You surely know it's a rational curve. What sort of information regarding the relation between $a$ and $b$ do you want?

There are plenty of nonconstant entire functions solving $$a(t)^3 + b(t)^3=c(t)^3.$$ For your favourite non-constant entire function $a(t)$, set $b(t)=a(t)$ and $c(t)=2^{1/3}a(t)$.

In that case should the ``elliptic logarithm'' of a rational point have this property then the answer to the original question would be `yes'. I don't think this can be the case but proving it suddenly looks like hard work :-(

Thanks in advance, Wadim. I'm not at all familiar with Yoshida's book.

Like W and W Chandraskeharan gives only the transformation law for $\tau\mapsto−1/\tau$ and not for general elements of $\Gamma_0(4)$. But they do give a reference to the old treatise of Tannery and Molk.

Kevin, repeated doubling of a point doesn't produce the subgroup generated by that point. What is needed is a theorem to the effect that if $\xi$ is irrational then the set of $2^n\xi$ in $\mathbb{R}/\mathbb{Z}$ meets every neighbourhood of the origin in $\mathbb{R}/\mathbb{Z}$. I'm sure this is true but can't see an immediate proof.

A nice proof of Schur's theorem is at M. Mirzakhani `A simple proof of a theorem of Schur' Amer. Math. Monthly 105 (1998), 260-262.

One can get all dimensions up to $n(n+1)/2$ by using subalgebras of upper triangular matrices. We can alo get some larger examples by the construction $(A\ B;0\ D)$ where $A$ and $D$ run through given subalgebras of $M_k$ and $M_{n-k}$ and $B$ is arbitrary. Some dimensions are not accessible by these constructions, e.g., dimension $8$ when $n=3$. Are there any subalgebras with these dimensions?

The integral $\int_0^\infty e^{-t}\log t\,dt$ equals $\Gamma'(1)$. This can be evauated as $-\gamma$ using the infinite product for the gamma function.

Vaserstein gave a more elementary proof of the Quillen-Suslin theorem. This was included in the second and third editions of Lang's Algebra.

Alas, although Whittaker and Watson prove the formula for the substitution $\tau\mapsto-1/\tau$ they do not give explicit formulae for the general transformation from $\Gamma_0(4)$.