# Questions tagged [modular-forms]

Questions about modular forms and related areas

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### Level vs. conductor of a supercuspidal representation

What is the relation between level and conductor of a supercuspidal representation of $\operatorname{GL}_2(\mathbb{Q}_p)$ for some prime $p$? Proposition 3.4 in Loeffler and Weinstein - On the ...
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### Recurrence relation between $a(p^{2n})$'s

Let $f$ be a Hecke eigenform of weight $k,$ and $a(p^n)$ be the $p^{n}$th Fourier co-efficient of $f$. I need to a determine a recurrence relation (Hecke type) between $s(n)=a(p^{2n})'s$. We can write ...
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### Analogous theorem for Hilbert modular forms

I have studied modular forms and saw a correspondence like a newform correspond to a automorphic representation of $\mathrm{GL}_n(\mathbb{A_Q})$. Does any similar result holds for Hilbert modular ...
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### Conceptual meaning of a non-linear relation connecting $6$ Mordell integrals?

Define Mordell integral by $$\phi_\alpha(\theta)=\int\limits_0^\infty\frac{\cos\pi \theta x}{\cosh \pi x}\,e^{-\pi \alpha x^2}dx.\tag{1}$$ There are a lot of linear relations connecting integrals ...
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### Properties of Mod $\ell^m$ Galois representation associated to modular form

(Sorry for my poor english..) Let $F(z)\in S_{2k}(SL_2(\mathbb{Z})$) be a newform and $\ell$ be a prime larger than $3$. Let $K$ be a some number field and $v$ be a prime of $K$ over $\ell$. Let $K_v$...
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### Corollaries of the halo conjecture that do not involve the eigencurve

In the theory of p-adic modular forms there is a certain construction called the Coleman-Mazur eigencurve. The spectral halo conjecture roughly states that if you remove a closed subdisc of the weight ...
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### Eichler Shimura in higher genera

Let $\mathcal{M}$ denote the moduli stack of smooth elliptic curves over $\mathbb{C}$. There is a local system, $\mathcal{H}$, on $\mathcal{M}$ with fibre $H^{1}(E,\mathbb{C})$ at the $\mathbb{C}$-...
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### Generalisation of modular forms

I am looking for a generalisation of a modular form that transforms as something like: $f(\frac{a \tau+b}{c \tau+d}) = (c \tau+d)^k c^k f(\tau)$ I understand this cannot be literally true, as the ...
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### Hodge-Tate map in families

I've been working with the Hodge-Tate map in the context of modular forms, but I don't understand really what it is. For an elliptic curve $E$ over $K$, a number field, the Hodge-Tate decomposition is ...
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### Modular form attached to a hecke character of cubic/quartic extension of rational numbers

We know that for each Hecke character of a quadratic extension of $\mathbb{Q}$, we can define a modular form (in fact a cusp form). We can find this construction in the book Topics in Classical ...
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### Shortest possible reasonably self-contained formulation of the modularity theorem

This is question in mathematical exposition, not research, I hope this is ok. I am writing a book about great theorems. My question is: what is the shortest formulation of the modularity theorem, ...
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### Can the Petersson inner product $\langle f(z), f(2z) \rangle$ be zero?

Suppose $f$ is a weight $k$ cuspidal Hecke eigenform on $\Gamma_0(N)$. Then $f(2z)$ is a weight $k$ cuspform on $\Gamma_0(2N)$. Is it possible that $f(z)$ and $f(2z)$ can be orthogonal (regarded as ...
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### Congruences of modular forms modulo other modular forms

Congruences between modular forms are certainly a big topic in number theory, maybe with $$E_{p-1}\equiv 1 \mod p \qquad \text{for a prime }p\geq 5$$ as the easiest example. Sometimes, $p$ might be ...
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### Classical Hecke operators and Hecke algebra of type $A_1$

What's the relation between the classical Hecke operators (as defined in J. P. Serre's A course in arithmetic chapter 6) and the Hecke algebra of type $A_1$, i.e. the algebra generated by the vertices ...
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### Inside the construction of the Frey curve

Consider the frey curve $E\mathrel: y^2=x(x-a^{p})(x+b^{p})$ with conductor $N =2\prod_{p|(abc)^{2p}}p$. Frey assume that $p$ does not divide $(abc)^{2p}$ so the level of the cusp form predict by ...
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### Hecke operators - why is it well-defined?

I have a basic question concerning Hecke Operators in spaces of Modular Forms. I am followinf these notes: http://www.few.vu.nl/~sdn249/modularforms16/Notes.pdf. In page 49, the author writes ...
Say there is a (non-holomorphic) real function $Z(\tau,\bar{\tau})$ which obeys the (non-holomorphic) modularity conditions of some weight $k$ Z(\tau+1,\bar{\tau}+1)=Z(\tau,\bar{\tau}),\qquad Z(-1/\...
### Upper bound of summation $\sum_{m < \frac{1}{2}X} \frac{|a(m_1m_2^2)|}{m_1m_2^2} \log\frac{X}{m}$
I am studying the paper M. Ram Murty, V. Kumar Murty: Mean values of derivatives of modular $L$-series, Ann. of Math. (2) 133 (1991), no. 3, 447-475. Let $L(s)=\sum_{n=1}^{\infty} \frac{a(m)}{m^s}$ ...