Questions tagged [modular-forms]

Questions about modular forms and related areas

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Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1

Consider an elliptic curve $E$ defined over $\mathbb Q$. Assume that the rank of $E(\mathbb Q)$ is $\geq2$. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank $=$ algebraic ...
H A Helfgott's user avatar
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24 votes
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868 views

Nekrasov-Okounkov hook length formula

I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls the Nekrasov-...
Dianbin Bao's user avatar
23 votes
0 answers
788 views

Eichler-Shimura over Totally Real Fields

By Eichler-Shimura over totally real fields I mean the following conjecture. Conjecture. Let $K$ be a totally real field. Let $f$ be a Hilbert eigenform with rational eigenvalues, of parallel weight $...
Siksek's user avatar
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21 votes
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890 views

Deciding whether a given power series is modular or not

The degree 3 modular equation for the Jacobi modular invariant $$ \lambda(q)=\biggl(\frac{\sum_{n\in\mathbb Z}q^{(n+1/2)^2}}{\sum_{n\in\mathbb Z}q^{n^2}}\biggr)^4 $$ is given by $$ (\alpha^2+\beta^2+6\...
Wadim Zudilin's user avatar
20 votes
0 answers
1k views

How to approach the Mazur-Wiles paper on Iwasawa theory?

I would like to read and understand the Mazur-Wiles paper on Iwasawa theory: "Class Fields of Abelian Extensions of $\Bbb Q$". What would be the right way to approach this paper? Currently, my ...
Asvin's user avatar
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19 votes
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Does this variant of a theorem of Hasse (really due to Gauss) have an "elementary" proof?

BACKGROUND Here are 3 theorems of varying difficulty. Let $M$ be the $Z/2$ subspace of $Z/2[[x]]$ spanned by $f^k$, with the $k>0$ and odd, and $f=x+x^9+x^{25}+x^{49}+\cdots$. For $g$ in $M$, let $...
paul Monsky's user avatar
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17 votes
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Trying to reconcile two facts about the Appell-Lerch sum learned from Polishchuk and Zwegers

One of the key characters in the thesis of Zwegers is the modular correction $\tilde\mu(u,v;\tau)=\mu(u,v;\tau)+\frac i2R(u-v;\tau)$ of the Lerch sum $\mu(u,v;\tau)=\frac{e^{\pi i u}}{\vartheta(v;\tau)...
მამუკა ჯიბლაძე's user avatar
17 votes
0 answers
768 views

How many Hecke operators span the Hecke algebra?

This is a generalisation of my earlier question about generators for the level 1 Hecke algebra. Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb{Z})$, and $k \ge 1$ an integer. ...
David Loeffler's user avatar
16 votes
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263 views

Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?

In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction: It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...
Asvin's user avatar
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16 votes
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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 $k$ ...
Bruno Joyal's user avatar
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16 votes
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Structure of the algebra of mod $p$ modular forms

Let me first define the algebra $M$ I am talking about: let us fix a prime $p$, an integer $N$ not divisible by $p$. For $k$ an integer, let me call $N_k$ the $\mathbb{Z}$-module of modular forms of ...
Joël's user avatar
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16 votes
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Deligne's letter to Jean-Pierre Serre

I'm looking for another letter of Pierre Deligne, this time to Jean-Pierre Serre (from around 1974 I think), in which he proves that the Galois representation associated to a certain Hecke eigenform ...
Przemyslaw Chojecki's user avatar
15 votes
0 answers
701 views

Wherefore art thou a Borcherds Product?

This question essentially asks how can one recognize (or rule out) that a generating function of combinatorial origin may be given as a Borcherds type product. I'll start with a motivational example: ...
Gjergji Zaimi's user avatar
14 votes
0 answers
885 views

Relation between Igusa tower and $p$-adic modular forms

As the title suggests, my question is devoted to understand (and maybe get some good references) the relation between the Igusa tower for a modular curves and $p$, or maybe $T$-adic modular forms. I ...
rime's user avatar
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What's the dimension of the space of CM cusp forms?

I would guess that the following is very well known, but I don't know the answer and I couldn't find anything with some googling. Let $\Gamma \subset \mathrm{SL}(2,\mathbf Z)$ be a congruence ...
Dan Petersen's user avatar
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13 votes
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310 views

Work of Atkin on the 26th power of eta

The 26th power of the Dedekind $\eta$ function has been mentioned several times here on MO: A 14th and 26th-power Dedekind eta function identity? What's the status of the following relationship ...
მამუკა ჯიბლაძე's user avatar
12 votes
0 answers
524 views

Additive and multiplicative convolution deeply related in modular forms

From the fact spaces of modular forms are finite dimensional, from the decomposition in Hecke eigenforms, and from the duplication formula for $\Gamma(s)$ there are a lot of identities mixing additive ...
reuns's user avatar
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12 votes
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630 views

The operator $\left(q\frac{d}{dq}\right)^s$ and fractional derivatives of modular forms

Recall the notion of a "nearly holomorphic modular form" introduced by Shimura: A function $f : \mathfrak h \to \mathbb C$ is said to be nearly holomorphic of level $\Gamma_1(N)$, weight $k$ and ...
Bruno Joyal's user avatar
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12 votes
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484 views

Weyl law for Maass forms with nontrivial character

The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...
Peter Humphries's user avatar
12 votes
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1k views

A conceptual proof of Jacobi's product formula for $\Delta$ ?

Let $\Delta$ be the unique normalized cusp form of weight 12 and level $1$. Then Jacobi's wel-known formula states: $$\Delta(z) = q \prod_{n=1}(1-q^n)^{24},$$ where $q=e^{2 i \pi z}$. For a graduate ...
Joël's user avatar
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11 votes
0 answers
217 views

Representation of the space of lattices in $\Bbb R^n$

The space of 2D lattices in $\Bbb R^2$ can be represented with the two Eisenstein series $G_4$ and $G_6$. Each lattice uniquely maps to a point in $\Bbb C^2$ using these two invariants, and the points ...
Mike Battaglia's user avatar
11 votes
0 answers
359 views

What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal endomorphism ring?

Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added ...
Vesselin Dimitrov's user avatar
11 votes
0 answers
226 views

Eisenstein series for non congruence subgoups

What is the present status of the Eisenstein series for noncongruence subgroups? I am aware of work of A. Scholl and Rohrlich work on the subject. Is there any specific examples that has been ...
debargha's user avatar
  • 248
11 votes
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533 views

Zero-free theta functions in the upper half plane

Problem $1$. Which full rank lattices $\Lambda \subset \mathbb R^d$ have their corresponding theta function $\theta_{\Lambda}(\tau):= \sum_{\bf n \in \Lambda } e^{\pi i \tau ||n||^2} $ zero-free in ...
Sinai Robins's user avatar
11 votes
0 answers
444 views

Linear eta product identities - how many are there?

For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n}) $, let for brevity $e_k:=\eta(q^k)$. With this notation, a blog entry of Michael ...
Wolfgang's user avatar
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11 votes
0 answers
568 views

Sums of four squares and the modular invariant

The question accounts my curiosity only, and may not be as deep as I think. One of recent talks at our local seminar was devoted to the proof of the classical formula $$ F(q)=\sum_{n=0}^\infty r_4(n)...
Wadim Zudilin's user avatar
11 votes
0 answers
454 views

Is the Gouvea-Mazur problem related to symmetric square $L$-functions?

Here's an idea that I've found appealing but have never been able to get anywhere with. One way to frame the Gouvea-Mazur question (for lack of a better term, since the original conjecture by the ...
Ramsey's user avatar
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11 votes
0 answers
570 views

What's known about the mod 2 reduction of the level l Jacobi modular equation?

Motivation: Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
paul Monsky's user avatar
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10 votes
0 answers
207 views

Are topological theta series (taking values in tmf(N)) of lattices good for anything?

I'm going to start with Mike Hopkins' great survey article in the ICM on topological modular forms (https://arxiv.org/abs/math/0212397). In it, he outlines a construction, for even unimodular lattices,...
Mike's user avatar
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0 answers
145 views

Is there some precise way in which modular cocycle $c/(c\tau +d)$ "is" the generator of $H^1(\mathcal{M}_{ell}, \omega^2) \cong \mathbb{Z}/12$?

It is semiclassical that the extension class $\text{Ext}^1(\omega^{-1},\omega)\cong H^1(\mathcal{M}_{ell},\omega^2)$ over the modular stack $\mathcal{M}_{ell}/\mathbb{Z}$ is nonvanishing, being cyclic ...
xir's user avatar
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10 votes
0 answers
349 views

Riemann–Hilbert-type problem

Let $P$ be a fixed pentagon in the hyperbolic plane $\mathbb H^2$ with all the angles equal to $\pi / 3$. Let $w_1, w_2, \dots, w_5$ be the sides of $P$ going in the counterclockwise order. We are ...
Misha's user avatar
  • 121
10 votes
0 answers
320 views

The mod 3 reduction of some powers of delta

Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix k>0 and ...
paul Monsky's user avatar
  • 5,412
9 votes
0 answers
197 views

Unexpected patterns on the graph of an L-function on the critical line

Let $L(s)$ be the $L$-function associated to the (only) classical modular form of weight $26$ and level $1$. The completed L-function $\Lambda(s)=2(2\pi)^{-s}\Gamma(s) L(s)$ is symmetric with respect ...
LeechLattice's user avatar
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9 votes
0 answers
389 views

Generalizing Ramanujan's and the Chudnovskys' 1/pi formula (Part 1)

Some years ago, I asked in MSE a question about the Chudnovsky brothers pi formula. Later, I asked in MO a related question. The former was unanswered until a few days ago when L. Miller gave me a ...
Tito Piezas III's user avatar
9 votes
0 answers
146 views

Terminology question for the cohomology of the Hilbert modular group

Let $\Gamma$ be the Hilbert modular group of determinant one matrices with entries in the ring of integers of a real quadratic field $F$, and let $M$ be a $\Gamma$-module. Is there a standard name for ...
Henri Darmon's user avatar
9 votes
0 answers
259 views

How explain these remarkable empirical observations about mod 3 modular forms of levels 1 and 5?

Define $b(n)$ by $b(0)= 0$, $b(3n)= 9b(n)$, $b(3n+1)= 9b(n) + 1$, $b(3n+2)= 9b(n) + 3$. (This sequence does not show up in the OEIS, but the similar Moser-de Bruijn sequence A000695 appears in many ...
paul Monsky's user avatar
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9 votes
0 answers
210 views

Surjectivity of reduction for Hilbert modular forms

Fix a totally real field $K$, a level $\mathfrak{n}$, a (parallel) weight $k\geq 2$ and a primitive ray class character $\chi$ modulo $\mathfrak{n}$. Then one can form the space $S_k(\mathfrak{n},\...
fretty's user avatar
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9 votes
0 answers
423 views

Rankin-Selberg for Maass form GL(3)xGL(2)

Let $F$ be a Maass cusp form for $\mathrm{SL}(3,\mathbb{Z})$ (level 1 trivial character). Let $g$ be a Maass cusp form for $\Gamma_0(N)$ with character $\chi$ mod $N$. For convenience, you may assume ...
7-adic's user avatar
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9 votes
0 answers
534 views

Twisted equivariant modular forms?

I'd like to know where I can find information about a class of objects which I think deserve to be called twisted equivariant modular forms. Let me guess a definition, indicate how it can be made more ...
Qiaochu Yuan's user avatar
9 votes
0 answers
720 views

Modular interpretation of Ramanujan theta operator?

I'm a beginner to the theory of modular forms trying to understand a certain construction from the point of view of elliptic curves. Let $f(q) = \sum a_n q^n$ be a formal power series. Define $\theta ...
Akhil Mathew's user avatar
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9 votes
0 answers
305 views

congruences of level 1 and level p modular forms

I've been carrying out some experiments on the computer and I noticed the following congruence phenomenon: fixing a prime $p$, it seems that any modular form over $SL_2(\mathbb{Z})$ and of weight $k \...
Nadim Rustom's user avatar
9 votes
0 answers
226 views

Big image theorems for products of modular forms?

In his 1975 Inventiones paper "On $\ell$-adic representations attached to modular forms", Ken Ribet shows that if $f_1, f_2$ are any two cuspidal modular eigenforms for $\operatorname{SL}_2(\mathbb{Z})...
crocodile's user avatar
  • 519
9 votes
0 answers
515 views

Why is this vector space related to modular forms?

In the course of doing some calculations on a project I am working on, I came across the following presentation of a vector space. It is generated by homogenous polynomials of even degree $n$ over a ...
Jim Conant's user avatar
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9 votes
0 answers
575 views

Tameness criterion in the reducible case

Dear MO, This is a follow up to a previous question here in MO, but I will make this question self-contained for convenience. Those already familiar with the following paper [G] by Gross can safely ...
Álvaro Lozano-Robledo's user avatar
9 votes
0 answers
553 views

Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation

Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
paul Monsky's user avatar
  • 5,412
8 votes
0 answers
121 views

Finding a rational point of large height on an elliptic curve knowing a real approximation

Let $y^2=x(x^2+n)$ be an elliptic curve with $n\in\Bbb Z$ (the same question can of course be asked for a general e.c). I know (e.g. it has rank 1) that there exists a nontrivial rational point $(r,s)$...
Henri Cohen's user avatar
  • 11.5k
8 votes
0 answers
274 views

On rational Ramanujan-type series for $1/\pi$

A Ramanujan-type series for $1/\pi$ is a series of the following form $$ \sum_{n=0}^{\infty} \frac{(\frac12)_n(\frac{1}{s})_n(1-\frac{1}{s})_n}{(1)_n^3} (bn+a) z^n=\frac{1}{\pi}, $$ where $(c)_n=c(c+1)...
Jesús Guillera's user avatar
8 votes
0 answers
293 views

Genus=2 theta functions, Arnold's relation, and KZ connection

Let $C_5:=\{{(z_1 \dots, z_5) \in (\mathbb{C})^5 | z_i \neq z_j \forall i\neq j }\}$ be the configuration space of five distinct ordered points in $\mathbb{C}$. Arnold showed that the holomorphic one ...
shehryar sikander's user avatar
8 votes
0 answers
594 views

Riemann hypothesis for the Hecke operators and modular forms

Let $f(z) = \sum_{n=1}^\infty a(n) e^{2i \pi nz}$ be an eigenform of $S_k(\Gamma_0(N))$. Since the Hecke operator acts by $T_p f = a_p f$ the Riemann hypothesis for $f$'s L-function is $$ \!\!\! \!\...
reuns's user avatar
  • 3,405
8 votes
1 answer
706 views

Hecke operator which changes character

In This MO question, Werner said that Hecke operator "changes" characters. I'm looking for any explicit theory of this kind, about Hecke operator with characters. Actually, there are somewhat ...
Seewoo Lee's user avatar
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