Questions tagged [modular-forms]

Questions about modular forms and related areas

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What is known about the sum x^{n^2}/n?

It follows from a general theorem of Honda that the formal group with the logarithm $$ x+x^{2^s}/2+x^{3^s}/3+x^{4^s}/4+\cdots $$ has integer coefficients. I became interested in it because its $p$-...
მამუკა ჯიბლაძე's user avatar
29 votes
3 answers
3k views

Modular forms of fractional weight

Modular forms of integral weight are prominent in number theory. Furthermore, there are $\theta$-functions and the $\eta$-function, having weight 1/2, which also have a rich theory. But I have never ...
wood's user avatar
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5 votes
3 answers
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On level $10$ of the McKay-Thompson series of the Monster

(For brevity, the level-6 functions have been migrated to another post.) I. Level-10 functions Given the Dedekind eta function $\eta(\tau)$. To recall, for level-6, $$j_{6A} = \left(\sqrt{j_{6B}} + \...
Tito Piezas III's user avatar
9 votes
2 answers
958 views

About a Ramanujan-Sato formula of level 10, a recurrence, and $\zeta(5)$?

I. Level 6 This is a long shot, but I am curious where it leads. Given the Dedekind eta function $\eta(\tau),$ define, $$\begin{aligned} j_{6A}(\tau) &= \Big(\sqrt{j_{6B}(\tau)} - \frac{1}{\sqrt{...
Tito Piezas III's user avatar
66 votes
8 answers
12k views

Why are powers of $\exp(\pi\sqrt{163})$ almost integers?

I've been prodded to ask a question expanding this one on Ramanujan's constant $R=\exp(\pi\sqrt{163})$. Recall that $R$ is very close to an integer; specifically $R=262537412640768744 - \epsilon$ ...
Michael Lugo's user avatar
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47 votes
6 answers
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Algebraic Attacks on the Odd Perfect Number Problem

The odd perfect number problem likely needs no introduction. Recent progress (where by recent I mean roughly the last two centuries) seems to have focused on providing restrictions on an odd perfect ...
Cam McLeman's user avatar
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23 votes
3 answers
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Why are values of Eisenstein $E_2^*$ algebraic integers?

I'm looking for a proof that the following term is an algebraic integer whenever $\tau_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$: $$A_N:=\sqrt{-N}\cdot\frac{E_2(\...
L. Milla's user avatar
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15 votes
1 answer
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When is the image of a 2-dim l-adic representation associated to a modular form open

I know the following theorems by Serre: 1, The 2-dim l-adic representation associated to a non-CM elliptic curve is open. 2, The 2-dim l-adic representation associated the weight-12 cusp form $\...
user42690's user avatar
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14 votes
3 answers
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Convergence of L-series

I remember to have read that the L-function of an elliptic curve, which a priori only converges for $\Re s > \frac{3}{2}$ also converges at $s=1$ provided that the $L$-function satisfies the ...
wood's user avatar
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14 votes
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Complex Multiplication and algebraic integers

Let $q=e^{2\pi i\tau}$ and $$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty\frac{nq^n}{1-q^n}$$ be the Eisenstein Series of weight $2$ and let $E_2^*(\tau) = E_2(\tau) - \frac{3}{\pi\cdot Im(\tau)}$ be the ...
L. Milla's user avatar
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13 votes
1 answer
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Reference for: CM Hilbert Modular forms arise from Hecke characters

For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
unramified's user avatar
6 votes
1 answer
578 views

How to compute Coefficients in Chudnovsky's Formula?

My aim is to understand all three coefficients arising in the Chudnovsky-Formula (see also Question 300385). Two of them are easily computed, but I failed with the third: It is known that for all $\...
L. Milla's user avatar
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4 votes
1 answer
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Transformations of Ramanujan's 1/pi formulas $\sum_{n=0}^{\infty} s(n)\frac{An+ B}{C^n}$ and Monster moonshine functions

Someone with many papers on Ramanujan's work asked me how I managed to find the closed-forms for the binomial sums of level $10$ in a recent MO post. (A colleague of his wasn't able to find them.) I ...
Tito Piezas III's user avatar
3 votes
0 answers
194 views

Ramanujan's pi formulas with a twist (nine years later)

(Note: The second method described here continues this post.) About nine years ago, I made an MO post "Ramanujan's pi formulas with a twist". An answer was informative, but not completely ...
Tito Piezas III's user avatar
127 votes
13 answers
29k views

Why are modular forms interesting?

Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about ...
51 votes
3 answers
11k views

What is the difference between an automorphic form and a modular form?

This is more of a question about terminology than about math. The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it ...
David Corwin's user avatar
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48 votes
2 answers
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Why should I care about topological modular forms?

There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on ...
Doug P's user avatar
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41 votes
2 answers
9k views

Intuition behind the Eichler-Shimura relation?

The modular curve $X_0(N)$ has good reduction at all primes $p$ not dividing $N$. At such a prime, the Eichler-Shimura relation expresses the Hecke operator $T_p$ (as an element of the ring of ...
Qiaochu Yuan's user avatar
34 votes
4 answers
3k views

$A_5$-extension of number fields unramified everywhere

So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...
Olivier's user avatar
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28 votes
1 answer
2k views

Can we use the Rogers-Ramanujan cfrac to parameterize the Fermat quintic $x^5+y^5=1$?

Define $\color{blue}{q=e^{2\pi i \tau}}$ and Dedekind eta function $\eta(\tau)$. Note: I found these relations empirically, but their consistent forms suggest they can be rigorously proven. I. $p=2$...
Tito Piezas III's user avatar
27 votes
4 answers
2k views

Which quaternary quadratic form represents $n$ the greatest number of times?

Let $Q$ be a four-variable positive-definite quadratic form with integer coefficients and let $r_{Q}(n)$ be the number of representations of $n$ by $Q$. The theory of modular forms explains how $r_{Q}(...
Jeremy Rouse's user avatar
26 votes
6 answers
6k views

Proofs of Jacobi's four-square theorem

What are the nicest proofs of Jacobi’s four-square theorem you know? How much can they be streamlined? How are they related to each other? I know of essentially three aproaches. Modular forms, as in,...
H A Helfgott's user avatar
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26 votes
6 answers
4k views

Where can I find a comprehensive list of equations for small genus modular curves?

Does there exist anywhere a comprehensive list of small genus modular curves $X_G$, for G a subgroup of GL(2,Z/(n))$? (say genus <= 2), together with equations? I'm particularly interested in genus ...
David Zureick-Brown's user avatar
23 votes
1 answer
2k views

Ramanujan's pi formulas with a twist

Given the binomial function $\binom{n}{k}$. 1. Define the following sequences, $$\begin{aligned} u_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\ u_2(k) &...
Tito Piezas III's user avatar
23 votes
3 answers
4k views

Relation between Hecke Operator and Hecke Algebra

In the study of number theory (and in other branches of mathematics) presence of Hecke Algebra and Hecke Operator is very prominent. One of the many ways to define the Hecke Operator $T(p)$ is in ...
Dipramit Majumdar's user avatar
19 votes
4 answers
2k views

Details for the action of the braid group B_3 on modular forms

I'm reading Terry Gannon's Moonshine Beyond the Monster, and in section 2.4.3 he hints at (but does not explicitly describe) a way to extend the action of $SL_2(\mathbb{Z})$ on modular forms to an ...
Qiaochu Yuan's user avatar
19 votes
2 answers
2k views

Generators of the graded ring of modular forms

Let $\Gamma$ be a finite-index subgroup of $\operatorname{SL}_2(\mathbb{Z})$. I've seen it stated (in a comment in the code of a computer program) that the graded ring $$ M(\Gamma, \mathbb{C}) = \...
David Loeffler's user avatar
19 votes
1 answer
3k views

Questions about the "universal elliptic curve" over the affine $j$-line punctured at 0 and 1728

So my question refers to families of elliptic curves over the $\mathbb{A}^1_\mathbb{C}\setminus\{0,1728\}$ whose fiber above a point $j$ has $j$-invariant equal to $j$ (I understand it's not universal)...
Will Chen's user avatar
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15 votes
1 answer
640 views

Is $\eta(\tau)^2$ a modular form of weight 1 on $\Gamma(12)$?

As we know, the Dedekind eta function $\eta(\tau)$ acquires a phase $\exp(2\pi i/24)$ under the modular transformation: $\tau \rightarrow \tau+1$. Therefore $\eta(\tau)^2$ is invariant under $\tau \...
Franklin Wu's user avatar
13 votes
3 answers
2k views

Do L-functions exist for Half-integral weight modular forms?

Classically, we can attach $L$-functions (with properties like, analytic continuation, functional equation) to Dirichlet characters, Hecke eigenforms, etc... My question is: can one attach $L$-...
N. Kumar's user avatar
  • 582
12 votes
2 answers
2k views

Extensions of the modularity theorem

In 1995 (if I'm not mistaken) Taylor and Wiles proved that all semistable elliptic curves over $\mathbb{Q}$ are modular. This result was extended to all elliptic curves in 2001 by Breuil, Conrad, ...
Eugene's user avatar
  • 1,438
12 votes
3 answers
2k views

Kronecker's Jugendtraum for real quadratic fields?

Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was ...
Adam Harris's user avatar
  • 1,895
11 votes
1 answer
650 views

Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?

Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion $$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$ and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke ...
D_S's user avatar
  • 6,100
11 votes
1 answer
860 views

Up to $10^6$: $\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$ (Number of partitions with no even part repeated )

Up to $10^6$: $\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$ A001935 Number of partitions with no even part repeated Is this true in general? It would mean relation between restricted partitions ...
joro's user avatar
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10 votes
3 answers
534 views

On the Klein quartic and the similar $a^2b+b^2c+c^2a$?

Given the Ramanujan theta function, $$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$ Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}$. I. Degree 5 \begin{align} a &= q^{11/...
Tito Piezas III's user avatar
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
3 answers
730 views

Asymptotic formulas for Monster-related modular functions?

Define the following, $$j(\tau) = \Big(\tfrac{E_4(\tau)}{\eta^8(\tau)}\Big)^3 = {1 \over q} + 744 + \color{blue}{196884} q + 21493760 q^2 + 864299970 q^3 + \cdots \tag{1}$$ $$j_{2A}(\tau) =\Big(\big(...
Tito Piezas III's user avatar
8 votes
3 answers
2k views

Numerical evaluation of the Petersson product of elliptic modular forms

It is known how to compute the Fourier expansion of elliptic modular forms using modular symbols, and it is known how to get numerical evaluations of $L$-functions of various type ; it's possible to ...
8 votes
1 answer
791 views

Omitting primes from a Hecke algebra

Let $N \ge 1, k \ge 2$ be integers, and $M_k(\Gamma_1(N))$ the space of weight k modular forms of level $\Gamma_1(N)$. Let $\mathbb{T}$ be the $\mathbb{Z}$-subalgebra of $\operatorname{End} M_k(\...
David Loeffler's user avatar
8 votes
2 answers
989 views

When do the Galois reps of modular forms have open image?

Suppose f is a newform (with coefficients generating some number field E), and $\rho_{f,\lambda}: {\rm Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) \to {\rm GL}_2(E_\lambda)$ the associated Galois rep (...
David Loeffler's user avatar
8 votes
3 answers
1k views

Modular form on $\Gamma_0(N)$

I recently asked this question on Math.StackExchange with no answer so far. So I thought maybe I can find an answer here. Let $M(k,\Gamma_0(N))$ be a space of modular forms of weight $k$ on $\Gamma_0(...
user31009's user avatar
  • 193
7 votes
2 answers
1k views

Duality of eta product identities: a new idea?

Looking at the collection of Eta Function Product Identities by Michael Somos, it seems like generally those identities come in pairs: let's call two eta product identities $\sum\limits_{i=1}^r a_iP_i=...
Wolfgang's user avatar
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6 votes
1 answer
1k views

How did Gauss characterize the metrical relations in the uniform (4 4 4) tiling of the hyperbolic unit disk?

My purpose is to verify an historical hypothesis I have on Gauss's tesselation of the unit disk as described in John Stilwell "Mathematics and its history". Looking at the relevant pages in ...
user2554's user avatar
  • 1,869
5 votes
1 answer
625 views

Ternary quadratic form theta series as Hecke eigenforms and class number one

At Simple comparison of positive ternary quadratic form representation counts Jeremy answered: "The reason is that the theta series for the sums of three squares form is an eigenfunction for all the ...
Will Jagy's user avatar
  • 25.3k
4 votes
1 answer
339 views

q-series related to d(n)=# of divisors of n

Let $\sigma_m (n)=\sum_{d|n}d^m$, and $d(n)=\sigma_0(n)$ as stated in the title. My question is: Does the q-series $f(q)=\sum_{n\ge 1} d(n)q^n$ gives something like a modular form (i.e. with some ...
Ted Mao's user avatar
  • 443
4 votes
1 answer
465 views

Index of the Hecke algebra with operators omitted

This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler. Let $N$ be a positive integer. For a finite set of primes $\Sigma$, let $\mathbb T^{\Sigma}$ be the $\mathbb ...
Olivier's user avatar
  • 10.2k
4 votes
2 answers
495 views

Finding a sextic analogue to the solvable octic $\frac{(x + 1)^6(x^2 + x + 7)}x = -k^3$ where $e^{(\pi/3)\sqrt{d}}\approx k^3+41.999999999999\dots$

I. Degree 8 Assume the $j_i$ to be free parameters. The following octics in $x$ belong to $8T43,$ have group $\text{PGL}(2,7)$, and order $2\times168 = 336.$ \begin{align} {j_1}\; &=\frac{(x^2 + ...
Tito Piezas III's user avatar
4 votes
1 answer
242 views

Does the modular form associated to cubic twist of a elliptic curve $E$ corresponds to some twist of $f_E$?

Let $E$ be an elliptic curve defined over $\Bbb Q$ and $f_E$ be the modular form associated with the elliptic curve $E$. Suppose the elliptic curve $E^D$ is a quadratic twist of $E$. I understand that ...
SUNIL PASUPULATI's user avatar
3 votes
3 answers
316 views

Solving solvable septics using only cubics?

After the satisfying resolution of my question on the Kondo-Brumer quintic, I decided to revisit my old post on septic equations. I. Solution by eta quotients The septic mentioned in that post may not ...
Tito Piezas III's user avatar
2 votes
1 answer
266 views

On Elkies' $\text{9T32}$ nonic and a shared property with j-function formulas

I. First Set Before going to Elkies' nonic, we start with something a bit simpler. There is a list of j-function formulas in this MSE post. For example, for prime levels $p = 5,7,13,$ we have, $$j=\...
Tito Piezas III's user avatar