A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group.

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a conjecture of two equivalent q-continued fractions related to the reciprocal of the Göllnitz-Gordon continued fraction A111374-OEIS

Given the square of the nome $q=e^{2i\pi\tau}$ and ramanujan theta function $f(a,b)=\sum_{k=-\infty}^{\infty}a^{k(k+1)/2}b^{k(k-1)/2}$ with $|q|\lt1$, define, ...
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Genus of a modular curve mod p

I am reading a paper that states the following without reference: The modular curve $X_0(\mathcal{l})/p$, $l \neq p$ has genus $[\mathcal{l}/12]$. Does anyone know where this comes from? I am ...
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Motivation for the definition of modular forms

I've just started to study modular forms and I was wondering about how one would motivate the definition. I agree that $f\left( \frac{az + b}{c z + d} \right) = (cz + d)^k f(z)$ is an interesting ...
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proof of a lemma in Serre's book “A Course in Arithmetic” p 82

Im trying to understuand the proof of this Lemma Let $\Omega$ be a lattice in $\Bbb C.$ The series :$$L:=\underset{0\neq\rho\in\Omega}{\sum}\frac{1}{|\rho|^t}$$ is absolutely convergent for ...
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How are holomorphic and real-analytic Eisenstein series related?

The holomorphic Eisenstein series can be given as $$G_{2k}(z)=\sum_{(c,d)\in{\bf Z}^2\backslash(0,0)}\frac{1}{(cz+d)^{2k}}$$ while the real-analytic Eisenstein series is ...
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Interpretation of enhanced elliptic curves

In "A first course in modular forms" (Diamond-Shurman) the author defines something called an 'enhanced elliptic curve' for the congruence subgroups $\Gamma_0(N), \Gamma_1(N)$ and $\Gamma(N)$. For ...
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$f$ weakly modular. How to prove that the order of vanishing $v_p(f)$ is well defined for $p\in SL_2(\mathbb{Z}) \setminus H$

I'm studying the proof of Lemma 1.6.6 in this Lecture notes page 16 Lemma If $f$ is weakly modular of weight $k,$ the order of vanishing $ v_p(f)$ is well defined for $p\in SL_2(\mathbb{Z}) ...
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Good description of orbits of upper half plane under $SL_2 (Z)$

It's known that $SL_2(Z)$ acts on $H=\{z\, |\, Im(z)>0\}$, is there a good description of orbits of $i$ and $w$, other than directly write down $=\{ \frac{ac|z|^2+bc\bar z+adz+bd}{c^2|z|^2+dc\bar z ...
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Using Modularity Theorem and Ribet's Theorem to disprove existence of rational solutions

This is likely overly optimistic, but can one take the results from the Modularity theorem and Ribet's theorem, and distill these down to an undergrad math level of a way to check if certain rational ...
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51 views

Dimension of the space of cusp forms

I guess this is rather basic, and as much a reality check as a question. We have the classical equations for the dimension of the space of modular forms of weight $k$ for $\Gamma(1)$: $$\dim ...
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199 views

How to solve the Brioschi quintic in terms of elliptic functions?

Given the Brioschi quintic $$w^{5}-10cw^{3}+45c^{2}w-c^2=0$$ I'm interested in seeing different ways of solving it in terms of elliptic functions or theta functions.
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Rogers-Ramanujan continued fraction in terms of theta functions?

The Rogers-Ramanujan cfrac is, $$r = r(\tau)= \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$ If $q = \exp(2\pi i \tau)$, then it is known that, $$\frac{1}{r}-r ...
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On the j-function $j\big(\frac{1+\sqrt{-31}}{2}\big)$ and special cubics

Given the j-function, $$j(\tau) = {1 \over q} + 744 + 196884 q + 21493760 q^2 + \dots$$ It seems that for $\tau = \frac{1+\sqrt{-d}}{2}$ and $d=12n+7$, then it has the form, $$j(\tau) = ...
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Some special Hecke groups

Any reference for the Hecke group generated by $z \mapsto -\frac 1 z, \quad z \mapsto z + \frac 2 p$ for $p$ an odd prime? I am conscious that these groups are not discrete. I am interested by the ...
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How to prove that the Peterson inner product of $E_k$ with a cusp eigenform is $0$?

How to prove that the Peterson inner product of $E_k$ and $f$ is $0$ where $f \in S_k$ is an eigenform for all of the level $1$ Hecke operators?
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Why does the valence formula imply $M_k=0$ for $k <0$?

I'm studying Modular Forms and in the notes I'm reading the author states the following result, known as the valence formula: "Let $f$ be a non-zero weakly modular meromorphic form of weight $k$ ...
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Finding the value 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 ...
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Transitive action of $SL_2(\mathbb{R})$ on $\mathbb{H}$

I'm studying Modular Forms and I'm not understanding why the action of $SL_2(\mathbb{R})$ on $\mathbb{H}$ defined by $\begin{pmatrix} a & b \\ c & d \end{pmatrix}z=\frac{az+b}{cd+d}$ is ...
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connection between odd primes and a certain q-series

I posed a conjecture about odd primes and a certain q-serieshere.I thought it would be more appropriate ,if I could ask the converse of the aforementioned problem . Is $p$ an odd prime iff ...
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Question about the sign of a certain 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 ...
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conjecture about primes and a certain q-series.

Using wolfram Mathematica ,I observed an interesting and surprising property concerning prime numbers and q-series which I could not prove.Yet there is strong evidence supporting it. I would be happy ...
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A theta function around its natural boundary

Let $q = e^{2\pi i\tau}$, if $$\psi(q^2)=\sum_{n=0}^{\infty} q^{n(n+1)}$$ is one of ramanujan theta functions,is it possible to evaluate the limit $$\lim_{q\rightarrow 1} (1-q){\psi^2(q^2)}$$ In fact ...
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What is the limit $\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$

If $\vartheta_{2}(q)$ is jacobi's theta function, what is the limit $$\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$$ for the nome $q$. I would like to know whether the limit exists or not. If it does, ...
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Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some ...
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a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction for the square of ramanujan's octic continued fraction which I discovered using certain three term ...
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the zeros of theta function?

Recall that the theta function with character $(a,b)\in \mathbb{R}^2$ is defined by $$ \vartheta_{a,b}(z, \tau) :=\sum^\infty_{n=-\infty} e^{\pi i (n + a)^{2} \tau + 2 \pi i(n + a)(z + b)},\quad a,b ...
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Can Eisenstein Series output complex numbers?

The 2nd Eisenstein Series, defined by, $$E_{2}(\tau)=1-24\sum_{n=0}^{\infty} \frac{nq^{2n}}{1-q^{2n}},$$ where $q=e^{i\pi \tau}$ is the nome acts on upper-half plane. Must it always output real ...
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Explicit descriptions of the modular curves $Y(5)$ and $Y_1(5)$

The modular curves $Y(5)$ and $Y_1(5)$ associated to the congruence subgroups $\Gamma(5)$ and $\Gamma_1(5)$ are of genus 0. As such, they are nothing but the Riemann sphere $\mathbb P^1$ minus a ...
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Why are rational numbers required in cusps of congruence subgroups?

While we consider the action of congruence subgroups on $\mathbb{H}$ (the upper half plane), we compactify using an additional point at infinity, that is fine. But why do we add even all rational ...
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Why only congruence subgroups for modular forms?

When we define modular forms, why do we restrict ourselves to congrumence subgroups? Why not any subgroup of finite index? Or, even more generally why not any subgroup? Is it just a matter of ...
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Ramanujan's tau function identity

While studying Ramanujan's tau function, I observed that the function satisfies a beautiful identity that I had not seen previously in the literature. Let $\tau(n)$ be Ramanujan's tau function, such ...
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The ratio of jacobi theta functions

Let $q=e^{2\pi i\tau}$. If $\theta_2$ and $\theta_3$ are jacobi theta functions , is it true that the ratio of the two functions can be expressed as a continued fraction of the form $$ ...
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a q-continued fraction related to the octahedral group

Let $q=e^{2\pi i\tau}$. If $u(\tau)$ is Ramanujan's octic continued fraction, $$u(\tau)=\cfrac{\sqrt{2}\,q^{1/8}}{1+\cfrac{q}{1+q+\cfrac{q^2}{1+q^2+\cfrac{q^3}{1+q^3+\ddots}}}}$$ is it true that ...
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Holomorphicity and modularity of Jacobi forms

I am reading this paper Suerconformal algebras and mock theta functions by T. Eguchi and K. Hikami. In section 3.2, the authors define a function $$J(z;w;\tau) = ...
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Fourier coefficients of eigenforms

How does one prove that the fourier coefficients of a normalized eigenform for Hecke operators $T_p$ on $S_k(N)$ all lie in a fixed number field? If the proof is lengthy, a reference to a book that ...
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What is the idea behind the new/old form theory of modular forms

I am interested in the theory of modular forms, since they have nice transformation laws and a connection to arithmetic using the Euler product which one can form using the theory of Hecke operators. ...
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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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A new $q$-continued fraction of order $12$

I think I may have discovered a $q$-continued fraction of order $12$ with a form different from that established by Mahadeva Naika. Let $q=e^{2i \pi \tau}=\exp(2i \pi \tau)$, then, $$\begin{aligned} ...
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What does holomorphic at the cusp infinity means

In the usual theory of classical modular forms, the modular forms defined to be "holomorphic at the cusp infinity". I do not know what this should mean? can anyone explain it for me? Thanks
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Finding references about modular forms and symmetric power $L$-functions

I want to write an introduction to my thesis which is about modular forms and symmetric power $L$-functions. Could you give me good references to these two topics as simple as possible to get an ...
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Counting points on the Klein quartic

In Moreno's book "Algebraic Curves over Finite Fields", he mentions the following in passing with no further comments ($K$ denotes the Klein quartic defined by $X^3 Y + Y^3 Z + Z^3 X = 0$): The ...
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On the definition of modular forms

In many books, I see people defining modular forms to be holomorphic/meromorphic functions in the upper half plane such that it is invariant under the $|_k$ action of the group ...
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Quotient of the upper-half plane as a projective line

In the discussion of the Belyi theorem here one pointed out that $\mathbb{H}^{*}/SL(2, \mathbb{Z})$ is isomorphic to the (complex) projective line $\mathbb{P}^1$, where $\mathbb{H}^{*}=\mathbb{H} \cup ...
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Extend a map to a 1-cocycle

Let $\Gamma=PSL(2,\mathbb{Z})$ be the modular group with the usual presentation $\Gamma=\langle S,U,T|\ S^2=U^3=1, T=US\rangle$ where ...
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Modular curves over finite fields

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 ...
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What is the exact value of $\eta(6i)$?

Let $\eta(\tau)$ be the Dedekind eta function. In his Lost Notebook, Ramanujan played around with a related function and came up with some of the nice evaluations, $$\begin{aligned} \eta(i) &= ...
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Multiplicative property for the coefficients of cusp form

This question might seen as a duplicate of this, however my aim is to understand the theory which lies beneath the computations of Sage. Let $\Gamma_0(4)$ be a congruence subgroup of ...
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$\Gamma_0(4)$ has no torsion except $-1$

Let $\Gamma_0(4)$ be a congruence subgroup of $SL(2,\mathbb{Z})$ defined as $$\Gamma_0(4)=\{M=\begin{pmatrix} a &b\\ c& d \end{pmatrix}\in SL(2,\mathbb{Z})\mid c=0\bmod 4\}.$$ How to prove ...
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Coefficient in the Fourier expansion of the cusp form

Ideal of cusp for $\Gamma_{0}(4)$ is principal and generated by $f(z)=η(2z)^{12}=q+\sum a(n)q^n $, this is discussed here. How one can compute the coefficient $a(n)$ when $n$ is rather large ? for ...
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$\Gamma(2)$ and $\Gamma_0(4)$ are conjugate

Show that the groups $\Gamma(2)$ and $\Gamma_0(4)$ are conjugate to each other in $SL(2,\mathbb{R})$, where $\Gamma(2)$ and $\Gamma_0(4)$ is congruence subgroups of the modular group Is it the ...