Questions tagged [quasimodular-forms]
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14
questions
3
votes
1
answer
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Eisenstein $E_2$ at imaginary quadratic arguments
In the paper On Epstein's zeta-function, Chowla and Selberg give a formula for evaluating the Dedekind eta function
$$\eta (\tau)=e^{\pi i\tau/12}\prod_{n=1}^\infty (1-e^{2\pi i n\tau}),\quad \Im\tau\...
- 11.8k
4
votes
0
answers
150
views
Logarithmic vector-valued modular functions and quasimodular forms with misleading modular weights
I have a somewhat imprecise question about functions with reasonably nice modular transformations that don't seem to fit nicely into what I understand of the plain vanilla theory of modular and ...
- 390
6
votes
1
answer
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How to compute Coefficients in Chudnovsky's Formula?
My aim is to understand all three coefficients arising in the Chudnovsky-Formula (see also Question 300385). Two of them are easily computed, but I failed with the third:
It is known that for all $\...
- 598
14
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2
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Complex Multiplication and algebraic integers
Let $q=e^{2\pi i\tau}$ and
$$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty\frac{nq^n}{1-q^n}$$ be the Eisenstein Series of weight $2$
and let $E_2^*(\tau) = E_2(\tau) - \frac{3}{\pi\cdot Im(\tau)}$ be the ...
- 598
2
votes
0
answers
151
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theta function with a low bound in the sum
I encounter a sum similar to the Jacobi Theta function except there is a lower bound $-J$ with $J\geq 0$:
\begin{equation}
f(q,x)=\sum_{n=-J}^\infty q^{n^2}x^n.
\end{equation}
My question is whether ...
- 99
21
votes
1
answer
1k
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Why does this quasi-modular function have integral values?
It is a well-known result that the modular function $1728J(\tau) := \frac{1728E_4(\tau)^3}{E_4(\tau)^3-E_6(\tau)^2}$ has integral values if $\tau$ has class number 1 - for example at $\tau_{163}:=\...
- 598
14
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2
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Modular equations for quasimodular forms
This problem is motivated by this question and by teaching
modular polynomials for the classical modular invariant $j(\tau)$.
The latter implies
that if we consider the fields of modular functions $\...
- 1,021
16
votes
3
answers
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How do modular forms of half-integral weight relate to the (quasi-modular) Eisenstein series?
The Eisenstein series
$$
G_{2k} = \sum_{(m,n) \neq (0,0)} \frac{1}{(m + n\tau)^{2k}}
$$
are modular forms (if $k>1$) of weight $2k$ and quasi-modular if $k=1$. It is clear that given modular forms $...
- 17.5k
16
votes
0
answers
395
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Power series which are $p$-adic modular forms for all $p$
Suppose that, for some integer $k$, a series $f(q) \in \mathbb Q \otimes \mathbb Z[[q]]$ has the property that for every prime $p$, $f(q)$ is the $q$-expansion of a $p$-adic modular form of weight $k$ ...
- 3,860
4
votes
2
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What literature is known about MacMahon's generalized sum-of-divisors function?
MacMahon in the paper Divisors of Numbers and their Continuations in the Theory of Partitions defines several generalized notions of the sum-of-divisors function; for example, if we write $a_{n,k}$ ...
CommunityBot
- 1
4
votes
1
answer
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Algebraic independence of $E_2$, $E_4$ and $E_6$
In "M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, Prog. Math. 129,
165-172 (1995)" there is a proposition stating essentially that $E_2$, $E_4$ and $E_6$ are ...
- 20.5k
1
vote
1
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313
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$f,f',...,f^{(j)}$ is $\mathbb C$-linearly independent if $f$ is a modular form
Conjecture: Let be $f$ a modular form of weight $k$ and $j$ a strictly positive integer, then the set $f,f',...,f^{(j)}$ is $\mathbb C$-linearly independent in $A$.
Is that conjecture true or false? ...
- 39.3k
9
votes
1
answer
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How do the rings of level $N$ quasi-modular forms related to the rings of modular forms?
It is well known that the graded algebra $\mathcal{M}(1)$ of Modular forms for $\Gamma = PSL_2(\mathbb{Z})$ is the polynomial algebra
$$
\mathcal{M}(1) = \mathbb{C}[E_4, E_6]
$$
where $E_4$ and $E_6$ ...
- 136
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votes
2
answers
2k
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Binomial coefficients and derivatives of modular forms
Let $E_2$, $E_4$, and $E_6$ denote the standard Eisenstein series.
The usual variables $q=e^{2\pi i\tau}$ allow us to regard the
$E_n$'s as functions on either the upper half plane or the unit
disk ...
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