Questions about modular forms and related areas

**15**

votes

**2**answers

316 views

### Can something finite over $\mathbb{C}(q)$ be a modular form?

If $f\in\mathbf{C}[[q]]$ is non-constant, and algebraic over $\mathbf{C}[q]$ (in the sense that it is a root of a polynomial with coefficients in in $\mathbf{C}[q]$) then can $f$ be the $q$-expansion ...

**10**

votes

**1**answer

239 views

### Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?

Fix $N\ge4$. Let $Y_1(N)$ and $X_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (...

**2**

votes

**1**answer

156 views

### Restriction to the diagonal of Hilbert eigenforms

Do you know of any reference that discusses whether the restriction to the diagonal of a Hilbert eigenform is an (elliptic) eigenform?

**5**

votes

**2**answers

223 views

### Definition of Hecke operators on orthogonal modular forms

In his paper Automorphic forms with singularities on Grassmannians, Borcherds poses Problem 16.5:
"Describe how the correspondence in this paper behaves under
the
action of Hecke operators."
Since ...

**3**

votes

**1**answer

95 views

### 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 \...

**4**

votes

**1**answer

220 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 ...

**3**

votes

**3**answers

334 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 ...

**2**

votes

**0**answers

53 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 ...

**15**

votes

**2**answers

548 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 ...

**0**

votes

**0**answers

55 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 ...

**1**

vote

**0**answers

72 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(\...

**5**

votes

**0**answers

84 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 ...

**3**

votes

**1**answer

319 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....

**6**

votes

**1**answer

227 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{...

**2**

votes

**0**answers

104 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 ...

**1**

vote

**0**answers

58 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 ...

**2**

votes

**2**answers

313 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 ...

**2**

votes

**1**answer

118 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 ...

**4**

votes

**1**answer

202 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 ...

**1**

vote

**0**answers

108 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 $\...

**10**

votes

**1**answer

237 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}...

**4**

votes

**1**answer

202 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 ...

**10**

votes

**1**answer

462 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 ...

**2**

votes

**0**answers

75 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 ...

**2**

votes

**0**answers

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 ...

**4**

votes

**1**answer

178 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 ...

**3**

votes

**0**answers

171 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 ...

**4**

votes

**1**answer

137 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 ...

**3**

votes

**1**answer

276 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(...

**2**

votes

**1**answer

124 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 ...

**2**

votes

**1**answer

159 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 ...

**18**

votes

**1**answer

754 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 ...

**20**

votes

**3**answers

760 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 $X(p)$,...

**4**

votes

**2**answers

216 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 ...

**7**

votes

**0**answers

139 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 ...

**4**

votes

**1**answer

175 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 ...

**0**

votes

**0**answers

70 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 ...

**12**

votes

**1**answer

347 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 ...

**4**

votes

**0**answers

412 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) ...

**5**

votes

**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 $2000$...

**2**

votes

**1**answer

205 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 ...

**6**

votes

**0**answers

147 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 $\...

**3**

votes

**1**answer

180 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{\...

**2**

votes

**0**answers

171 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 $\...

**3**

votes

**1**answer

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 ...

**14**

votes

**1**answer

245 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 ...

**5**

votes

**2**answers

313 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^{\...

**3**

votes

**0**answers

284 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
$$X_{0}(N\cdot {p}^{2})\rightarrow E$$
...

**6**

votes

**1**answer

227 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 ...

**7**

votes

**0**answers

123 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 ...