All Questions
Tagged with analytic-number-theory modular-forms
131
questions
3
votes
0
answers
42
views
How to prove a property of the product of Eisenstein Series
I want to prove that, if $$ E_i(\tau) = \frac{1}{2\zeta(2i)}\sum_{(m,n)\in\mathbb{Z}^2-\{(0,0)\}}(m\tau+n)^{-k}$$
is the normalized Eisenstein Series of weight $i$, then $E_i(\tau)E_j(\tau)\neq E_{i+j}...
- 39
2
votes
1
answer
108
views
Degree of extension of the field of coefficients of modular forms
I am beginning to study modular forms and I came across an inequality defining the bound for the degree of extension of the field of coefficients of a modular form $f\in S_k(\Gamma_1(N))$. This goes ...
- 720
3
votes
1
answer
99
views
Confusion about definition Petersson product
I'm taking a course on modular forms, but my background in analysis is not that strong (I have taken complex analysis and measure theory before however). Therefore I'm a bit confused about the ...
0
votes
0
answers
167
views
Rankin-Selberg Convolution of newforms with different levels
Let $f \in \mathscr{S}_{k}(N,\chi)$ and $g \in \mathscr{S}_{k}(M,\psi)$ be newforms with $(N,M) = 1$. For $\Re(s) > 1$, I was able to derive an integral representation for the Rankin-Selberg ...
- 111
0
votes
0
answers
37
views
Is any power of a theta function over primes a modular form?
Apologies for the vague title but I wasn't sure how to word this. To preface my question, let's recall the theta function:
$$\theta(\tau) = \sum_{n \in \mathbb Z} e^{i \pi n^2 \tau}$$
This function is ...
0
votes
1
answer
98
views
Relationships between Riemann Xi and Jacobi Theta functions
I asked ChatGPT about the relationship between the Riemann Xi function and the Jacobi theta functions. ChatGPT provided the following relation:
$\xi\left(\frac{1}{2} + it\right) = \frac{1}{2} \left( \...
- 553
0
votes
0
answers
60
views
Hurwitz formula for Eisenstein series and other closed forms.
I have several questions about the Eisenstein series: $G_k(\tau)=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}\frac{1}{(m+n\tau)^k}$ where {m,n} does not equal {0,0} and k is even and bigger that ...
2
votes
2
answers
141
views
Modular functions and modular forms
A modular form of weight $k$ as a holomorphic function satisfying $f(\gamma z)=(cz+d)^kf(z)$ (automorphy relation of weight $k$) on the upper-half plane and also holomorphic at cusps.
A modular ...
- 1,442
3
votes
1
answer
161
views
An automorphic function with no poles is constant.
Daniel Bump calls $f$ an automorphic function if it satisfies the formula
$$f\left(\frac{az+b}{cz+d}\right)=f(z)$$
where $\begin{pmatrix}
a&b\\
c&d
\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})$
...
- 430
1
vote
1
answer
111
views
Show that $f(\rho)=0$ for any modular form of weight $k$ if $\rho=e^{\frac{2\pi i}{3}}$ and $3\nmid k$.
In Exercise 1.3.3 from Automorphic Forms and Representations, Daniel Bump gives the hint as follows:
Observe that $\gamma(\rho)=\rho$ where $\gamma= \begin{pmatrix} 1&1\\
-1& \\ \end{pmatrix}...
- 430
2
votes
1
answer
76
views
Lindeloff bound on average for Hecke Maass L functions
I have heard that a Lindeloff-type bound for Hecke Maass form L functions exist on average. However, I am unable to find it in literature nor understand the precise statement. Is it something like $$\...
- 417
1
vote
0
answers
46
views
How do you evaluate the $q$-series of expressions like $j(\tau)^{-1/5}$?
Klein's $j$-invariant (or the modular function $j$) appears in all introductory texts to modular forms but I have not seen a treatment of how to efficiently work out coefficients for the $q$-expansion ...
- 1,776
0
votes
0
answers
51
views
Definition of CM form
Could someone please tell me how we define CM modular form and non-CM modular forms in the most basic way?Also, could you provide the references to your definition?
I have tried searching for the ...
- 1,132
0
votes
0
answers
71
views
Lambert series and higher order derivatives of the Euler Function
It is well known that:
$$\sum_{n=1}^\infty \frac{n q^n}{1-q^n} = -q \frac{d}{dq} \Phi(q)$$
Where $\Phi$ is defined as follows for $|q|<1$:
$$\Phi(q) = \prod_{n=1}^\infty (1-q^n)$$
Is there any ...
- 1,919
3
votes
1
answer
256
views
Why is the Eisenstein series $G_2$ a quasimodular form?
For even $k \geq 4$, the Eisenstein series
\begin{align*}
G_k(\tau) &= \sum_{(n, m)\in \mathbb{Z}^2} \frac{1}{(m + n\tau)^k}
\end{align*}
(omitting the term $(n, m) = (0, 0)$) is a modular form ...
- 25.6k