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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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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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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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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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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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 $\operatorname{SL}_2(\mathbb{...
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Problems about complex torus when reading D-S's A First Course in Modular Forms
Authors's Reasoning:
Definition 1.3.4. A nonzero holomorphic homomorphism between complex tori is called an isogeny.
In particular, every holomorphic isomorphism is an isogeny. Every isogeny surjects ...
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$\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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The issue of defining the action of the double coset operator on the divisor group of a modular curve
This problem comes from the end of Section 5.1 in GTM 228 A First Course in Modular Forms. Let $\Gamma_1$ and $\Gamma_2$ be congruence congruence subgroups of SL$_2(\mathbb{Z})$ and $\alpha\in$ GL$_2^+...
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What is the order of the torsion subgroup of $𝐻_1(\Gamma_1(193),\Bbb{Z})$?
I'm trying to compute the torsion subgroup of the first integral homology group $𝐻_1(\Gamma_1(193),\Bbb{Z})$
This group arises from modular symbols associated to the congruence subgroup $\Gamma_1(193)...
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Cusp forms and $L$-functions of imaginary quadradic fields
I read somewhere that every $L$-function of an imaginary quadratic field $K$ with Grössencharacter is the $L$-function of a cusp form for a certain congruence group of $\operatorname{SL}_2(\Bbb Z)$. ...
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Structure of the anemic Hecke Algebra $\mathbb{T}_1^0(N)$ acting on cusp forms.
Let $N \geq 1$ be an integer. Denote by $\mathbb{T}_1(N)$ the image of the full Hecke algebra $\mathbb{Z}[T_p]_{p \in \mathbb{P}}$ acting on $S_2(\Gamma_1(N))$ and by $\mathbb{T}_1^0(N)$ the image of ...
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$\dim M_2(\Gamma_0(p))$ equals $\dim M_{p+1}(\Gamma(1))$ equals the number of supersingular elliptic curves mod $p$
Let $p \ge 5$ be a prime. Then the following three positive integers are equal: the dimension of $M_2(\Gamma_0(p))$, the dimension of $M_{p+1}(\Gamma(1))$, and the number of supersingular elliptic ...
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$\int_1^\infty (f(iy)-a_0)(y^s+(-1)^\frac{k}{2}y^{k-s})\frac{dy}{y}$ is entire on $\mathbb{C}$, where $f$ is a modular form of weight $k$.
The question comes from the theory of $L$-series of modular forms.
Setup
Suppose $f=\sum_{n\geq0}a_nq^n$ is a modular form of weight $k\geq 4$ w.r.t. the full group $\text{SL}_2(\mathbb{Z})$. We ...
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How to show that $\frac{K(k')}{K(k)}=\sqrt{7} \implies k=k_{7}=\frac{\sqrt{2}}{8}(3-\sqrt{7})$?
$$K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}\tag{1}$$ is the
complete elliptic of the first kind and $k'=\sqrt{1-k^2}$ the
complementary modulus. We are interested in solving:
$$\frac{K(k')}{...
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Limiting case in the proof of Isekis transformation formula
In Chapter 3.5 of 'Modular Function and Dirichlet Series in Number Theory' Apostol introduces the function $$\Lambda(\alpha, \beta, z) = \sum_{r = 0}^{\infty} \{ \lambda((r+\alpha)z - i\beta)+\lambda((...
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How to show $\sum_{n=0}^{\infty}\frac{(6n+1)(2n)!^3}{2^{8n}n!^6}=\frac{4}{\pi}$?
Context
We continue trying to understand Ramanujan's series for $1/\pi$ so this question has this two important references:
"Modular equations and approximations to $π$". By S. Ramanujan. ...
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The analogue of the modular discriminant for exponent values other than 24
The Ramanujan tau function $\tau$ is given by the coefficients of the power series expansion of the modular discriminant $\Delta$:
$$
\sum_{n=1}^{+\infty} \tau(n)\,q^n := q\,\prod_{i=1}^{+\infty}(1-q^...
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Kernel of the residue map $ \mathsf{PSL}_2(k[[t]]) \to \mathsf{PSL}_2(k) $
Let $k$ be a finite field, and let $ k[[t]] $ be the ring of formal power series over $k$. There is a very natural map $ \mathsf{PSL}(2,k[[t]]) \to \mathsf{PSL}(2,k) $ given by evaluation $ t \mapsto ...
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Structure of Hecke algebras for weight 2 cusp forms
Let $p$ be a rational prime and $S_2(\Gamma_0(p))$ the space of weight 2 cusp forms of level $\Gamma_0(p)$.
Let $\mathbb{T}:=\mathbb{Z}[\{T_n,\langle n\rangle\ |\ n\in \mathbb{Z}_{>0}\}]\subset \...
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Equivalence of Hecke operator definitions
I was looking for a reference of the proof that the double coset definition and the sublattice definition for Hecke operators are equivalent. I know the proof, but I'm doing TA in a modular forms ...
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Specializations of the q-Pochhammer symbol and their modularity
The $q$-Pochhammer symbol is defined as
$$(a;q)_{n} = \prod_{i=0}^{n-1}(1-aq^i)$$ where $a\in\mathbb{C}$ and $n\in\mathbb{N}\cup\{\infty\}$
Let $q=e^{2\pi i\tau}$. I am aware that $(q;q)_{\infty}$ is ...
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Congruences between CM Modular forms of different weight for special primes
While playing around with modular forms with CM, I found something interesting:
Set $f_{27.2.a.a}:=q\prod_{n=1}^\infty (1-q^{3n})^2(1-q^{9n})^2=\sum_{n=1}^\infty \alpha_n q^n$ to be a cusp form of $\...
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Perfect cube coefficients of 8-th power of Euler pentagonal function.
Thanks for your reading. For Euler's pentagonal numbers, we know they can be attached to modular form
$\begin{align}
\prod_{n=1}^{\infty}\left(1-x^n\right)&=\sum_{k=-\infty}^{\infty}(-1)^k x^{k(3 ...
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Fourier Series of Automorphic Poincare Series
I've been trying to verify the Kuznetsov trace formula for weight zero, general level, and at different cusps (as far as I've looked there's not a clear write-up of this in the literature). I've ...
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A problem about Eisenstein series for congruence subgroup $\Gamma_1(N)$
I'm new to Modular forms and am now working with the Eisenstein series for the congruence subgroup $\Gamma_1(8)$. I used SageMath to obtain the Eisenstein series of weight 10 and got 6 Eisenstein ...
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