Who proved it first?

During Joel Specter’s thesis defense, he started out by remarking that the q-expansion:

\displaystyle{f = q \prod_{n=1}^{\infty} (1 - q^n)(1 - q^{23 n}) = \sum a_n q^n}

is a weight one modular forms of level \Gamma_1(23), and moreover, for p prime, a_p is equal to the number of roots of

x^3 - x + 1

modulo p minus one. He attributed this result to Hecke. But is it really due to Hecke, or is this more classical? Let’s consider the following claims:

  1. The form f is a modular form of the given weight and level.
  2. If p is not a square modulo 23, then a_p = 0.
  3. If p is a square modulo 23, and x^3 - x + 1 has three roots modulo p, then a_p= 2.
  4. If p is a square modulo 23, and x^3 - x + 1 is irreducible modulo p, then a_p = -1.

At when point in history could these results be proved?

Let’s first start with Euler, who proved that

\displaystyle{ \prod_{n=1}^{\infty} (1 - q^n) = \sum_{-\infty}^{\infty} q^{(3n^2+n)/2} (-1)^{n}}
Using this, one immediately sees that

\displaystyle{f = \sum \sum q^{\frac{1}{24} \left((6n+1)^2 + 23 (6m+1)^2 \right)} (-1)^{n+m}}

This exhibits f as a sum of theta series. With a little care, one can moreover show that

\displaystyle{2f = \sum \sum q^{x^2 + x y + 6 y^2} - \sum \sum q^{2 x^2 + x y + 3 y^2}}.

This is not entirely tautological, but nothing that Gauss couldn’t prove using facts about the class group of binary quadratic forms of discriminant -23. The fact that f is a modular form of the appropriate weight and level surely follows from known results about Dedekind’s \eta function, which covers (1). From the description in terms of theta functions, the claim (2) is also transparent. So what remains? Using elementary number theory, we are reduced to showing that a prime p with (p/23) = +1 is principal in the ring of integers of \mathbf{Q}(\sqrt{-23}) if and only if p splits completely in the Galois closure H of x^3 - x + 1.

Suppose that K = \mathbf{Q}(\sqrt{-23}) \subset H. What is clear enough is that primes p with (p/23) = + 1 split in K, and those which split principally can be represented by the form x^2 + xy + 6y^2 in essentially a unique way up to the obvious automorphisms. Moreover, the class group of \mathrm{SL}_2(\mathbf{Z}) equivalent forms has order 3, and the other \mathrm{GL}_2(\mathbf{Z}) equivalence class is given by 2x^2 + xy + 3y^2. In particular, the primes which split non-principally in K are represented by the binary quadratic form 2 x^2 + xy + 3y^2 essentially uniquely. From Minkowski’s bound, one can see that H has trivial class group. In particular, if x^3 - x + 1 has three roots modulo p, then the norm of the corresponding ideal to K is also principal and has norm p = x^2 + xy + 6y^2. This is enough to prove (3).

So the only fact which would not obviously be easy to prove in the 19th century is (4), namely, that if p = x^2 + xy + 6y^2, then p splits completely in H. The most general statement along these lines was proved by Furtwängler (a student of Hilbert) in 1911 — note that this is a different (and easier?) statement than the triviality of the transfer map, which was not proved until 1930 (also by Furtwängler), after other foundational results in class field theory had been dispensed with by Tagaki (another student of Hilbert!). Yet we are not dealing with a general field, but the much more specific case of an imaginary quadratic field, which had been previously studied by Kronecker and Weber in connection with the Jugendtraum. I don’t know how much Kronecker could actually prove about (for example) the splitting of primes in the extension of an imaginary quadratic field given by the singular value j(\tau). Some of my readers surely have a better understanding of history than I do. Does this result follow from theorems known before 1911? Who proved it first?

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