The content of this note was the topic of a lecture by Günter Harder at the School on Automorphic Forms, Trieste 2000. The actual problem comes from the article A little bit of number theory by Langlands.

The problem is about a connection between two quite different objects. The first object is the following pair of positive definite quadratic forms: $$ P(x,y,u,v) = x^2 + xy + 3y^2 + u^2 + uv + 3v^2 $$ $$ Q(x,y,u,v) = 2(x^2 + y^2 + u^2 + v^2) + 2xu + xv + yu - 2yv $$ The second object is the elliptic curve $$ E: y^2 + y = x^3 - x^2 - 10x - 20. $$

To each of our objects we now associate a series of integers. For each integer $k \ge 0$ define $$ n(P,k) = | \{(a,b,c,d) \in {\mathbb Z}^4: P(a,b,c,d) = k\} |, $$ $$ n(Q,k) = | \{(a,b,c,d) \in {\mathbb Z}^4: Q(a,b,c,d) = k\} |. $$

As a matter of fact, these integers are divisible by $4$ for any $k \ge 1$ because of the transformations $(a,b,c,d) \to (-a,-b,-c,-d)$ and $(a,b,c,d) \to (c,d,a,b)$.

For any prime $p \ne 11$ we now put $$ a_p = |E({\mathbb F}_p)| - (p+1),$$ where $E({\mathbb F}_p)$ is the elliptic curve over ${\mathbb F}_p$ defined above.

Then Langlands claims

**For any prime $p \ne 11$, we have **
$ 4a_p = n(P,p) - n(Q,p).$

The "classical" explanation proceeds as follows: Given the series of integers $n(P,p)$ and $n(Q,p)$, we form the generating series $$ \Theta_P(q) = \sum \limits_{k=0}^\infty n(P,k) q^k = 1 + 4q + 4q^2 + 8q^3 + \ldots, $$ $$ \Theta_Q(q) = \sum \limits_{k=0}^\infty n(Q,k) q^k = 1 + 12q^2 + 12q^3 + \ldots. $$ If we put $q = e^{2\pi i z}$ for $z$ in the upper half plane, then $\Theta_P$ and $\Theta_Q$ become ${\mathbb Z}$-periodic holomorphic functions on the upper half plane. As a matter of fact, the classical theory of modular forms shows that the function $$ f(z) = \frac14 (\Theta_P(q) - \Theta_Q(q)) = q - 2q^2 - q^3 + 2q^4 + q^5 + 2q^6 - 2q^7 - 2q^9 - 2q^{10} + q^{11} - 2q^{12} \ldots $$ is a modular form (in fact a cusp form since it vanishes at $\infty$) of weight $2$ for $\Gamma_0(11)$. More precisely, we have $ f(z) = \eta(z)^2 \eta(11z)^2,$ where $\eta(z)$ is Dedekind's eta function, a modular form of weight $\frac12$.

Now we have connected the quadratic forms to a cusp form for $\Gamma_0(11)$. This group has two orbits on the projective line over the rationals, which means that the associated Riemann surface can be compactified by adding twocusps: the result is a compact Riemann surface $X_0(11)$ of genus $1$. Already Fricke has given a model for this Riemann surface: he found that $X_0(11) \simeq E$ for the elliptic curve defined above.

Now consider the space of cusp forms for $\Gamma_0(11)$. There are Hecke operators $T_p$ acting on it, and since it has dimension $1$, we must have $T_p f = \lambda_p f$ for certain eigenvalues $\lambda_p \in {\mathbb Z}$. A classical result due to Hecke then predicts that the eigenvalue $\lambda_p$ is the $p$-th coefficient in the $q$-expansion of $f(z)$. Eichler-Shimura finally tells us that $\lambda_p = a_p$. Putting everything together gives Langlands' claim.

Way back then I asked Harder how all this follows from the general Langlands conjecture, and he replied that he did not know. Langlands himself said his examples came "from 16 of Jacquet-Langlands". So here's my question:

* Does anyone here know how to dream up concrete results like the one above from Langlands' conjectures, or from "16 of Jacquet-Langlands"?*

How to dream up Langlands' programme from results like the one above?$\endgroup$ – Chandan Singh Dalawat Jul 23 '10 at 3:01