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13

There's a lot one could say, but I'll try to be brief. Roughly the idea (just like with the zeta functions) is that L-functions provide a way to analytically study arithmetic objects. Specifically a lot of interesting data is encoded in the location of the zeroes and poles of L-functions, and because L-functions are analytic objects, you can now use ...


4

A main possibly non-intuitive usage of "form" is as a somewhat particular type of map/function. Traditionally, the word function was used in a more restrained way and it was mainly used for real and complex functions, only. For example, classically in (real) functional analysis one would have: A function would be a map from $\mathbb{R}$ to $\mathbb{R}...


2

Here is the general picture. Let $f \in \mathcal{M}_k(N,\chi)$ be a modular form of weight $k$, level $N$, and nebentypus $\chi$. Suppose that $f$ has the Fourier expansion \[f(z) = \sum_{m = 0}^{\infty} a_f(m) e(mz).\] Then by replacing $z$ with $nz$, we see that \[f(nz) = \sum_{m = 0}^{\infty} a_f(m) e(mnz) = \sum_{m = 0}^{\infty} a_{f_n}(m) e(mz)\] with \[...


1

I'm not a Mathistorian, but... Likely it originally meant its English meaning of "appearance", and it still does in most usages. Quadratic forms have a very specific appearance, namely a homogenous quadratic polynomial. Modular forms are functions satisfying a certain form of equation and some other conditions. Conjunctive/disjunctive/Skolem normal forms are ...


1

So I'm not sure about references specific to Hauptmoduln (except specialized research articles), so perhaps you would be better served by reading up on modular functions, forms and curves. I suggest Milne's online notes which are excellent, or Diamond and Shurmon's book which is also excellent. As far as your specific and interesting question regarding the ...


1

I think the argument is something like this. Note that $t=t(\tau)$ is a meromorphic modular function for $\Gamma_1(6)$. Therefore, if $E(t)(f(t)−\zeta(3))$ is a polynomial in $t$, then $E(\tau)(f(\tau)-\zeta(3))$ would be a meromorphic modular function for $\Gamma_1(6)$. As you said, $E(\tau)$ is a modular form of weight $2$ for $\Gamma_1(6)$. It follows ...



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