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Identities for Prime Coefficients of Certain Cusp Forms

While working with Fourier expansions of cusp forms of congruence subgroups of the modular group, I observed the following patterns in their prime coefficients. Let $a(n)$ be the Fourier coefficients ...
Madhusudhan Raman's user avatar
0 votes
0 answers
81 views

Computing elliptic periods from modular form

How are the periods of a modular elliptic curve computed as path integrals of its associated normalized weight 2 cusp form on the modular curve? Please provide specific paths for both periods and cite ...
kindasorta's user avatar
  • 2,907
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 ...
sendit's user avatar
  • 177
2 votes
0 answers
286 views

Is Sturm's theorem able to do these?

$\newcommand{\Ord}{\operatorname{Ord}}$Let $p$ be a positive integer and $F(q)=\sum A(m)q^m$ be a formal power series with integer coefficients. Then $\Ord_p(F(q))$ is defined by $$\Ord_p(F(q)):=\min\{...
T. Amdeberhan's user avatar
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 ...
user avatar
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, ...
Anton Hilado's user avatar
  • 3,309
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 ...
Seewoo Lee's user avatar
  • 2,215
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 ...
Adithya Chakravarthy's user avatar
2 votes
0 answers
490 views

On quasi-modular forms with integer Fourier coefficients

It is well-known that the ring $M$ of modular forms has the structure $M=\mathbb{C}[E_4,E_6]$, where $E_k$ are the Eisenstein series. It is also known that one can define the concept of quasi-modular ...
Yuji Tachikawa's user avatar
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 ...
Davood Khajehpour's user avatar
2 votes
0 answers
245 views

Ambiguity about the exact definition of coefficients of modular forms

You can see the parts after my questions in the boxes. I received the answer to my first question in the comments. I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
Tireless and hardworking's user avatar
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 ...
Krishnarjun's user avatar
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/...
xir's user avatar
  • 2,044
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 ...
user100603's user avatar
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 ...
D_S's user avatar
  • 6,180
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?
Shimrod's user avatar
  • 2,375
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 ...
efs's user avatar
  • 3,107
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\...
LWW's user avatar
  • 663
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(\...
Shimrod's user avatar
  • 2,375
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 ...
Alex Saad's user avatar
  • 661
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. ...
R.T.'s user avatar
  • 123
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 ...
Kimball's user avatar
  • 6,039
1 vote
2 answers
334 views

(Reference) A Shimura operator acting on Hermitian modular forms

In his book Arithmeticity in the theory of automorphic forms (http://bookstore.ams.org/surv-82-s) Shimura introduces at page 146 an operator $\Delta_p^q$ which should act on nearly holomorphic modular ...
Angelo Rendina's user avatar
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 ...
Angelo Rendina's user avatar
2 votes
0 answers
146 views

English proof for decomposition of modular polynomials

Let $\Phi_n(X,Y)$ be the modular polynomial that are the canonical equations for the modular curve $X_0(n)$. They parameterise pairs of elliptic curves related by a cyclic isogeny of degree $n$. I am ...
BlackAdder's user avatar
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 $f_\lambda$ $(\...
Med's user avatar
  • 400
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 ...
M.Souf's user avatar
  • 433
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 + \...
user105068's user avatar
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(\...
Stabilo's user avatar
  • 1,479
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 ...
The Thin Whistler's user avatar
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 ...
user avatar
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 ...
Dr. Pi's user avatar
  • 3,062
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(...
Olivier's user avatar
  • 10.9k
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 ...
Eins Null's user avatar
  • 1,629
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 ...
Pig's user avatar
  • 809
0 votes
1 answer
211 views

Question about sign change of Hecke eigenvalues

I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$-functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper ...
Khadija Mbarki's user avatar
2 votes
0 answers
159 views

The dimension of the space of automorphic forms with multiplier system

Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{Z})$ and $\vartheta$ a multiplier system of weight $k$ for $\Gamma$, by which we mean a function $\vartheta:\Gamma \rightarrow \mathbb{C}$ ...
M.Souf's user avatar
  • 433
2 votes
1 answer
223 views

Level-Lowering in Weight 2

Let $N$ and $p$ be relatively prime integers with $p$ a prime. Suppose $f$ is a weight $k=2$ (normalized, cuspidal, etc) newform of level $\Gamma_1(N) \cap \Gamma_0(p)$. I seem to recall the existence ...
Jeff H's user avatar
  • 1,422
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 ...
MichalisN's user avatar
  • 363
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 ...
David Loeffler's user avatar
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, ...
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 ...
Siksek's user avatar
  • 3,142
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)-\...
DCT's user avatar
  • 1,537
1 vote
6 answers
1k views

List of structure theorems for vector valued Siegel modular forms (esp. of genus 2)

What are Siegel modular forms? We start with defining their common domains $\mathbb{H}_g$ as the set of symmetric $g \times g$ matrices with positive definite imaginary parts. The symplectic group $...
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, ...
David Corwin's user avatar
  • 15.4k
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 $...
Tom's user avatar
  • 85
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 ...
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 ...
paul Monsky's user avatar
  • 5,422
1 vote
0 answers
162 views

Construction of RM abelian variety from eigenform

Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...
David Corwin's user avatar
  • 15.4k
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 ...
Jeff H's user avatar
  • 1,422