Proving Hurwitz Identity: Modular Forms & Beyond

[binbagsss]

Hi,

My notes say that hurwitz identity currently has no elementary proof?

One way to prove the identity is through modular forms: to consider Eisenstein series, $E_4^2$ and $E_8$ , note that the dimension of space of modular functions of weight 8 is one, find the constant of proportionality to be $1$ by comparing the first coefficient which is identically one (sufficing here since the dimension of the space is one) and so the identity comes out by setting these two Eisenstein series equal.

What other proofs are there?

Thanks

 

[jvicens]

for your question!

There are actually several other ways to prove the Hurwitz identity, also known as the Hurwitz formula. One approach is through the use of theta functions, which are a type of modular form. By manipulating theta functions and using certain identities, one can arrive at the Hurwitz formula.

Another proof uses the theory of elliptic curves. By considering the Weierstrass form of elliptic curves and using the addition formula for points on the curve, one can derive the Hurwitz formula.

There is also a combinatorial proof of the Hurwitz formula, which uses the concept of partitions and counting certain types of objects. This proof is more accessible to those without a background in advanced mathematics.

Overall, the Hurwitz identity is a fundamental result in mathematics and has been proven using various techniques. Each proof offers a unique perspective on the identity and highlights its significance in different areas of mathematics.