, then primes should be denoted by q, b, or d. – Michael Lugo Jun 30 2010 at 20:11
@Micheal, while it is a silly convention, I have been known to use the letters p, q, b, d to stand for the vertices of a rectangle... – Mariano Suárez-Alvarez Jul 2 2010 at 14:22
I thought that the $K$ in $K$-theory was for Klass. Is this not the case? – Emerton Jul 6 2010 at 22:41

The possibly incorrect folk understanding (which may be just the background you assume for your question) is that Weil set the tradition in place by choosing $\ell$ or $l$ as the prime different from $p$. He did this (at least) when considering Galois action on torsion (or cohomology) of elliptic curves and/or Abelian varieties and through the French school of algebraic geometry and number theory it propagated universally. Or so the folklore goes.

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 If it was Weil (which seems plausible), I wonder what led him to make such a choice of letters. As I just commented to quim, I may have to venture further into his collected papers. – Jim Humphreys Jul 6 2010 at 16:11

Kummer used $\lambda$ for a prime (he also used q), see here. I have no clue whether Weil introduced $\ell$ on his own or was following a previous tradition, but $\lambda$ is really close to $\ell$...

I don't know if Kummer introduced $\lambda$ or it was already in use by Jacobi, Dirichlet, or someone else. Does anybody know?

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 This further look back into history is a good reminder of how irrational the development of notation has been in mathematics. Still, someone must have been first to use (and maybe rationalize) the symbol $\ell$ in the way indicated. It might have been Weil, so maybe I will give his collected papers a try when I get courageous. – Jim Humphreys Jul 6 2010 at 16:09