# Questions tagged [hecke-operators]

The hecke-operators tag has no usage guidance.

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### 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(\...

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### 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\,\...

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### 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 ...

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### 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 ...

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### 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) \...

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### 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 ...

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### 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 \...

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### 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 ...

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### 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 $M_k(\Gamma_0(N))$. ...

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### 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{...

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### 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 $\...

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### Origin of Hecke operators

What is the original paper in which Erich Hecke had first introduced the Hecke operators?

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### 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 $\newcommand{\End}{\operatorname{End}}\End_{\mathbb Q}(J_0(p))$ ...

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### 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 \...

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### 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 ...

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### 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)...

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### 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 ...

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### 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 ...

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### 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}{...

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### 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 $\...

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### 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 ...

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### 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 ...

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### 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) = \...

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### 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 ...

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### 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 ...

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### 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
$$ \!\!\! \!\...

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### 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 ...

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### 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
...

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### 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 (...

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### 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?

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### 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(\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...