Let be a lattice. The Weierstrass -function is defined as

It is -invariant, meromorphic, and has a double pole at each lattice point and no other poles. Its Laurent expansion at the origin is

where . Its derivative is

The functions and satisfy

in terms of the quantities and . Now if we take , where is in the upper half-plane, then , where is the weight Eisenstein series. These Eisenstein series, or rather the normalized Eisenstein series , satisfy certain relations, such as: (wikipedia)

In most basic texts on modular forms, these identities are derived by proving and exploiting the fact that the space of modular forms of weight is finite-dimensional. For instance, the fact that and have the same value at , combined with , implies . However, there is another, more “hands on” way to derive these identities.

Let us prove that . Substituting Laurent series at the origin in the equation , we see after some rearranging that the function

is identically , and therefore

Now

Thus, for instance, . Making these substitutions in the first equation and factoring out , we see that

Since , we have .

In fact, we have a bijection between and the set of infinituples of complex numbers satisfying the infinite system of equations :

Luckily, the values of and determine all of the others, and the ring is generated by and , and is in fact isomorphic to . This means precisely that is in bijection with the closed points of .

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This proof is beautiful! Thanks for bringing this much joyal to our lives!