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1answer
52 views

Eisenstein series is a modular form

I want to prove that the Eisenstein series $$G_k(z)= \sum_{c,d}(cz+d)^{-k}, (c,d)\in \mathbb{Z}^2 \setminus \{(0,0)\}$$ satisfies the relationship $$G_k((az+b)/(cz+d))= (cz+d)^k\sum_{(c',d')} ...
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1answer
52 views

Operations with $\text{SL}_2(\mathbb{Z})$

We define $\Omega :=\left\lbrace z \in \mathbb{H}\colon -\frac{1}{2} \leq \operatorname{Re}z \leq \frac{1}{2} \wedge |z| \geq 1\right\rbrace$. I want to show that the following holds: $$ \forall \, ...
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0answers
45 views

$(\sigma_0^{-1} SL(2,\mathbb{Z}) \sigma) \cap SL(2, \mathbb{Z})$ is a right coset of $\Gamma_0(m)$ in $SL(2, \mathbb{Z})$

Let $\Gamma_0(m)$ be the subgroup of $SL(2,\mathbb{Z})$ which is defined as follows: $$\Gamma_0(m)=\left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL(2, \mathbb{Z} : c ...
2
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1answer
67 views

History of the Coefficients of Elliptic Curves — Why $a_6$? [duplicate]

I would like to know what is the motivation behind the naming convention of the Weierstrass form of elliptic curves given as $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ I can see that $a_1,a_2,a_3,a_4$ ...
4
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1answer
49 views

Identities of Hecke operators

While studying, I recently came across the following interesting problem. Let's say that the (level one) weight $k$ modular forms $M_k(\Gamma(1))$ have dimension $d$. We know by the ring structure ...
4
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1answer
81 views

Undergraduate Introduction to Modular Forms

What are the best introductory texts (or lecture notes) on modular forms aimed at an advanced undergraduate audience (for a student with a course in complex analysis and two courses in algebra and ...
2
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1answer
46 views

Hecke equivariance in Poincaré duality.

Consider the first singular homology and cohomology groups of a modular curve, $H^1(X,\mathbb{Z})$ and $H_1(X,\mathbb{Z})$. The Hecke algebra acts on both of them and they are dual to each other under ...
1
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1answer
53 views

Holomorphic Eisenstein Series

$$G_k (z)=\sum_{\displaystyle(m,n) \neq (0,0)} \frac{1}{(mz+n)^k}$$ For $k \geq 3$ how to show that $G_k (z)=0$ for odd $k$ and $G_k (\tau z)=(cz+d)^k G_k (z)$ for $\tau \in SL_2(Z), \tau = \left( ...
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1answer
82 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
2
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1answer
65 views

Rewriting $\tau(p)\Delta(\tau)$ when $p$ is prime

$p$ is a prime, and $\tau$ is Ramanujan's tau function: $$p^{11}\Delta(p\tau)+\frac{1}{p}\sum_{k=0}^{p-1}\Delta\bigg(\frac{\tau + ...
2
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0answers
36 views

eisenstein part of theta function

If $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the $\textbf{theta series}$ ...
2
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1answer
86 views

Modular Form, but Cusp Form in Disguise

Suppose I have a holomorphic modular form $f \in M_k(\Gamma_0(N), \chi)$ with $k \in \mathbb Z^+$ and $f = \sum_{n=1}^\infty a(n)q^n$. Considering the $q$-expansion, one may suspicious that $f$ is ...
2
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1answer
57 views

Why a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$?

It is said that a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$. Why a modular form is a highest weight vector of a ...
3
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1answer
51 views

Elliptic points on modular curves and a uniformising parameter

I'm working with the modular curves $X_0(p)$ for $p=2,5$, with $$f_2=q\prod_{n\geq1}(1+q^n)^{24}$$ and $$f_5=q\prod_{n\geq1}(1+q^n+q^{2n}+q^{3n}+q^{4n})^6$$ being inverses of the Hauptmodul and ...
3
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1answer
47 views

Diamond Operator and Fourier Coefficients

Let $d\in(\mathbb{Z}/N\mathbb{Z})^\times$, then one has the Diamond operator $\langle d \rangle$ acting on $M_k(\Gamma_1(N))$ via ...
0
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1answer
28 views

Diamond operator on a meromorphic function

Is there a standard way in which the diamond operator for modular forms is defined for an arbitrary meromorphic function?. I know the definition for a weight k modular form, but since I usually wont ...
1
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1answer
28 views

Weber function takes on real values?

Why does the Weber function take on real values? In the middle of the proof, the author argues... "cleary it is real-valued." I don't follow the argument. Info about the Weber function ...
2
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1answer
70 views

Equidistribution of lattice points on spheres in dimensions $d \ge 4$ (Pommerenke's theorem)

I am looking for either an English translation of: Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden. Acta Arith. 5, 227-257 (1959) Or a reference in ...
2
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0answers
15 views

Is it true that $\theta_{1,1}^{4N} \in J_{2N,2N}(2N)$?

I need examples of Jacobi forms for full congruence subgroups $\Gamma(N) $ of $SL(2,Z)$. As a particular case, take the theta function $\theta_{1,1}(t,z) := \sum_{n\in\mathbb{Z}} exp(\pi it(n + ...
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0answers
34 views

What does $\Gamma_\infty$ denote in modular forms

I am catching up on some notes for an online modular forms course. The slides have just suddenly started using the notation $\Gamma_\infty$, without even defining $\Gamma_k$ (except for $k=1$), never ...
2
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1answer
85 views

Hilbert modular forms and Hecke operators over Q

Let F be a totally real field. We know that we can define a Hecke operator $T_\mathfrak{m}$ on the space of Hilbert modular forms over $F$, say with some level structure, for any ideal $\mathfrak{m}$ ...
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2answers
428 views

Proving a complex function is continuous.

I've recently started complex analysis but I have very little background in complex numbers and to make sure I don't fall behind I'm doing some extra exercises one of which is Show $f$ is continuous ...
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0answers
135 views

The conjecture $A159634(n) = \psi(2n)/3$ where $\psi$ is the Dedekind psi function.

Steven Finch is the author of the OEIS-sequence A159634 which is defined as "Coefficient for dimensions of spaces of modular & cusp forms of weight $k/2$, level $4n$ >and trivial character, ...
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1answer
145 views

Proving addition and multiplication

(1)Show that addition and multiplication mod n are associative operations. (2)Show that there are both an additive and a multiplicative identity. (3)Show that multiplication distributes over ...
5
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2answers
118 views

Relation: Modular Forms and hyperbolic geometry, or, why do they map from $\mathbb{H}$?

In my very young mathematical career, I have worked a lot with modular forms. Recently, I worked as a teaching assistant in a course about geometry. At the end of the course, we dealt with hyperbolic ...
6
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0answers
109 views

Proving that $\sigma_7(n) = \sigma_3(n) + 120 \sum_{m=1}^{n-1} \sigma_3(m)\sigma_3(n-m)$ without using modular forms?

This problem appears as a (starred!) exercise in D. Zagier's notes on modular forms. I have to admit that I have no idea how to do it. Here, $\sigma_k(n) =\sum_{d\mid n} d^k$, as usual. This ...
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1answer
64 views

Semistable Elliptic Curves

For a general representation $\rho: G_{\mathbb{Q}} \rightarrow \operatorname{GL}(V)$, where $V$ is a two dimensional $\overline{\mathbb{F}}_p$ vector space, the level $N(\rho)$ in Serre's conjecture ...
2
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2answers
155 views

Hypergeometric formulas for the j-function

Given the j-function $j(\tau)$ and elliptic lambda function $\lambda(\tau)$. Define, $$g = -1+2\frac{\,_2F_1\big(\tfrac{1}{2},\tfrac{1}{2},1,\,\lambda(2\tau)\big) ...
2
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0answers
176 views

Learning roadmap for p-adic modular forms and eigenvarieties.

What are some good sources for breaking into the field of p-adic modular forms? It was suggested to me to read Katz' paper "P-adic Properties of Modular Schemes and Modular Forms". I have found this ...
1
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1answer
63 views

Definition of Automorphic form

I am looking at how one maps the space of cusp forms $S_k(N,\psi)$ into the space of automorphic forms on $GL_2$ over the Adeles. And I have a question about the definition of $K$-finiteness. I am ...
3
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0answers
52 views

Are the Fourier coefficients of a new form real?

Let $f\in S^\text{new}_k(Γ_0(N))$ be a $\text{newform}$ . Are all its Fourier coefficients real? Of course the Hecke operators $T_n$ are selfadjoint for $(n,N)=1$, but is it also true for all $n$?
5
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1answer
134 views

How should I think about Ihara's Lemma?

I am trying to learn about Ihara's lemma because I see it mentioned in many papers in arithmetic geometry. To be more precise, I would like to know: What is the significance of this result? Why is ...
3
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1answer
36 views

Size of automorphism group of element of $Y_0(5)$

The modular curve $Y_0(5)$ parametrizes elliptic curves $E$ with an isogeny of degree five. So an element of $Y_0(5)$ can be interpreted as $E \xrightarrow{\phi} E'$. Suppose we are working over a ...
1
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1answer
94 views

The natural boundary of the modular $\lambda$ function is the real axis

I want to solve the following problem from Ahlfors' text: Show that the function $\lambda(\tau)$ introduced in Chap. 7, Sec. 3.4, has the real axis as a natural boundary. Here $\lambda(\tau)$ is ...
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5answers
457 views

Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), ...
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2answers
175 views

Introductive Book on Modular Forms

I'm looking for an introductive book on Modular Forms and their applications to Algebraic Geometry and Algebraic Number Theory. Some Ideas? Explaining you my prerequisites, I've a good knowledge of ...
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3answers
105 views

How to show that $\Delta\left(\frac{az+b}{cz+d}\right)=(cz+d)^{12}\Delta(z)$?

Let $(a, b; c, d) \in SL_2(\mathbb{Z})$ and $\Delta(z)=q\prod_{n=1}^{\infty} (1-q^n)^{24}$, $q=e^{2\pi i z}$. How to show that $\Delta(\frac{az+b}{cz+d})=(cz+d)^{12}\Delta(z)$? Thank you very much. I ...
5
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1answer
118 views

Approximation of $e^{\pi\sqrt{n}}$ using Ramanujan's Class Invariants

From Wikipedia article we get $\displaystyle \begin{aligned}e^{\pi\sqrt{43}} &\approx 884736743.999777466 \\ e^{\pi\sqrt{67}} &\approx 147197952743.999998662454\\ ...
4
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1answer
138 views

Where does Klein's j-invariant take the values 0 and 1, and with what multiplicities?

I tried to solve the following problem from Ahlfors' text, please verify my solution: Where does the function $$J(\tau)=\frac{4}{27} \frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2} $$ ...
3
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1answer
138 views

Monstrous Moonshine for $M_{24}$?

This is connected to my MO post "Monstrous Moonshine for $M_{24}$ and K3?". In page 44 of this paper, eqn(7.16) and (7.19) yield, ...
3
votes
1answer
71 views

Why is the modular $\lambda$ function a quotient of two meromorphic functions in the U.H.P.?

In Ahlfors' complex analysis text, page 278 it says: It is quite clear from (9) that $\lambda(\tau)$ is the quotient of two analytic functions in the upper half plane $\text{Im} \tau > 0$. ...
3
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0answers
44 views

Automorphisms of elliptic curves with a point of order N

Probably a stupid question, but I'm trying to figure out the following: Suppose we have an element of $X_1(N)$, i.e. an elliptic curve $E$ with a point $p$ of order $N$. If we have another such curve ...
0
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0answers
87 views

Congruence subgroups , widths of cusps and q-expansions

If $\Gamma$ is a congruence subgroup and $c$ is a cusp of $\Gamma$ then by definition to say that a weakly modular form $f \in M_k(\Gamma)$ is holomorphic at $c$ is that I can write $f[\gamma]_k$(z) ...
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0answers
9 views

No translations in $RS^{-1}$

Let $G$ be a subgroup of $\Gamma(1)$ of finite index and $G_0$ be the subgroup of translations in $G$. Let $R$ be a coset of representatives for $G_0$ in $G$. Assume that $p \in \mathbb{Q}$ is not ...
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0answers
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Determining cosets of representatives

Is there an algorithm to determine the cosets of representatives of a congruence subgroup of a modular form? I know what cosets of representatives and when given a set, I know how to check that they ...
0
votes
1answer
178 views

Find the cusps for the congruence subgroup $\Gamma_0(p)$

Find the cusps for the congruence subgroup $\Gamma_0(p)$. How does one go about doing this? I know the definition of a cusp - the orbit for the action of $G$, in this case $\Gamma_0(p)$, on $\mathbb ...
4
votes
2answers
127 views

Topology of the completed upper-half plane

Define the topology on $\mathbb{H}^* : = \mathbb{H} ∪ \mathbb{Q} ∪\infty$ by taking a basis of open sets around $\infty$ to be $S_{\epsilon} : = \{ z ∈ H : Im ( z ) > 1 /\epsilon \}∪\infty$ , and ...
2
votes
1answer
124 views

Calculating index of a subgroup

Compute the index $[Γ( 1 ) ′ : Γ_0 ( N ) ′ ]$ where $Γ(1)' := SL(2,\mathbb{Z})$ $Γ_0(N)':= \{ \begin{pmatrix} a & b\\ c & d\\ \end{pmatrix} \in Γ(1)' : c \equiv 0 \mod{N} \} $ I'm ...
0
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1answer
141 views

Principal congruence subgroups

Why the index of the principal congruence subgroup of level 2, defined as $$ \Gamma(2)=\left \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb{P}SL(2,\mathbb{Z}): \begin{pmatrix} a ...
4
votes
1answer
201 views

Galois representations attached to modular forms of weight one

Why are the Galois representations attached to weight one eigenforms by Serre-Deligne complex representations? In weight $k\geq 2$, the Galois representations constructed by Eichler-Shimura and ...