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.

In mathematics, 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, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology and string theory (Wikipedia).

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How to prove that $\exists u_1\in P_n$ s.t $y[u_1]$ is minimal and there are finitely many $u_1=(g_1,...,g_n) \in \Bbb Z^n$ s.t $\gcd(g_1,...,g_n)=1$

If $P_n=\{y\in M(n,\Bbb R)|$ $y$ is positive definite and symmetric $\}$ then for a fixed $y \in P_n$ consider the set $A=\{y[u]|u \in \Bbb Z^n\}$ where $y[u]=u^tyu$ is clearly positive. Now, how to prove that $\exists u_1\in P_n$ s.t $y[u_1]$ is…
Ri-Li
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How to interpret the symbol $L( \frac{1}{2}, \pi \times \chi)$?

I am trying to interpret the symbol $L( \frac{1}{2}, \pi \times \chi)$ where $\chi = \mathbb{A}^\times / \mathbb{Q}^\times$ and $\pi$ is a cuspidal representation of $GL_2( \mathbb{A})$ (where $\mathbb{A}$ are the adeles over $\mathbb{Q}$),…
cactus314
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A function which is repressnted in terms of Modular Discriminant

Prove following property of modular discriminant. If p is a prime, define $F_{p}(\tau)= p^{11} \Delta(p \tau) + 1/p \sum_{k=0}^{p-1} \Delta(\frac{\tau +k}{p})$. Question : Prove that $F_p(\frac{1}{-\tau})= {\tau}^{12}F_{p}(\tau)$ So, I tried using…
Avenger
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