Questions tagged [modular-forms]

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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Factorization of modular forms using zeros and poles

In complex analysis, it is possible to decompose any given entire function into a product of linear factors using the zeros, using the Weierstrass factorization theorem. For example, $\sin(z)$ is ...
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$M_k(\Gamma_1(N)) = \bigoplus_{\chi \mod N} M_k(N, \chi)$

I am stuck with the following statement in the study of modular forms: $$ M_k(\Gamma_1(N)) = \bigoplus_{\chi \mod N} M_k(N, \chi), $$ where $\Gamma_1(N) := \left\{\begin{pmatrix}a & b\\ c & d\...
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Pseudo-periodicity of analytic self-maps of the upper half-plane

I have a couple of questions, in increasing order of softness: Consider an analytic map of the upper-half plane into itself $f: \mathbb{H}\to\mathbb{H}$. When this function is $1$-periodic, i.e., $f(z+...
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Compute the Fourier expansion of adelic Eisenstein series associated to the classical holomorphic Eisenstein series.

For each place $v$ of $\mathbf{Q}$, define $\Phi_v:(\mathbf{Q}_v)^2\to\mathbf{C}$ by $$ \Phi_v(x,y)=\begin{cases} \mathbb{I}_{\mathbf{Z}_v}(x)\mathbb{I}_{\mathbf{Z}_v}(y)&\text{if $v<\infty$},\\...
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How to calculate the order of cusps of half-integral weight cusp?

I have a problem when reading a paper of K. Ono "Distribution of the Partition Function Modulo m". In the proof of Theorem 8, he states that $$\frac{(\Delta(z)^{\delta(m,1)}\ |\ U(m))\ |\ V(...
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Applications of the Modularity Theorem

Besides its role in proving Fermat's Last Theorem and its well-known consequence that the $L$-function of an elliptic curve is defined at 1 (so that, in particular, the BSD Conjecture makes sense), ...
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Strong multiplicity one theorem for new forms of different levels

Let $f, g$ be newforms of level $N_f, N_g$ (possibly different) with $N_f, N_g |N$, and assume that they have same eigenvalue for Hecke operators $T_p$ with $(p, N) = 1$. Then $N_f = N_g$ and $g = cf$ ...
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Lehmer's formula for Ramanujan's tau numbers [closed]

Ramanujan tau function $τ(q)$ together with Dedekind eta function $η(q)$ is defined by $τ(q)=η²⁴(q)=q∏_{k=1}^{∞}(1-q^{k})=∑_{n=1}ⁿτ(n)qⁿ$. where $τ(n)$ are Ramanujan's tau numbers. Recently I came ...
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Ramanujan Identities between Eisenstein Series $E₄$ and $E₆$ and Elliptic Integral of the Second Kind $K$ [closed]

Do you know how to prove the following Ramanujan identities: $E₄=((16)/(π⁴))(k⁴-k²+1)K⁴$ and $E₆=((32)/(π⁶))(2k⁶-3k⁴-3k²+2)K⁶$ where $E₄$ and $E₆$ are the famous Eisenstein Series with variable $q$ ...
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Writing the automorphic form $\phi_f$ of a newform $f$ as a pure tensor

Let $\mathbb{A}$ be the adele ring of $\mathbb{Q}$. Let $f\in S_k(\Gamma_0(N),\chi)$ (possibly a newform) and $\phi_f:\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A})\to \mathbb{C}$ be its ...
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Verifying Modularity for the Theta Function

I was trying to verify modularity of the theta function $$\theta(z) = \sum_{t \in \mathbb{Z}}e^{2\pi it^{2}z}$$ for $\Gamma_{0}(4)$ directly. I know the factor of automorphy should be $$j(\gamma,z) = \...
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Does the L-series of a modular form that is not a cusp form make sense?

If $f$ is a modular form and we let $a_n$ be the Fourier coefficients, then the $L$-Series associated to $f$ is $$ L(s,f)=\sum_n\frac{a_n}{n^s} $$ Usually, we only define this for cusp forms that is ...
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Kusznetov trace formula deduction in Iwaniec-Kowalski

I don't see how the authors deduce Theorem 16.9 of Iwaniec-Kowalski. They say in the line just before that they, after Cauchy-Schwarz, put (16.56) and (16.69) in (16.43) and it comes out, but I can't ...
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$F=\{z\in\Bbb{H}:\ |z|>1,\ 2|\Re(z)|<\lambda\}$ is fundamental domain for $G_\lambda$(the subgroup of $SL(2,\Bbb{R})$ generated by $S$ and $T_\lambda$

Let $0<\lambda<2$ be a real number and $G_\lambda$ be the subgroup of $SL(2,\Bbb{R})$ generated by $S=\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$ and $T_\lambda=\begin{pmatrix}1 & \...
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Let R be the equivalence relation on Z defined by xRy if x^2 ≡ y^2 (mod 5). Give a partition on Z with respect to R.

Let R be the equivalence relation on Z defined by xRy if x^2 ≡ y^2(mod 5). Give a partition on Z with respect to R. (Hint: You may use the following results: For any x ∈ Z, x^2 ≡ 0 (mod 5) if and only ...
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Are there exist infinitely many odd numbers and even numbers in p(an+b)?

The main question is: Are there exist infinitely many odd numbers and even numbers in $p(an+b)$? Where $an+b\ (n\geq1)$ is an arbitrary arithmetic sequence with $a\in\mathbb{Z}_{>0}$, $b\in\{0,\...
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Why are there only two elliptic points on the modular curve $Y(1)$?

I am trying to understand the Corollary 2.3.4 of "A first course in Modular forms", here the authors claim the following statement: The elliptic points for $SL_2(\mathbb{Z})$ are $SL_2(\...
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Generators of $M_{2k}(\Gamma_0(p))$, $p$ is prime

We know that $M_{2k}(SL(2,\mathbb{Z}))$, the space of holomorphic modular forms of weight $2k \geq 4$, is spanned by $E_4(\tau)^i E_6(\tau)^j$ so that $4i+6j=2k$. Is there a similar type of result for ...
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Are imaginary quadratic numbers fixed by a $2 \times 2$ integral matrix?

In Zagier's 1-2-3 of Modular Forms, the following is written. For context, $\mathfrak{z}$ is a CM point in the upper-half plane, though this is not relevant for the question. By definition, $\...
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How to compute this double summation series

Have anybody ever met this kind of series: $$\sum_{n_1,n_2=1,n_1\neq n_2}^{\infty}\frac{n_1n_2\left(n_1^2-n_2^2\right)q^{\frac{n_1-n_2}{2}}}{\left(1-q^{n_1}\right)\left(1-q^{n_2}\right)\left(1-q^{-n_1-...
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Alternative definition of Normalized Eisenstein Series

Prove that for $k \geq 4$, $E_k(z) = \displaystyle\frac{1}{2} \displaystyle\sum_{m, n \in \mathbb{Z}\\ gcd(m,n) = 1} (mz + n)^{-k}$. The usual definition of $E_k(Z)$ is $E_k(Z) = \displaystyle\frac{...
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Is a totally positive unit necessarily a square?

Let $F$ be a totally real number field with narrow class number one which means every non-zero ideal has a totally positive generator. Let $\mathfrak{o}$ be the ring of integers, $\mathfrak{o}_+$ the ...
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Relation between Petersson inner product and cup product

I was wondering how to define Petersson inner product for Hilbert modular forms. I had a discussion with my supervisor who vaguely suggested that in the case of elliptic modular forms the Petersson ...
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Eisenstein series satisfying a set of differential equations

A set of interesting physical quantities can be expressed as some linear combination of the following combinations of Eisenstein series, $$ \mathbb{E}_0 \equiv 1, \qquad \mathbb{E}_{2k} \equiv \sum_{\...
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Close form of a series

Does the following series have a close form? $$\sum_{n=1}^{\infty}\frac{n^3q^{\frac{3n}{2}}}{\left(1-q^n\right)^3}$$ The final result may be related with combinations of Jacobi theta function, ...
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What is the importance of RH for $L$-functions of modular forms?

The Reimann Hypothesis (RH) for $L$-functions of modular forms states that all the non-trivial zeroes of an $L$-function of a modular form must lie on the critical line. My question is: why is this ...
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Proving $\Delta E_4 $ is a simultaneous eigenfunction of weight 16 for $ SL(2, \mathbb Z)$

Let $\Gamma$ be the $SL(2, \mathbb Z)$ group, $M_k(\Gamma)$ the space of modular forms of weight $k$ and $S_k(\Gamma)$ the subspace of cusp forms. It is known (for ex. Neal Koblitz's book "...
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A base change theorem

In (Katz's '$p$-adic properties of modular forms and modular schemes', 1972) Theorem 1.7.1., we use the following fact: To show $K\otimes H^0(M_n,\omega^{\otimes k})\cong H^0(M_n,\omega^{\otimes k}\...
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Fourier coefficients of a modular form of higher level at a cusp other than $i\infty$

I've been trying to learn a bit about modular forms, and mainly using the Sherman-Diamond textbook. Now, looking at modular forms of higher level where we might have more than one cusp of the ...
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Proving $48\sum\limits_{n\ge1}{e^{2n}(1+e^{4n})\over(1-e^{4n})^2}=24\pi^2\sum\limits_{n\ge1}{e^{\pi^2n}(1+e^{2\pi^2n})\over(1-e^{2\pi^2n})^2}+\pi^2-2$

I am looking for a direct proof of the identity $$2\sum_{n\ge1}\frac{e^{2n}(1+e^{4n})}{(1-e^{4n})^2}=\pi^2\sum_{n\ge1}\frac{e^{\pi^2n}(1+e^{2\pi^2n})}{(1-e^{2\pi^2n})^2}+\frac{\pi^2-2}{24}\tag1$$ ...
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About holomorphic functions on the upper half plane with respect to SO(2).

Let $\mathbb{H}$ be the upper half plane. For $g=\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in SL_2(\mathbb{R})$ and $z \in \mathbb{H}$, set $J(g,z)=cz+d$. Let $SO(2)$ be the special ...
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Galois group of field of definition which is generated by $N$-torsion points of elliptic curve

We consider two variables $t$, $u$, and the elliptic curve $E:y^{2}=x^{3}+tx+u$ which is defined over the function field $\mathcal{K}=\mathbb{C}(t,u)$. For integer $N>1$, we define the field of ...
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Questions on the congruence subgroup

I have two questions related to congruence subgroups. Let $\Gamma=\Gamma_0(N)=\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \subset SL_2(\mathbb{Z}) \mid c \equiv 0 \pmod{N} \}$ be a ...
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morphism between universal elliptic curves

I was trying to go through an exercise ($7.9.3$) in the book "A first course in modular forms" (Diamond, Shurman). We are trying to proof that the hecke operators $T_p$ are defined over $\...
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Modular interpretation of the modular curve associated to $\Gamma_0(N)\cap\Gamma_1(p^s)$

Let $p$ prime, $N\ge 1$ with $p\nmid N$, $s\ge 1$. Let $\Phi_s=\Gamma_0(N)\cap\Gamma_1(p^s)$. Let $Y_s$ be the open curve $\mathcal{H}/\Phi_s$, where $\mathcal{H}$ is the complex upper half plane. Why ...
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Modular cycles?

It is well known that cocycles (differential forms) and cycles share many properties through duality (e.g., de Rham). I've been reading about modular forms recently and I came with a very naive ...
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Sum of 3 squares formula using modular forms

I am trying to prove the sums of 3 squares formula in Cohen's 1975 paper 'Sums Involving the Values at Negative Integers of L-Functions of Quadratic Characters'. That is, for $H(N)$ the Hurwitz class ...
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Explicit definition of the cusps of a congruence subgroup of the symplectic group

I'll begin by defining the notion of a cusp of a congruence subgroup of $\textrm{SL}_2(\mathbb{Z})$: $\textrm{SL}_2(\mathbb{Z})$ has a natural action on the compactified upper half complex plane $\...
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Background behind Eichler's discovery of the relationship of a modular form with an elliptic curve

In Fraenkel's Love and Math (and Richard Taylor's Modular Arithmetic IAS Post https://www.ias.edu/ideas/2012/taylor-modular-arithmetic), specifically in Chapter 8 Magic Numbers, page 88., Fraenkel ...
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Motivation for modular forms in terms of functions on complex elliptic curves

Given a vector space $V$ over some field $F$, we'd like to consider rotations or scalings and whatnot, which come down to defining functions $f: V \rightarrow V$ on our space which respect certain ...
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Elliptic curves, $j$-invariant and example of $j(\Lambda)=0$

First, consider $\Lambda=\mathbb{Z}\bigoplus\omega\mathbb{Z}$ with $\omega$ the third root of unity in the upper half plane. I know that the lattice is such that $g_2(\Lambda)=0$, where $g_2$ is the ...
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About the proof of the eigenspaces of Y(4) in Kohnen's 'Newforms of half-integral weight'

Currently, I'm reading Kohnen's 'Newforms of half-integral weight'. In the proof of proposition 1 at page 7, Kohnen wrote '' Thus, if $f \in S_{k+1 / 2}(N, \chi)$, we find $$ f|Q=\frac{1}{2} f|(Q+\...
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Commutativity of T(n) and R(n) as functions on lattices (Lang Introduction to Modular Forms)

I am currently reading through Lang's Introduction to Modular Forms. In chapter II, he introduces the Hecke Operator as follows. Let $\mathcal{L}$ be the free abelian group generated by the lattices ...
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Residual Mod 3 Representation Attached to Elliptic Curve is Not Induced By a Certain Galois Group

In Cornell, Silverman, Stevens, "Modular Forms and Fermat's Last Theorem," Edixhoven, the author of the chapter on Serre's Conjecture, asserts a few times, beginning on page 234 of the text,...
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Reference request: Katz modular forms modulo $p$ at cusps vs classical modular forms modulo $p$ at cusps

I, like the author of this post, am severely lacking the background to make the connection between reducing modular forms' $q$-expansions modulo $p$ at various cusps, and $q$-expansions of modular ...
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Hecke algebra of $(\mathrm{GL}(2, \mathbb{Q}), \mathrm{GL}(2, \mathbb{Z}))$

In the Wikipedia article about the Hecke algebra of a locally compact group, it is noted that if we take $(\mathrm{GL}(2, \mathbb{Q}), \mathrm{GL}(2, \mathbb{Z}))$ as the pair $(G,K)$ of a unimodular ...
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Neighbourhood of certain point in H is homeomorphic to image in modular curve (Diamond & Shurman 2.2.1)

I am trying to solve exercise 2.2.1 in Diamond and Shurman: Let $\Gamma$ be a congruence subgroup of $SL_2(\mathbb{Z})$, and $\pi:\mathcal{H}\to Y(\Gamma)$ the quotient map from the complex upper half ...
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2 votes
1 answer
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Number of zeroes of a modular form on $\Gamma_0(N)$ on fundamental domain

It is a well-known result that a modular form of weight $k$ on ${\rm SL}_2({\bf Z})$ has $k/12$ zeros on any fundamental domain of the action on the upper half-plane. The proof is complex-analytic in ...
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3 votes
2 answers
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Representations of a ternary quadratic form, modular forms of wheight 3/2 and Eisenstein series

Let $Q(x_1,...,x_n)=\sum_{i,j=1}^na_{ij}x_ix_j$ be a positive-definite quadratic form with $a_{ij}\in\Bbb{Z}$. In another question I ask about the following claim: There is some $L>0$ (depending ...
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Twist of a rational modular form (e.g. as in Tunnell‘s theorem)

I am working on the proof of Tunnell‘s theorem and I am a bit stuck at the application of Waldspurger‘s theorem. Tunnell applies this theorem to give an expression for $L(E_n,1)$. I see that $L(E_n,s)$...
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