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.
1,512 questions
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Why elliptic curves instead of Edwards curves?
I have been wondering this for a while; I get how elliptic curves of the Weierstrass form $y^2=4x^3-g_2x-g_3$ have the lowest degree and are the simplest way to study a genus $1$ curve. But why aren’t ...
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How to prove this Dedekind eta function identity?
I'm studying modular forms and reading Tom M. Apostol's book. I'm studying this modualr form: $\eta(\tau)=e^{\frac{\pi i\tau}{12}}\prod_{n=1}^{\infty}(1-e^{2\pi in\tau}).$ I am trying to prove the ...
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Sturm bound and the modular forms of elliptic curves
Let $p$ be a prime, and let $E/\mathbb{Q}$ and $F/\mathbb{Q}$ be elliptic curves. Suppose the modular forms $f$ attached to $E$ and $g$ attached to $F$ have $q$-expansions that agree modulo $p$ in ...
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If modular form satisfies $a_n(f_E) \equiv a_n(f_F) \pmod{p}$ for all $n \geq 1$, are the semisimplification isomorphic as $G_{\mathbb{Q}}$-modules?
Let $p$ be a prime. Let $E$ and $F$ be elliptic curves over $\mathbb{Q}$, and let $f_E$ and $f_F$ denote the associated modular forms. Suppose that all Fourier coefficients are congruent modulo $p$, i....
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References on efficiently applying Hecke operators to Eta Quotients
I will share the formula for the Hecke operator as given in A First Course in Modular Forms by Fred Diamond and Jerry Shurman.
Proposition 5.2.1.
Let $N \in \mathbb{Z}^+$, let $\Gamma_1 = \Gamma_2 = \...
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Crystalline $p$-adic Galois representations associated to modular forms
Let $K$ be a number field, and let $G_K$ be its absolute Galois group. Let $f$ be a normalized Hecke eigenform of weight $k( \geq 2)$, level $N$, and character $\chi$. Let $p$ be a rational prime with ...
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Mod $p$ reduction of $X(p)$. [closed]
Let $X(p)$ be the mudular curve over ${\Bbb Q}(\zeta_p)$. Suppose ${\frak X}(p)$ is the regular model over ${\Bbb Z}(\zeta_p)$. If we consider the mod ${\frak p}$ reduction of ${\frak X}(p)$ over ${\...
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A step regarding calculations with Maass forms
I have a question about a step in the proof of Proposition 3.3.2 in Goldfeld's book Automorphic forms and L-functions for the group $\textrm{GL}_n(\mathbb R)$.
If $f$ is a Maass form for $\textrm{SL}...
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Change-of-sign of the $GL_2$ Maass form Fourier coefficients $\rho_f(m)$ and $\rho_f(-m)$
Suppose that $f$ is a half-integral-weight $GL_2$ Maass form of level $q$ and nebentypus $\chi$. If $\rho_f(m)$ denotes the $m^{th}$ Fourier coefficient of $f$, then it is known that $$\rho_f(-m)=\pm\...
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An $\operatorname{SL}_2(\mathbb{Z})$-invariant meromorphic function with large poles
I am learning about modular forms and modular functions, and in working out the nitty-gritty details, I have stumbled on the following question:
Is it possible to find an $\operatorname{SL}_2(\mathbb{...
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Closed form of the integral $\int_{0}^{i\infty} \frac{E_4(z)}{\sqrt{j(z)}}dz$
When playing around with modular form integrations, one may accidentally come across the following mysterious integral:
\begin{align*}
I=\int_0^{i\infty}\frac{E_4(z)}{\sqrt{j(z)}}dz\approx 0....
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Problems about complex torus when reading D-S's A First Course in Modular Forms
Authors's Reasoning:
Definition 1.3.4. A nonzero holomorphic homomorphism between complex tori is called an isogeny.
In particular, every holomorphic isomorphism is an isogeny. Every isogeny surjects ...
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A representation of $S(q)=\sum_{n\ge1} n q^{n^2}$ via Zwegers’ $\mu$-function and its modular completion
I am looking at the Lambert-type series
$$
S(q)=\sum_{n\ge1} n\,q^{n^2}\qquad (q=e^{2\pi i\tau},\ \Im\tau>0),
$$
and I found that it admits a representation connected to Zwegers’ $\mu$-function.
I ...
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$\lambda$-torsion representations and mod $\lambda$ coefficients of primitive forms
Let $f \in S_2(\Gamma_0(N))$ be a primitive form (newform + normalized eigenform), $A_f$ the attached abelian variety over $\mathbb{Q}$, $K$ the number field of $f$ and suppose that the coefficients ...
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Modular-like behavior of function formally similar to theta function
I've been looking at the function
$$f(z) = \sum_{n=1}^\infty \left(n-\frac{1}{2}\right) e^{i\pi \left(n-\frac{1}{2}\right)^2 z}$$
for $z$ in the upper half-plane, which is formally related to the half-...
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