Questions about modular forms and related areas

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### Number of vectors of fixed norm

Let $P$ and $Q$ be two even, unimodular, positive definite quadratic forms of rank $n$. Let $r_{k}(P)$ be the number of vectors of norm $k$, in symbols:
$$
r_k(P)=\textrm{cardinality of }\{v\in \...

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120 views

### Hilbert modular forms twist-equivalent to their conjugates

Let $L / K$ be a solvable (or cyclic) Galois extension of totally real fields, and let $f$ be a Hilbert modular newform over $L$.
Suppose that, for every $\sigma \in Gal(L / K)$, the conjugate ...

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**3**answers

311 views

### Jacobi's theorem on sums of two squares (reference request)

One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals
$$4(d_1(n)-d_3(n)),$$
where the function $d_i$ counts the number ...

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**0**answers

48 views

### automorphic forms associated with symmetries of vertices of uniform honeycombs in hyperbolic space

Is there a catalogue of automorphic forms (modular/Maass/Siegel/Hilbert...) which lists them in terms of Poincaré series associated with the symmetries of the vertices of uniform honeycombs in ...

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523 views

### Why is 12 the smallest weight for which a cusp forms exists

I have already asked this question on MathStackExchange (here) but did not receive an answer, therefore I am trying it here:
On wikipedia (here) I have read the following:
Twelve is the smallest ...

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47 views

### Is there an analogue of Fourier series using infinite series of Jacobi elliptic functions to model doubly-periodic functions?

We can use Fourier series to model general singly-periodic functions. Infinite sums of orthogonal trigonometric polynomials can approximate any singly-periodic function that is continuous along its ...

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69 views

### Adjoint of U_p and Atkin-Lehner theory

In the below everything is quoted from Miyake's "Modular Forms".
Let $p \mid N$. We have Hecke operator
$T(p)\in \mathscr{R}(N)$ (pg. 135) on $S_{k}(\Gamma_0(N))$ given by
$$T(p) = \Gamma_0(N)\big(\...

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79 views

### Calculate Ramanujan's class invariant by using modular equation of degree $5$

Let $$K(k):=\int_{0}^{\frac{\pi}{2}}\frac{d\phi}{\sqrt{1-k^2\sin^2\phi}}=\frac{\pi}{2}{ _2F_1\bigg(\frac{1}{2},\frac{1}{2},1;k^2 \bigg)}$$
where $0<k<1$
Let $K, K′, L$ and $L′$ denote the ...

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**1**answer

250 views

### Elementary proof of Ramanujan's “most beautiful identity”

Ramanujan presented many identities, Hardy chose one
which for him represented the best of Ramanujan. There are many proofs for this identity.
(for example, H. H. Chan’s proof, M. Hirschhorn's proof....

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**1**answer

221 views

### Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$
It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity $\eta(-\frac{...

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96 views

### q-expansion principle for Γ(N)

I would like to know if there's a $q$-expansion principle for $\Gamma(N)$.
Namely, let $f$ be a weight $k$ weakly holomorphic modular form for $\Gamma(N)$ whose $q$-expansion at infinity has ...

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53 views

### What is the relation between roots of classical and Atkin modular polynomial?

Modular polynomials are a good tools in elliptic curves. I need to find the roots of $l$-th classical modular polynomial $\Phi_l(X,Y)$ over the prime field. Unfortunately the coefficients of these ...

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310 views

### How many integer solutions of $a^2+b^2=c^2+d^2+n$ are there?

Are there any references to study the integer solutions (existence and how many) of Diophantine equations like $a^2+b^2=c^2+d^2+2$, $a^2+b^2=c^2+d^2+3$, $a^2+b^2=c^2+d^2+5$...? Actually, I can prove ...

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116 views

### Clarification request on sign changes of Hecke eigenvalues

In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in ...

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181 views

### Does viewing an Eisenstein series as a sum over cusps explain the antagonism between Eisenstein serieses and cusp forms?

I'm trying to understand the relationship between various aspects of the concept of "Eisenstein series" (as discussed for example in Diamond & Shurman's "A First Course in Modular Forms"), in ...

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103 views

### Kitagawa's p-adic modular symbols for different weights: a confusing observation

References are to K. Kitagawa, "On standard $p$-adic $L$-functions of families of elliptic cusp forms", Contemp. Math. 165, 1994.
Let $\mathcal O$ be the ring of integers in a finite extension of $\...

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**1**answer

235 views

### On $\eta(6z)\eta(18z)$ and the splitting / modularity of $x^3 - 2$

Consider one of the simplest non-abelian examples of modularity. Let $$\eta(6z)\eta(18z) = q\prod_{n=1}^\infty (1 - q^{6n})(1 - q^{18n}) = q - q^7 - q^{13} -q^{19} + q^{25} + 2q^{31} - q^{37} + 2q^{43}...

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196 views

### Non-vanishing of L-function of modular form

There is a theorem by Langlands and Shalika (link) that the L-function of a cuspidal automorphic representation does not vanish on the line $\mathrm{Re}( s)=1$ (in their normalization which might be ...

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**1**answer

456 views

### What is the motivation behind Ramanujan's conjecture?

One motivation I have seen given for Ramanujan's conjecture for the estimate
$$ |a_p|< C p^{k - \frac{1}{2}} $$
for the Fourier coefficients of a cusp form of weight $2k$ is that it allows one to ...

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74 views

### Split multiplicative galois representation and specialization

My questions stems from my attempt to understand the paper of Greenberg and Stevens about the Mazur-tate-Teitelbaum conjecture (you can find the paper here). To understand this question you probably ...

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148 views

### Comparison of algebraic and analytic q-expansion

I would like to check that algebraic and analytic q-expansion of a modular form coincide.
I'm thinking about modular forms as global sections of some sheaf on modular curves. If $X$ is a modular ...

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**1**answer

175 views

### Special fibre of the modular curve $X(N)$

Let $N$ be an integer $\geq 3$ and $X(N)\rightarrow \mathrm{Spec } \mathbb{Z}[1/N]$ is the projective smooth modular curve defined in Deligne-Rappoport. Is there an exemple of $N$ for which the ...

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146 views

### Equivalent definitions of the Hasse invariant

As probably many others before me, I got stuck in verifying all the nice properties of the Hasse invariant.
Let me start by recalling one definition:
Let $E\to S$ be an elliptic curve in ...

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**1**answer

135 views

### Congruence Primes and Modular Degrees

Let $\mathcal{S}=S_2(\Gamma_0(N) \cap \mathbf{Z} [[ q ]]$ be the set of cusp forms of weight $2$ on $\Gamma_0(N)$ with integral coefficients.
Let $f \in \mathcal{S}$ be a normalized newform, so it ...

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**1**answer

218 views

### An electronic copy of Vishik's work on $p$-adic $L$-functions for modular forms

This question is very simple.
Would someone be so nice as to send me an electronic copy of M. M. Vishik, Non-Archimedean measures connected with Dirichlet series, Mat. Sb. (N.S.), 1976, Volume 99(...

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**1**answer

119 views

### Maass form properties and their fourier coefficients

Some Maass form can be written ($K_{iR}$ is the K-Bessel function):
$$f(x+iy)=\sum_{n \ne 0}^{\infty} a_n \sqrt{y} \;K_{iR}(2\pi |n| y) \; e^{2 i\pi nx}$$
with the $a_n$ multiplicative, but inversly ...

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156 views

### Applications of Level Lowering

What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...

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741 views

### Sphere packings : what next after the recent breakthrough of Viazovska (et al.)?

Given the march 2016 breakthrough concerning sphere packings by Viazovska for the case of dimension 8, and by Cohn, Kumar, Miller, Radchenko and Viazovska for the case of dimension 24, it follows that ...

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750 views

### Does X(13) have potentially good reduction at 13?

The complete level modular curve $X(p)$ does not have potentially good reduction at $p$ for any $p \neq 2,3,5,7,13$ because then there are cusp forms on $X_0(p)$ showing up in the cohomology of $X(p)$,...

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211 views

### Field cut out by a CM modular form is imaginary

Let $f=\sum_{n=1}^\infty a_nq^n$ be a newform of level $N$ and weight $k\ge 2$. Suppose that $f$ is a CM modular form in the sense of §3 of Ribet's paper Galois representations attached to eigenforms ...

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134 views

### The Fricke involution and expansions at infinity

Let $p$ be prime, and $f$ be a modular form for $\Gamma_0 (p)$ whose expansion at infinity has coefficients in ${\mathbb Z}\left[1/p\right]$. I'd like a down to earth proof that the same holds true ...

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166 views

### Computing coefficients for the slash operator of a modular form

Suppose $f$ is a classical modular form of weight $r$ for a (congruence) group $\Gamma$. Let $\gamma$ be any matrix in $\operatorname{SL}_2(\mathbb{Z})$. Then the slash operator $|_\gamma$ is usually ...

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67 views

### the shifted convolution sums and the sub convexity problem for l functions

in the paper of gergely harcos, an additive problem in the fourier coefficients of cusp forms, a bound for the shifted convolution sums for hecke eigenvalues was explicited and i thought that his ...

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336 views

### Is Faltings' $p$-adic Eichler-Shimura isomorphism the $p$-adic comparison isomorphism?

This is a question about Faltings' $p$-adic Eichler-Shimura isomorphism from his 1987 article "Hodge-Tate structures and Modular Forms".
Let $N\ge5$, $k\ge2$ be integers. Denote by $X(N)$ the proper ...

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409 views

### What is the best way to learn about Modular Forms?

I am a senior Mathematics Major, and I am interesting in learning about Modular Forms. I have a layman's general sense of what they are but I was wondering if there is a lecture(I am willing to pay) ...

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**1**answer

223 views

### Up to $2000$, $A145722(n-1) \equiv \sigma(4n-3) \pmod{5}$

A145722 is
Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.
Using the pari program and offset 0, up to $2000$...

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200 views

### Weight 12 cusp forms for $\Gamma_0(p)$

Let $S_k$ be the space of weight 12 cusp forms of $\Gamma_0(p)$, ($p$ prime), then Sage tells that $\dim S_k^{\text{new}}=\dim S_k-2$. Thus the old forms spans a 2-dimensional subspace. One of the ...

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145 views

### Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$:
$e^*\Omega^1_{E/X}$ and $\...

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**1**answer

169 views

### p-adic modular forms, Hecke algebra, deformation theory and modular curves.

Let $h^{ord}(N,\mathcal{O})$ be the $p$-ordinary Hecke algebra, and $\mathfrak{m}$ be a maximal ideal of the semi local ring $h^{ord}(N,\mathcal{O})$ corresponding to a residual representation $\bar{\...

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167 views

### Control theory for Kitagawa's $\Lambda$-adic modular symbols

Let $p$ be a prime, $\Gamma=1+p\mathbb Z_p$ and $\Lambda=\mathbb Z_p[[\Gamma]]$ the Iwasawa algebra. Let $\kappa\colon\Gamma\rightarrow\mathbb Z_p^\times$ be the inclusion. For a character $\...

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123 views

### How different can characters be for a sum of modular forms to still be in Gamma_0?

I have a modular form I am constructing out of sums and products of various dissected divisor-sum series, namely forms of the type $$f_i = \sum_{j=0}^\infty \sigma_1(36j+i) q^{36j+i}.$$
Each of these ...

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238 views

### Is special value of Epstein zeta function in 3 variables a period?

Kontsevich-Zagier's article "Periods" contains the following question
Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period?
($\sum'$ means we do not sum ...

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310 views

### Lifting the Hasse invariant mod $2$

Katz defines in Section 2.0 $p$-adic properties of modular schemes and modular forms the Hasse invariant as a mod $p$ modular form $A$ of weight $p-1$. In other words, it is a section of $\omega^{\...

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283 views

### Lifting a real quadratic twist of an Elliptic Curve to the modular curve

Let $E$ be an elliptic curve of conductor $N\cdot p^2$ over $\mathbb{Q}$, defined by the equation
$$y^2=x^3+p^2b\cdot x + p^3\cdot c$$
and parametrized by a map
$$X_{0}(N\cdot {p}^{2})\rightarrow E$$
...

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225 views

### Adelic and classical modular forms on quaternion algebras

Let $R$ be an Eichler order of an indefinite quaternion algebra $B/\mathbb{Q}$ (suppose B is not the collection of $2\times 2$ matrices) and $S$ the corresponding Shimura Curve. Modular forms of ...

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121 views

### Riemann-Roch for curves over Dedekind domains and base-change for modular forms

In p-adic properties of modular schemes and modular forms Katz formulates the following base change theorem as Theorem 1.7.1
Let $n\geq 3$ and $\overline{\mathcal{M}}_n$ be the compactified moduli ...

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407 views

### What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?

What is known about the structure of $J_{0}(N)$ over $\mathbb{Q}[\mu_{p^{\infty}}]$?
More generally, what do we know about $J_{0}(N)$ over
$\mathbb{Q}[\mu_{p^{\infty}},k^{1/p^{n}}]$, where $k\in\...

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**1**answer

170 views

### Polygonal Mersenne numbers [closed]

I posted the same question on Math SE since this one got put on hold.
Link to Math SE question:Polygonal Mersenne numbers
Polygonal numbers are of the form $\cfrac {n^2(s-2)-n(s-4)}{2}$, where $s$ ...

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**1**answer

155 views

### How to determine whether a power of eta function is a eigenform? [closed]

I find that it is complicated to do this from the definition. In fact, I know that $\eta^k(m z)$ is a eigenform for $k=1,2,3,4,6,8,12,24$ and $m=\frac{24}{gcd(k,24)}$. What I want to know is the cases ...

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491 views

### Analytic continuation for $L$-functions of elliptic curves

Let $E$ be an elliptic curve over a number field.
When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...