Questions tagged [elliptic-modular-form]

Classical (or “elliptic”) modular forms are functions in the complex upper half-plane which transform in a certain way under the action of a discrete subgroup of SL(2, R) such as SL(2,Z).

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2answers
102 views

Derive singular value $\lambda(\sqrt{2}i)=(\sqrt{2}-1)^2$

Does anyone know how to prove that the following special value of the Modular Lambda Function is correct? $$\lambda(\sqrt{2}i)=(\sqrt{2}-1)^2$$ I have a somewhat promising observation that might help ...
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1answer
46 views

Definition of the ring of weakly modular forms over $\mathbb{Z}_{(p)}$

I am an undergraduate in a small mathematics course focused on a single project. Right now we are in the preliminary stage, the part where our advisor has given us a high-level reference, referred us ...
1
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1answer
45 views

If $\wp(u) = \wp(v)$, then $u-v$ or $u+v$ is a period of $\wp$ [duplicate]

The Weierstrass $\wp$ function with an associated lattice $L$ is given by the following equation for $z \notin L:$ $$ \wp(z)=\frac{1}{z^{2}}+\sum_{\omega \in L \backslash\{0\}}\left[\frac{1}{(z-\omega)...
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1answer
108 views

Modularity of Ramanujan-Sato series

The Ramanujan-Sato series $$j^*(\tau)=432\frac{\sqrt{ j(\tau)}+\sqrt{j(\tau)-1728}}{\sqrt{ j(\tau)}-\sqrt{j(\tau)-1728}}=432\frac{E_4(\tau)^{\frac32}+E_6(\tau)}{E_4(\tau)^{\frac32}-E_6(\tau)} \\ = \...
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1answer
81 views

Hypergeometric functions and modular forms

May I please ask if it is possible to write Hypergeometric functions in terms of Jacobi theta functions? I am trying to bring the following Hypergeometric expression (pg.9, eq 4.3) into a known ...
2
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1answer
158 views

Meaning of modular form over finite index subgroups of modular group

I am studying modular forms and I have two questions on modular forms over finite index subgroups of the modular group. What is the role of the enhanced elliptic curve on number theory? Modular form ...
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0answers
76 views

Action of quotient of congruence subgroups on moduli space

Let $\Gamma(N) \leq \Gamma_{1}(N) \leq \Gamma_{0}(N)$ be the usual congruence subgroups of the modular group $SL_{2}(\mathbb{Z})$, with all containments normal. We have, e.g., the quotient group $\...
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1answer
117 views

Algebraic singular moduli

I am trying to understand a point made by Zagier in the paper Elliptic Modular Forms and Their Applications. Proposition 22 is the statement that if $\tau$ in the upper half plane $\mathcal{H}$ is a ...
3
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2answers
624 views

Transformations between the fundamental domains of Eisenstein series

I am self-learning elliptic functions and modular form but struggling with the very basics. I have programmed the Eisenstein series $$E_6 = \sum_{m,n \in \mathbb{Z}} \frac{1}{(m +n \tau)^6} $$ and ...
3
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1answer
285 views

Computing Eisenstein series and modular discriminant

I begin to self learn elliptic functions and modular forms by coding (from scratch) the special functions such as $E_6(\tau), E_6(\tau)$ (Eisenstein series), $g_2(\tau),g_3(\tau)$ (invariants in ...
3
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1answer
140 views

On definition of modular forms

I've been reading Complex Analysis by E. Freitag as alternative material of A First Course in Modular Forms by F. Diamond and I am struggling to comprehend some definitions. In the page 326, he made ...
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1answer
125 views

Why is $\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q \, \Pi_{n\geq1}(1+q^n)^{24}}$ [closed]

Where the Dedekind eta function, $$\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$$ and $q = \exp(2\pi i\tau)$. I cant seem to get the equality, am I missing some identities? Thanks!
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1answer
619 views

the corresponding modular form of an elliptic curve

The modularity theorem reveals the relationship between elliptic curves and modular forms. Is there a series of steps or an algorithm such that we can obtain the corresponding modular form, when given ...
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0answers
156 views

Jacobi Forms, Monstrous Moonshine, Hauptmodules: Relation between Specialization of q-Expansion

I have a weak jacobi form: $\phi(\tau,z)$ of weight $0$ and index $M=3$. We know it has a "q-expansion": $$\phi(\tau,z)=\sum_{n\geq 0,~l}c(n,l)q^ny^l$$ where $q=e^{2\pi i \tau}$ and $y=e^{2\pi i z}$...