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Hardy [19] commenting on Ramanujan's work on elliptic and modular functions says

We present only one of Ramanujan's modular equations.It is here that both the profundity and limitations of Ramanujan's knowledge stand out most sharply.

where fori=1and2with

This slightly rewritten form of entry 12(iii) of Chapter 19 of Ramanujan's * Second
Notebook* (see [7], where Berndt's proofs may be studied). One can think
of Ramanujan's quintic modular equation as an equation in the multiplier of
(5.13). The initial surprise is that it is solvable. The quintic modular relation
for
, , and the related equation for , are both
nonsolvable. The Galois group of the sixth-degree equation (see
(5.12)) over
is and is nonsolvable. Indeed both Hermite and Kronecker showed,
in the middle of the last century, that the solution of a general quintic may be
effected in terms of the solution of the 5th-order modular equation (5.12) and the
roots may thus be given in terms of the theta functions.

In fact, in general, the Galois group for of (5.11) has order and is never solvable for . The group is quite easy to compute, it is generated by two permutations. If

are both elements of the -group and induce permutations on the of Theorem 1. For any fixed
From (5.8) and (5.10) one sees that Ramanujan's modular equation can be rewritten
to give
solvable in terms of and . Thus, we can hope to find
an explicit solvable relation for in terms of and .
For **p=3**, is of degree 4 and is, of course, solvable. For **p=7**, Ramanujan
again helps us out, by providing a solvable seventh-order modular identity for the
closely related * eta function* defined by

This leads to the interesting problem of mechanically constructing these equations.
In principle, and to some extent in practice, this is a purely computational problem.
Modular equations can be computed fairly easily from (5.11) and even more easily in
the associated variables **u** and **v**. Because one knows a priori bounds on the
size of the (integer) coefficients of the equations one can perform these
calculations exactly. The coefficients of the equation, in the variables **u** and
**v**, grow at most like . (See [11].) Computing the solvable forms
and the associated computational problems are a little more intricate --- though
still in principle entirely mechanical. A word of caution however: in the
variables **u** and **v** the endecadic modular equation has largest coefficient 165, a
three digit integer. The endecadic modular equation for the intimately related
function **J** {Klein's * absolute invariant*) has coefficients as large as

The paucity of Ramanujan's background in complex analysis and group theory leaves open to speculation Ramanujan's methods. The proofs given by Berndt are difficult. In the seventh-order case, Berndt was aided by MACSYMA --- a sophisticated algebraic manipulation package. Berndt comments after giving the proof of various seventh-order modular identities:

Of course, the proof that we have given is quite unsatisfactory because it is a verification that could not have been achieved without knowledge of the result. Ramanujan obviously possessed a more natural, transparent, and ingenious proof.

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