The modular equation $\Phi_n(X,Y)$ is a polynomial in $\mathbf Z[X,Y]$ relating the modular invariant $j$ and the functions $j\left(\frac{a\tau+b}{c\tau+d}\right)$, where $ad-bc=n$.

For example, we have identically


It is often asserted that the coefficients of $\Phi_n(X,Y)$ are astronomically large even for small values of $n$. Why is this so? Is there some lower bound on the coefficients?


1 Answer 1


Paula Cohen, On the coefficients of the transformation polynomials for the elliptic modular function, Math. Proc. of the Cambridge Phil. Soc. 95, 389–402 (1984).

The largest coefficient $c_{m}$ in absolute value of $\Phi_{m}$ is of order $$c_{m}\simeq m^{km},$$ with $k$ a number of order unity ($k=9$ for $m=2^n$). So for $m=2^{10}=1024$ one has $c_m\simeq 10^{27743}$ --- astronomically large, indeed.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.