# Questions tagged [modular-forms]

A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group.

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### Factorization of modular forms using zeros and poles

In complex analysis, it is possible to decompose any given entire function into a product of linear factors using the zeros, using the Weierstrass factorization theorem. For example, $\sin(z)$ is ...
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### Applications of the Modularity Theorem

Besides its role in proving Fermat's Last Theorem and its well-known consequence that the $L$-function of an elliptic curve is defined at 1 (so that, in particular, the BSD Conjecture makes sense), ...
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### Strong multiplicity one theorem for new forms of different levels

Let $f, g$ be newforms of level $N_f, N_g$ (possibly different) with $N_f, N_g |N$, and assume that they have same eigenvalue for Hecke operators $T_p$ with $(p, N) = 1$. Then $N_f = N_g$ and $g = cf$ ...
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### Lehmer's formula for Ramanujan's tau numbers [closed]

Ramanujan tau function $τ(q)$ together with Dedekind eta function $η(q)$ is defined by $τ(q)=η²⁴(q)=q∏_{k=1}^{∞}(1-q^{k})=∑_{n=1}ⁿτ(n)qⁿ$. where $τ(n)$ are Ramanujan's tau numbers. Recently I came ...
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### Ramanujan Identities between Eisenstein Series $E₄$ and $E₆$ and Elliptic Integral of the Second Kind $K$ [closed]

Do you know how to prove the following Ramanujan identities: $E₄=((16)/(π⁴))(k⁴-k²+1)K⁴$ and $E₆=((32)/(π⁶))(2k⁶-3k⁴-3k²+2)K⁶$ where $E₄$ and $E₆$ are the famous Eisenstein Series with variable $q$ ...
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### Writing the automorphic form $\phi_f$ of a newform $f$ as a pure tensor

Let $\mathbb{A}$ be the adele ring of $\mathbb{Q}$. Let $f\in S_k(\Gamma_0(N),\chi)$ (possibly a newform) and $\phi_f:\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A})\to \mathbb{C}$ be its ...
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