Questions tagged [hecke-operators]
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41 questions
1 vote
1 answer
336 views
Infinite set of isogenous elliptic curves over $\bar{\mathbb{Q}}$ with finitely many reductions at every good prime
Let $I$ be an infinite set of elliptic curves over $\bar{\mathbb{Q}}$ with a set $S$ of good primes of $\bar{\mathbb{Z}}$ for all elliptic curves in $I$. For $E\in I, v\in S$, let $\bar E_v$ be the ...
- 85
1 vote
0 answers
152 views
Regarding operators in modular forms
If $f \in M_k(N, \chi)$ is a normalized integer weight Hecke eigenform for all $T_p$, is there a way to obtain $g \in M_k \left(4N, \chi \left( \frac{-4}{\cdot} \right) \right)$ via the use of ...
- 95
0 votes
0 answers
108 views
Regarding Hecke eigenforms
Let $f = \displaystyle{\sum_{n}} a(n) q^{n} \in M_{k}(N, \chi)$ be a normalized Hecke eigenform. We know that the coefficients a(n) follow a multiplicative relation, $a(n) a(m) = \displaystyle{\sum_{d ...
- 95
2 votes
0 answers
139 views
Regarding Rankin-Cohen bracket of two modular forms
How does U-operator act on Rankin-Cohen brackets of two modular forms $[f_{1},f_{2}]$ where $f_{1}, f_{2} \in M_{k_{i}}(N, \chi)$ for $i \in \{1,2\}$? Here, $U_{m}(f) = \frac{1}{m} \displaystyle{\...
- 95
3 votes
1 answer
247 views
Regarding normalized Hecke eigenforms
Consider the space of modular forms $M_{k}(\Gamma_{0}(tN), \chi_{t} \chi_{N})$ where $t, N > 1, \: (t, N) = 1$ and both are relatively prime to 6. Does the space always has a basis of normalized ...
- 95
1 vote
1 answer
440 views
What is the image of the Hecke operator $U_p$?
Let $N \geq 1$ be an integer and let $p$ be a prime not dividing $N$. For $r \geq 1$, let $M_2(\Gamma_0(Np^r))$ denote the space of weight $2$ modular forms of level $\Gamma_0(Np^r)$. Let $$U_p: M_2(\...
- 2,049
2 votes
0 answers
323 views
Hecke operators acting on the Weierstrass $\wp$-function
Roughly speaking, my question is the following: Let $\wp(\tau, z)$ be the Weierstrass $\wp$-function, where $\tau \in \mathbf{H}$ and $z \in \mathbf{C}$. If $p$ is a prime number, can we define $T_p\,\...
- 2,049
2 votes
0 answers
128 views
Manifestation of Hecke operator on the category of abelian varieties (or motives)
If we are given some postivie integer $N$ and a prime $p$, then we have the Hecke operator $T_p$ on modular forms, which is a cohomological manifestation of the Hecke correspondence $$X_0(N)\leftarrow ...
- 4,468
3 votes
1 answer
375 views
Hecke operators on universal elliptic curves
Suppose that $\tau \in \mathbf{H}$ belongs to the complex upper half plane. The quotient $\mathbf{C}/(\mathbf{Z}+\mathbf{Z}\tau)$ gives an elliptic curve over $\mathbf{C}$. Write this elliptic curve ...
- 2,049
3 votes
0 answers
108 views
Hecke operators for modular form with respect to $\Gamma_{\theta}(2)$ subgroup
The congruence subgroup $\Gamma_{\theta}(2)$ is defined as: $$\Gamma_{\theta}(2)=\left\{\gamma\in SL(2,\mathbb{Z})|\gamma\equiv\left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right) \...
- 31
6 votes
0 answers
490 views
p-adic Hecke operators in the Iwahori-Hecke algebra $C_c(J\backslash G(F)/J)$
$\DeclareMathOperator\ch{ch}$Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer. I shall use $\kappa(F)$ to denote ...
- 778
7 votes
1 answer
1k views
Explicit computation of the effect of the Atkin-Lehner operator/Fricke involution's effect on $q$-expansion
As a part of the research with which I am involved, I would like to understand how to compute the effect of the Atkin-Lehner operator/Fricke involution $W_2 = \begin{pmatrix} 0 & 1 \\ -2 & 0 \...
0 votes
0 answers
130 views
The specific connection between the Hecke operator and the t'Hooft Operator
As I was reading some articles concern about the Selberg trace formula and its general form, I have noticed that the Selberg trace formula and its general form can be understand as the energy spectrum ...
3 votes
1 answer
522 views
Integrality of Atkin-Lehner operator for $\Gamma_1(N)$
A result due to B. Conrad (http://math.stanford.edu/~conrad/papers/prasanna-inv.pdf, Theorem A.1) states that the Atkin-Lehner operator $w_{Q,k}$ is $\mathbb{Z}[1/Q]$-integral on $M_k(\Gamma_0(N))$. ...
- 917
1 vote
1 answer
264 views
Algebra of Hecke operators on $M_k(\mathrm{SL}_2\mathbb{Z})$ is an integral domain?
Let $M_k(\mathrm{SL}_2\mathbb{Z})$ be the space of modular forms of (integer) weight for the full modular group. Let $\mathbf{H}$ denote the Algebra generated by the Hecke operators $T_n$. Is $\mathbf{...
- 377
13 votes
1 answer
493 views
How does one compute the Hecke algebra acting on modular forms?
I asked this on mathstackexchange, but got no answer. Let $N\geq1$ be an integer, and let $\mathbb{T}$ be the Hecke algebra acting on the cusp forms of weight k and level $\Gamma_0(N)$. Then $\...
- 663
5 votes
1 answer
915 views
Origin of Hecke operators
What is the original paper in which Erich Hecke had first introduced the Hecke operators?
- 2,395
7 votes
3 answers
630 views
Endomorphism ring of $J_0(p)$ and Hecke operators
Let $J_0(p)$ be Jacobian of the modular curve $X_0(p)$ over $\mathbb Q$ where $p$ is a prime, consider the subring $\mathbb T$ inside $\newcommand{\End}{\operatorname{End}}\End_{\mathbb Q}(J_0(p))$ ...
- 7,442
2 votes
0 answers
71 views
Product of $V_N$ operator(index changing) and its adjoint on Jacobi forms
In Kohnen & Skoruppa's 1989 inventiones paper, page 549, the operator $V_N: J_{k,1}^\text{cusp} \longrightarrow J_{k,N}^\text{cusp}$ is defined by the action $$ \sum_{\substack{D<0,r \in \...
- 377
3 votes
1 answer
205 views
Computing double coset operators in a computer algebra system
I want to do double coset operators computations on modular forms of half integer weight and with character such as the trace operators that map modular forms of congruence subgroups $\Gamma_0(N)$ to ...
user35360
4 votes
0 answers
195 views
Hecke operators that lower level
I am working with weakly holomorphic modular functions (weight $=0$) $f \in M_0(\Gamma(N), \chi)$ of level $N$ with some character $\chi$ (We can ignore the character for now). Let $f \in M_0(\Gamma(N)...
user35360
9 votes
0 answers
301 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 ...
- 5,604
8 votes
1 answer
295 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 ...
- 6,124
5 votes
0 answers
151 views
Volumes of Hecke operators
Let $G=GL(2, F)$ and $K$ a maximal compact subgroup. Unramified Hecke operators are defined by the action of the double cosets $$T(n) = \bigcup_{\substack{ad=n, a>0 \\ a|d}} K \left( \begin{array}{...
- 551
3 votes
0 answers
353 views
Ramanujan conjecture in terms of representations
Given an automorphic representation, I would like to bound $\alpha_1^\nu(p) + \alpha_2^\nu(p)$ where the $\alpha_i$ are the Satake parameters of an automorphic form $f$ of, say, $GL_2$. So that $\...
- 609
6 votes
0 answers
332 views
Local L-factors for automorphic representations
For Hecke L-functions associated to a holomorphic cusp form $f$ of level $N$, the local factors can be decomposed into $$L_p(s, f) = (1-\lambda_f(p)p^{-s} + \chi_N(p)p^{-2s})^{-1}$$ where $\chi_N$ is ...
- 609
23 votes
2 answers
3k views
What is the matter with Hecke operators?
This question is inspired by some others on MathOverflow. Hecke operators are standardly defined by double cosets acting on automorphic forms, in an explicit way. However, what bother me is that ...
- 609
7 votes
1 answer
199 views
Is every eigenvector sequence for the Hecke operators a eigenform?
Let $f(n) = a_n$ be a sequence taking values in $\mathbb C$ for $n=1,2,...$. Let $T_m$ be the Hecke operators (of a fixed weight $k$) defined as usual in terms of the $a_n$. That is: $$T_m(f)(n) = \...
- 8,081
14 votes
2 answers
612 views
Distribution relation in the Euler system of Heegner points
I am trying to understand the details behind the so-called "distribution relations" between Heegner points on the modular curve $X_0(N)$, as given (for instance) in Gross's paper Kolyvagin's work on ...
- 329
3 votes
0 answers
324 views
Hecke operator acting on Siegel modular forms
Let $F,G$ be Hecke eigenforms of weight $k$ and genus $2$. For any Hecke operator $T$ (either $T_q$ or $T_{q^2}$) let $\lambda_T(\star)$ be the correspondent eigenvalue. Assume that there exists a ...
- 253
8 votes
0 answers
669 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 $$ \!\!\! \!\...
- 3,434
9 votes
1 answer
791 views
Hecke operator which changes character
In This MO question, Werner said that Hecke operator "changes" characters. I'm looking for any explicit theory of this kind, about Hecke operator with characters. Actually, there are somewhat ...
- 2,245
8 votes
1 answer
613 views
Reaching Hecke eigenvalues from a trace formula
I am interested in studying equidistribution of Hecke eigenvalues and proving statistical properties of arithmetical objects. On the road, I face the following problem: how to express sums of the form ...
- 5,501
18 votes
1 answer
938 views
Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?
Fix $N\ge4$. Let $Y_1(N)$ and $X_1(N)$ be the usual modular curves. I want to view them as schemes over $\mathbb Z$ representing the moduli functors of (usual or generalized) elliptic curves with (...
- 744
3 votes
1 answer
344 views
Restriction to the diagonal of Hilbert eigenforms
Do you know of any reference that discusses whether the restriction to the diagonal of a Hilbert eigenform is an (elliptic) eigenform?
- 855
2 votes
0 answers
319 views
Adjoint of U_p and Atkin-Lehner theory
In the below everything is quoted from Miyake's "Modular Forms". Let $p \mid N$. We have Hecke operator $T(p)\in \mathscr{R}(N)$ (pg. 135) on $S_{k}(\Gamma_0(N))$ given by $$T(p) = \Gamma_0(N)\big(\...
- 21
1 vote
0 answers
141 views
Question about expression of a sum of two Hecke eigenvalues
I did some computations but I am stuck in finding the exression of the sum $$\lambda_f(n^2)+\lambda_f(n)^2 $$ in terms of $\lambda_f(n),$ where $f$ is a modular form for the full modular group. Any ...
- 1,297
0 votes
1 answer
169 views
Expression of a sum of Hecke eigenvalues in terms of one Hecke eigenvalue
Let $f$ be a modular form of an even weight $k$ over the modular group $SL_2(Z).$ Denote $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ I am doing some calculations and I am stack in ...
- 1,297
6 votes
0 answers
141 views
divisibility by Bernoulli numbers of discriminant of Hecke algebra over the space of modular forms of level 1
For the space of modular forms and the space of cusp forms (here I only care about the level $1$ case), we have the action by Hecke algebras. Therefore, we can calculate the discriminant of this ...
- 61
6 votes
2 answers
666 views
Lower bound of Hecke eigenvalues of Maass form
If $f$ is a Maass form and $p$-Hecke eigenvalue (i.e. Hecke eigenvalue of usual Hecke operator $T_p$) of $f$ is $\lambda_f(p)$, do we know anything about lower bound of the sum$$S(x) = \sum_{x\le p\le ...
- 1,702
5 votes
1 answer
725 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 ...
- 26.5k