Questions about modular forms and related areas

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290 views

### Type of a modular form

Let $f$ be an arbitrary weight 1 newform. We know by Serre-Deligne that there is an odd 2-dimensional irreducible Artin representation $\rho$ such that $L_f(s)=L(\rho,s)$.
I was wondering how much ...

**8**

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300 views

### rings of modular functions on the upper half plane

Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index.
Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index.
Let $M_0(\Gamma_i)$ denote the ring of modular ...

**9**

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**1**answer

387 views

### Converse to Modularity I: weight 2 newforms

Since 2008 we have the following remarkable correspondence:
Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1
newforms
note: all Galois representations in this question are ment ...

**4**

votes

**1**answer

125 views

### Smoothness of Hecke algebras

First I will introduce some notation and definitions.
Fix a level $N$ (take $N=1$ if it makes things easier) and a prime $p$. Let $k$ be a finite field of characteristic $p$ and let $\mathcal{C}$ be ...

**2**

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**1**answer

123 views

### Which values of symmetric square $L$-functions are critical?

I've recently been learning about the special values of symmetric square $L$-functions of modular forms.
If $f$ is a cuspidal modular eigenform (of some weight $k \ge 2$) then its symmetric square ...

**3**

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55 views

### Is there a nice way to invert this expression?

Let us first define the Euler polynomials to be the polynomials $P_n(q)$ that satisfy
$$
\frac{qP_n(q)}{(1 - q)^{n+1}} = \Big(q\frac{d}{dq}\Big)^n\frac{q}{1 - q}.
$$
For example, $P_0(q) = P_1(q) = ...

**8**

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**2**answers

306 views

### Determining the Lambert series for $xq+x^2q^4+x^3q^9+…+x^nq^{n^2}+…$

I am trying to determine the polynomials $P_n(x)$ from
$$
xq+x^2q^4+x^3q^9+...+\ x^nq^{n^2}+...=\sum_{n\geqslant1}\frac{P_n(x)q^n}{1-xq^n};
$$
that is,
$$
\sum_{d|n}x^{\frac ...

**7**

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**1**answer

86 views

### Arithmetic Points are Dense on a Hida Family

I am reading the paper "Constancy of the Adjoint L-invariant" by H. Hida (http://www.math.ucla.edu/~hida/ConstP.pdf).
Correct me if I'm wrong, but I've read/heard that the arithmetic points $p \in ...

**9**

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138 views

### The operator $\left(q\frac{d}{dq}\right)^s$ and fractional derivatives of modular forms

Recall the notion of a "nearly holomorphic modular form" introduced by Shimura:
A function $f : \mathfrak h \to \mathbb C$ is said to be nearly
holomorphic of level $\Gamma_1(N)$, weight $k$ and ...

**2**

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49 views

### Questions about holomorphy and zeros of the symmetric power $L$-function

Let $f$ be a primitive form of an even weight $k$ for the full modular group and let $L(Sym^rf,s)$ be the symmetric $r$th $(r\geq 2)$ power $L$-function associated to $f.$ I have three questions ...

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107 views

### Need an explanation of a deduction

When I was reading the paper of Winfried Kohnen, Yuk-Kam Lau and Igore E. Shparlinski (ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS), I found this result (which is Theorem 2 of the ...

**0**

votes

**1**answer

85 views

### Question about sign change of Hecke eigenvalues

I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$-functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper ...

**9**

votes

**1**answer

444 views

### A question on Ramanujan's $1/\pi$ formulas

It is known that Ramanujan discovered a number of formulas for $1/\pi$. All of these formulas are of the form $$\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(1/2)_n(s)_n(1-s)_n}{(1)_n^3}(a+bn)z^n,$$where ...

**7**

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203 views

### Numerically double-checking formula with L-values

I'm working with a special case of Ichino's triple product formula, which for classical holomorphic newforms $f$, $g$ ,$h$ of weights $k$, $m-k$, $m$ (and central characters satisfying $\chi_f \chi_g ...

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159 views

### Morphism of Shimura varieties and differential equations

Is there a way of constructing a morphism between Shimura varieties using differential equations? Maybe, this looks like a completely ridiculous question, so I think that I should explain the context ...

**7**

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**1**answer

255 views

### Definition of p-adic modular forms

I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point.
He first describes p-adic modular forms of tame level N as functions on the Igusa ...

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88 views

### Question about expression of a sum of two Hecke eigenvalues

I did some computations but I am stuck in finding the exression of the sum
$$\lambda_f(n^2)+\lambda_f(n)^2 $$ in terms of $\lambda_f(n),$ where $f$ is a modular form for the full modular group. Any ...

**0**

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100 views

### Discussion for the sign of a specific sum

Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$
I want to ...

**0**

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**1**answer

68 views

### Expression of a sum of Hecke eigenvalues in terms of one Hecke eigenvalue

Let $f$ be a modular form of an even weight $k$ over the modular group $SL_2(Z).$ Denote $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ I am doing some calculations and I am stack in ...

**12**

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**2**answers

508 views

### Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes

Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible?
Since our polynomial's Galois group over $\mathbb{Q}$ is $S_5$, Chebotarev's density theorem implies that ...

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51 views

### Eisenstein part of the theta series of lattices in same genus

In Lynne Walling's paper "A Remark on Differences of Theta Series" (see http://www.sciencedirect.com/science/article/pii/S0022314X84710651), it is proved that the difference of theta series attached ...

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84 views

### Explicit formula for the product of three Hecke eigenvalues

I am interested in Hecke eigenvalues $\lambda_f(n)$ which are normalized Fourier coefficients of modular forms of an even weight $k$ for the full modular group. I often know that these coefficients ...

**2**

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**1**answer

209 views

### Congruence Number of 197A1

It is reported in this paper by Zagier, as well as in Sage, that the elliptic curve $E=197A1$ has congruence number 10. (Since $E$ has prime conductor, a theorem of Ribet ensures that the congruence ...

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165 views

### Modular factorization of Dedekind zeta functions

It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K=\zeta\prod_\chi L(s,\chi)$$
with the Dirichlet characters distinct and ...

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169 views

### Is the twisted symmetric fifth power $L$-function holomorphic?

Let $\pi$ be a Maass cusp form for SL($2,\mathbb Z$). Let $\omega$ be a primitive Dirichlet character.
Let us consider the $L-$ function
$$L(s,Sym^5 \pi \times \omega)$$ or $L(s,Sym^6 \pi \times ...

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75 views

### Find motivation for calculating $\int_{2}^{X} A^2(t) A(\alpha t)dt$

I read a thesis of Kong Kar Lun (student of Tsang K.M) about the some mean value theorems for certain errors terms in analytic number theory and in which he gave the asymptotic formulas of the ...

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45 views

### Interest to know explicit values of certain coefficients

Sorry if my question is stupid but it comes to my mind whenever I read about the theory of symmetric power $L$ functions. Out of curiosity, I did a web search and found only the explicit expression of ...

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378 views

### Three questions about modular forms frequently asked to me [closed]

I have three questions related to the theory of modular forms and it was frequently asked to me by my collegues and even my invited teacher in our seminars of the number theory at the faculty of ...

**2**

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**1**answer

138 views

### Modular property of indefinite degenerate theta series

Is there anything known about the (mock)modular properties, if any, of the following theta series,
$\sum_{n\in {\mathbb Z}^r_+} e^{2\pi i \langle b, n\rangle} q^{\frac12 \langle n,n\rangle}$,
where ...

**9**

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321 views

### Are there congruence subgroups other than $SL_2(\mathbb Z)$ with exactly 1 cusp?

Are there any congruence subgroups other than $SL_2(\mathbb Z)$ which have exactly 1 cusp? By congruence subgroup, I mean a subgroup of $SL_2(\mathbb Z)$ containing $\Gamma(N)$ for some $N$.
This ...

**2**

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**1**answer

144 views

### How to construct the symmetric power function from a modular form?

I want to understand how we construct from a modular form $f$ its symmetric power function $Sym^rf.$ I read that there is a particular representation that does this but I am not familiar with this ...

**0**

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**1**answer

111 views

### Complex plane mod lattice to elliptic curve correspondence generalization

If we observe the correspondence
$$\mathbb{C}/\Lambda \rightarrow E: Y^{2} = X^{3} - \frac{g_{2}(\Lambda)}{4}X - \frac{g_{3}(\Lambda)}{4},$$
we see the relationship between weight 4 and weight 6 ...

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125 views

### Continued fractions and modular forms

Let $q=e^{2\pi it}$. If $u(t)$ is Ramanujan's octic continued fraction, is it true that the generator of the octahedral group can be expressed as a continued fraction of the form
$$
...

**3**

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76 views

### Representations of $\mathbb{H}^{\times}$ and $\mathbb{H}^{\times}/\mathbb{R}^{\times}$

In an attempt to recapture Eichler's theta correspondence I have hit a stumbling block.
Let $D$ be a quaternion algebra over $\mathbb{Q}$, ramified at $p,\infty$. Also let $V_j = ...

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40 views

### Vector Valued Modular Forms with Monodromy

Is there a theory of vector valued modular objects with given weight, which represent the modular group further by permuting the entries (i.e. vector valued modular forms), but, which have a point of ...

**4**

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127 views

### Relations between modular functions of certain $q$-continued fractions

Given the j-function $j:=j(\tau)$, and $q=e^{2\pi i\tau} = \exp(2\pi i\tau)$ where, for convenience, we set $\tau=\sqrt{-n}$.
I. $\frac{A_2(q)}{A_1(q)} = \text{q-cfrac}:\;$ Icosahedral group
...

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190 views

### Are these two $q$-continued fractions equivalent?

In this MSE post, Nicco Mnisi defined a particular $q$-continued fraction of order $12$. More generally, define the cfrac found in Ramanujan's Notebooks, Vol III, Chap. 16, page 24, where $|ab|<1$ ...

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185 views

### Existence of Hecke operators with distinct eigenvalues?

Consider the space of modular forms $M_k(N)$. Any modular form $f \in M_k(N)$ is determined by a finite number of Fourier coefficients (e.g., Sturm's bound), thus there is a finite set of Hecke ...

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296 views

### On a claim of Zagier on extending a map to cocycle

Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ ...

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297 views

### Computing an eigencuspform in $S_2(\Gamma_0(1776))$

Consider
$$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$
the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then ...

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104 views

### References for modular curves over finite fields [closed]

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...

**2**

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101 views

### What is the image of the Ramanujan Delta function?

Consider the Ramanujan $\Delta$ function as a map from the upper half plane to the complex plane. We know that the image of $\Delta$ is unbounded and that it does not contain the point $0$. What else ...

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216 views

### Hodge–Tate structures of modular forms

The title refers to the paper of Faltings:
Hodge-Tate structures and modular forms.
Math. Ann. 278 (1987), no. 1-4, 133–149.
The main theorem in the paper says that the associated Galois rep to a ...

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64 views

### different definitions of epsilon constants for representations of GL(2) from modular forms

I ran into this question when trying to compute the Atkin-Lehner pseudo-eigenvalue of
newforms. Let $k \geq 2$, let $\omega$ be a Dirichlet character modulo $N$ and let $f \in S_k(N,\omega)$ be a ...

**6**

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272 views

### Fourier expansion of Eisenstein Series

I have been reading a bit about the Fourier expansion of Eisenstein series (weight 1/2). I came across the fact that the coefficients contain Modified Bessel functions.
Further reading I found ...

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**1**answer

128 views

### From an eigenfom with $\mathbf{Q}$-coefficients to $j$-invariants

Given a cuspidal, classical eigenform $f\in S_2(\Gamma_0(N))$ of weight $2$ and with $\mathbf{Q}$-coefficients is there a way of describing the set $J_f$ of $j$-invariants of the elliptic curves lying ...

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66 views

### The dimension of the space of automorphic forms with multiplier system

Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{Z})$ and $\vartheta$ a multiplier system of weight $k$ for $\Gamma$, by which we mean a function $\vartheta:\Gamma \rightarrow \mathbb{C}$ ...

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191 views

### A Siegel modular form related to the product of two eta functions

I am looking for a Siegel modular form of genus $2$ (living on the Siegel modular 3-fold $A_2=\mathrm{Sp}(4,\mathbb{Z})\backslash \mathfrak H_2$) which becomes "roughly" the product of two eta ...

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228 views

### The ring of modular forms for $\Gamma_0(11)$

Let $\mathcal M(11) = \oplus \mathcal M_k(11)$ be a graded algebra of modular forms for congruence group $\Gamma_0(11)$. I want to find generators and relations between them. I proved that $\dim ...

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105 views

### Endomorphism algebras of abelian surfaces with real multiplication

Given an abelian variety $A$ over a field $F$, one may consider the ring of endomorphisms $End(A)$, the ring of $F$-rational maps $A \to A$ respecting the group structure on $A$. We may also consider ...