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13
votes
0answers
271 views

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
votes
1answer
490 views

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 ...
1
vote
1answer
153 views

$f,f',…,f^{(j)}$ 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? ...
5
votes
1answer
418 views

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 $E_6$ ...
8
votes
2answers
1k views

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 ...
4
votes
2answers
590 views

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 $a_{n,k}$ ...
11
votes
3answers
2k views

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 $...
12
votes
1answer
681 views

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 $\...