A bijective challenge – E. Kowalski's blog

Étienne Fouvry, Philippe Michel and myself have just finished a new paper, which is available on my web page and will soon be also on arXiv. This was quite an extensive project, which also opens many new questions. I will discuss the general problem we consider, and the techniques we use, in other posts, but today I want to discuss a by-product that we found particularly nice (and amusing). It can be phrased as a rather elementary-looking challenge: given a prime number $p$ , and an element $t$ of $\mathbf{Z}/p\mathbf{Z}$ which is neither $0$ , $4$ or $-4$ modulo $p$ , let $N_p(t)$ be the number of solutions $(x,y)\in (\mathbf{Z}/p\mathbf{Z}-\{0\})^2$ of the congruence

$x+\frac{1}{x}+y+\frac{1}{y}+t=0.$

The challenge is to prove, bijectively if possible, that

$N_p(t)=N_p\Bigl(\frac{16}{t}\Bigr)$

and that

$N_p(t)=N_p\Bigl(\frac{4t-16}{t+4}\Bigr)=N_p\Bigl(\frac{4t+16}{t-4}\Bigr)$

if $p\equiv 1\pmod{4}$ .

This sounds simple and elegant enough that an elementary proof should exist, but our argument is a bit involved. First, the number $N_p(t)$ is the number of points modulo $p$ of the curve with equation above, whose projective (smooth) model is an elliptic curve, say $E_t$ , over $\mathbf{F}_p$ . Then we checked using Magma that $E_t$ and $E_{16/t}$ are isogenous over $\mathbf{F}_p$ , and this is well-known to imply that the two curves have the same nunmber of points modulo $p$ . The other two cases are similar, except that for

$\gamma(t)=\frac{4t-16}{t+4}\text{ or } \frac{4t+16}{t-4},$

the relevant isogenies are between $E_t$ and $\tilde{E}_{\gamma(t)}$ , where $\tilde{E}_t$ denotes the quadratic twist of $E_t$ by $-1$ . Hence the number of points are the same when $-1$ is a square modulo $p$ .

In the first case, the isogeny is of degree $4$ , and the others are of degree $8$ , so the formulas which define them are rather unwieldy, at least in the equivalent Weierstrass model.

The best explanation of this has probably to do with the relation between the family of elliptic curves and the modular curve $Y_0(8)$ (a relation whose existence follows from Beauville’s classification of stable families of elliptic curves over $\mathbf{P}^1$ with four singular fibers, as C. Hall pointed out), but we didn’t succeed in getting a proof of all our statements using that link. In fact, we almost expected to find the identities above already spelled out in some corner or other of the literature on modular curves and universal families of elliptic curvers thereon, but we did not find anything.

In the next post, I’ll come back to this to explain the link with our paper, which ostensibly is about estimates for sums of Fourier coefficients of modular forms multiplied with functions “of algebraic origin”. Kloosterman sums will enter the picture to make the connection (in more ways than one!), and we’ll see a rather elegant formula of Beltrami…

P.S. Here is a link to a transcript of a Magma session proving the existence of the isogenies which imply our formulas, and ending with the formula of the $4$ -isogeny, written in terms of the original curve.

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