All Questions
62 questions
22
votes
3
answers
7k
views
A recommended roadmap to Fermat's Last Theorem
I was inspired to undertake math as a career after watching a documentary on the proof of Fermat's Last Theorem. As such it's been a small goal of mine to understand Wiles et al's proof.
In a ...
- 1,458
22
votes
1
answer
2k
views
Reference request: The first cohomology of SL(2,Z) with coefficients in homogeneous polynomials
Let $H_k$ be the vector space of degree $k$ homogeneous polynomials in two variables.I'm looking for a reference for the fact that $H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus E_{...
- 4,898
18
votes
1
answer
562
views
Is special value of Epstein zeta function in 3 variables a period?
Kontsevich-Zagier's article "Periods" contains the following question
Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period?
($\sum'$ means we do not sum ...
- 809
16
votes
0
answers
11k
views
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 ...
14
votes
4
answers
3k
views
Jacobi's theorem on sums of two squares (reference request)
One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals
$$4(d_1(n)-d_3(n)),$$
where the function $d_i$ counts the number ...
- 3,062
13
votes
5
answers
4k
views
Brief Introduction to Modular Forms
What are the best introductory texts on modular forms that are suited for a brief six week course intended for advanced undergraduates? The students will be quite sharp and as far as prerequisites go, ...
13
votes
1
answer
1k
views
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 ...
- 659
11
votes
1
answer
698
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 ...
- 6,180
11
votes
1
answer
782
views
Atkin--Lehner operators in Hida theory
Let $p$ be a prime, and $F$ a $p$-adic Hida family of ordinary modular forms (of some tame level $N \ge 1$). I'd like to know whether, for $q$ a prime factor of $N$, the actions of the Atkin--Lehner ...
- 37k
11
votes
0
answers
576
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 ...
- 5,422
10
votes
1
answer
314
views
Coefficient bounds on cusp forms, half-integer weight
Let $f(\tau) = \sum_{n=1}^{\infty} a(n) q^n$ be a cusp form on $\Gamma_0(4N)$ of half-integer weight $k \ge 5/2.$ The Ramanujan-Petersson conjecture in this case is that $$a(n) \ll n^{(k-1)/2 + \...
- 101
10
votes
1
answer
550
views
Igusa's $\chi_{10}$ and Borcherds products
Igusa defined a genus 2 Siegel modular form $\chi_{10}$, which vanishes on the Humbert surface $G_{1}$ (the image of a "degenerate" Hilbert modular surface, the product of modular curves, ...
- 3,309
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
9
votes
2
answers
584
views
Number Fields Arising from Newforms
It is well-known that, given a normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$.
In their 1995 paper "Fermat's Last Theorem", Darmon, Diamond, and Taylor ...
- 1,422
9
votes
1
answer
638
views
Algebraic independence of $P,Q,R$ or $E_2,E_4,E_6$ over $\mathbb C(z)$
Let $P,Q,R$ be the Fourier series of the Eisenstein series $E_2,E_4,E_6$, that is,
$$
P(q)=1-24\sum_{n=1}^{\infty}\sigma_1(n)q^n,
$$
$$
Q(q)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n,
$$
$$
R(q)=1-504\...
- 663
9
votes
2
answers
3k
views
Transformation formulae for classical theta functions
I am looking for a reference for the transformation formulae
for the classical theta-functions
$$\theta_4(\tau)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}$$
and
$$\theta_2(\tau)=\sum_{n=-\infty}^\infty q^{...
- 20.8k
9
votes
1
answer
650
views
Sum of three squares as class numbers and Waldspurger's formula
It is known that the number of ways to express $n \in \mathbb{Z}_{\geq 0}$ as a sum of three squares (let's denote it as $r_3(n)$) can be expressed as Hurwitz-Kronecker class number (certain weighted ...
- 2,215
9
votes
0
answers
560
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 ...
- 5,422
8
votes
3
answers
3k
views
Computing the q-series of the j-invariant
It is a fundamental fact, often quoted these days in its connection with Monstrous Moonshine, that the q-expansion (i.e., the Laurent expansion in a neighborhood of $\tau = i\infty$) of the j-...
- 1,415
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
890
views
Weisinger's thesis
I am currently reading Atkin and Li's paper on Twists of newforms and Atkin-Lehner pseudo eigenvalues and one of the references there is to Weisinger's thesis:
Weisinger J., Some results on classical ...
- 363
8
votes
1
answer
247
views
Origin of definitions of ramified Hecke operators
Consider a classical space $M_k(N)$ of elliptic modular forms of weight $k$ for $\Gamma_0(N)$. The definition of an unramified Hecke operator $T_{p^m}$ in terms of double cosets is the disjoint union ...
- 6,039
7
votes
1
answer
281
views
Sato-Tate conjecture when Fourier coefficients are complex numbers
Let $k\geq 1$ and let $f=\sum_{n\geq 1}a(n)q^{n}$, $a(n)\in\mathbb{R}$, be a normalised cuspidal Hecke eigenform of weight $2k$ for $\Gamma_{0}(N)$ without complex multiplication. So the result of ...
- 433
7
votes
1
answer
232
views
Is anything known about this class of series involving the divisor function?
I hope it is OK to ask the following reference request. If my question is not suitable, please let me know and I will do my best to modify it!
Let $N\in\mathbb{N}$, let $q$ be a point in the open ...
- 661
6
votes
3
answers
966
views
congruences for Fourier coefficients of modular forms
Are there other good articles on congruences for Fourier coefficients of modular forms beside Swinnerton-Dyer's article in "Modular Functions of One Variable III"?
I am looking for generalisations ...
user19475
6
votes
2
answers
730
views
Definition of Hecke operators on orthogonal modular forms
In his paper Automorphic forms with singularities on Grassmannians, Borcherds poses Problem 16.5:
"Describe how the correspondence in this paper behaves under
the
action of Hecke operators."
Since ...
user94539
6
votes
2
answers
1k
views
Reference for universal elliptic curves
I've seen the following sentence come up in a few papers:
Consider the modular curve $Y_1(N)$ and let $E$ be the universal elliptic curve over $Y_1(N)$.
This comes up in Deligne's construction of ...
- 1,949
6
votes
1
answer
480
views
Twisted modular forms of half-integral weight
I am looking for references (or explainations) about the twist of modular forms of half-integral weight. I try to mimic the proof of the "integral weight case" to prove that the twist of
$$ \theta(\...
- 1,479
6
votes
2
answers
976
views
References for modular polynomials
I am teaching a graduate "classical" course on modular forms. I try to achieve the most elementary level for presenting modular polynomials. Serge Lang's "Elliptic functions" cover the topic quite ...
- 13.5k
6
votes
1
answer
380
views
Applications of Level Lowering
What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
- 1,629
6
votes
0
answers
456
views
Conditions under which an $\eta$-quotient becomes a **weak** modular form (reference request for theorems similar to Ligozat's theorem)
For any $z \in \mathcal{H}$, let $q = e^{2\pi iz}$; and the eta function is defined as
${\displaystyle \eta (q)
=q^{\frac {1}{24}}\prod _{n=1}^{\infty }\left(1-q^{n}\right).}$
By an $\eta$-quotient ...
6
votes
0
answers
465
views
Eisenstein series of Hilbert modular forms
I am reading Shimura's paper "The Special Values of the Zeta Functions Associated With Hilbert Modular Forms" and I do not exactly understand his definition of the Eisenstein series in section 3.
...
- 123
5
votes
1
answer
873
views
Origin of Hecke operators
What is the original paper in which Erich Hecke had first introduced the Hecke operators?
- 2,375
5
votes
3
answers
2k
views
Modular forms reference
If f is a weight 2 newform on $\Gamma_1(N)$ then there exists an abelian variety Af whose endomorphism algebra is isomorphic to the field generated by the coefficients of f.
I've seen this proven in ...
user1073
5
votes
1
answer
388
views
Results in an article by Siegel
Studying the Eisenstein cocycle by Sczech, I noticed that to understand its connection with the values at negative integers with zeta functions it is necessary to understand the resuts by Siegel in
...
- 3,107
5
votes
0
answers
174
views
Effective Hecke Equidistribution
In 1918 and 1920 Hecke introduced his L-functions attached to his Grössencharakteren (Hecke characters) and proved they are equidistributed in a sense to made precise momentarily. One can identify ...
- 177
5
votes
0
answers
247
views
Congruence of Fourier coefficients of Siegel cusp forms
Let $F$ be a Siegel cusp form of weight $2k$ and genus $2$ in the Maass subspace (i.e. the Saito-Kurokawa lift of some classical cusp form $f$ of weight $4k-2$); assume that $F$ and $f$ are Hecke ...
- 253
4
votes
2
answers
450
views
Is there a version of Serre's modularity conjecture for projective representations?
Serre's modularity conjecture asserts that a continuous odd irreducible representation $$\overline{\rho} : G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$$ must be modular, in the sense that ...
- 3,142
4
votes
3
answers
1k
views
Companion forms
What is the best known result concerning the existence of companion forms for classical modular forms? Gross' tameness criterion paper is always mentioned with a "unchecked compatibility" caveat? ...
- 173
4
votes
0
answers
294
views
Modular forms on $\Gamma(N)$
I'm wondering where I can find a good reference about what is known about modular forms (especially cuspidal eigenforms) of full principal level $\Gamma(N)$, in terms of their Hecke theory, old/...
- 2,044
4
votes
0
answers
252
views
Height pairings of Heegner points of nontrivial conductor
I am studying Gross's and Zagier's proof of the BSD conjecture for elliptic curves of rank $\leq 1.$ Their calculation essentially boils down to the following ingredients:
(1.) Finding a suitable ...
- 1,805
4
votes
0
answers
242
views
Evaluation of $E_{\ell,2}$ on supersingular curves over $\mathbb{F}_{p^2}$
As mentioned in an answer to Modularity of $E_2$ on congruence subgroups, there exist modular forms $E_{\ell,2}$ of level $\Gamma_{0}(\ell)$ and weight 2, with $q$-expansion $E_{\ell,2}(q)=E_{2}(q)-\...
- 1,537
3
votes
2
answers
384
views
Poles of the Rankin-Selberg zeta function associated to Hilbert cusp forms
Let $K$ a totaly real number field, $\mathcal{O}_K$ its ring of integre and $h$ the narrow class number of $K$. Let $\mathbf{f}$ a collection $(f_1, ..., f_h)$ of Hilbert cusp forms $f_\lambda$
$(\...
- 400
3
votes
1
answer
277
views
How do modular functions of level $N>1$ transform under the full modular group?
Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$
First question
What can we say in general about the factor $j(\...
- 2,375
3
votes
1
answer
444
views
Details about the $\mod p$ reduction map
Let $N$ be a natural number and let $\Gamma_1(N)$ be the congruence subgroup of $SL_2(\mathbb{Z})$. Let $M(N)$ denote the space of all integer weight holomorphic modular forms for $\Gamma_1(N)$ whose ...
- 589
3
votes
1
answer
425
views
An electronic copy of Vishik's work on $p$-adic $L$-functions for modular forms
This question is very simple.
Would someone be so nice as to send me an electronic copy of M. M. Vishik, Non-Archimedean measures connected with Dirichlet series, Mat. Sb. (N.S.), 1976, Volume 99(...
- 10.9k
3
votes
0
answers
152
views
Finiteness of points over the cyclotomic extension for modular forms
Let $\rho(f):G_\mathbb{Q} \rightarrow GL_2(K_f)$ be the Galois representation attached to some cuspidal modular form $f$ where $K_f$ is a finite extension of $\mathbb{Q}_p$.
Let $V_f$ be the vector ...
- 313
2
votes
1
answer
338
views
Finite Flat Group Schemes for Modular Forms of Higher Weight
Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from an abelian variety, ...
- 15.4k
2
votes
1
answer
243
views
$\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline
Is [Scholl, Motives for modular forms, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight?
Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing ...
user471019
2
votes
3
answers
912
views
Reference on generators of subgroups of symplectic groups
We should start with the definition of the symplectic group for an arbitrary ring $R$.
The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with $...
- 85