# Questions tagged [hecke-operators]

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25
questions

**12**

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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 $\...

**5**

votes

**1**answer

781 views

### Origin of Hecke operators

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

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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 \...

**3**

votes

**1**answer

92 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 ...

**4**

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114 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)...

**9**

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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 ...

**6**

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**1**answer

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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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110 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}{...

**3**

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216 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 $\...

**5**

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96 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 ...

**21**

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**2**answers

2k 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 ...

**5**

votes

**1**answer

109 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) = \...

**13**

votes

**2**answers

378 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 ...

**3**

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164 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 ...

**8**

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474 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
$$ \!\!\! \!\...

**6**

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**1**answer

267 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 ...

**6**

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**1**answer

340 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
...

**14**

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**1**answer

547 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 (...

**2**

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**1**answer

227 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?

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211 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(\...

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113 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 ...

**0**

votes

**1**answer

118 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 ...

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108 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 ...

**5**

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**2**answers

418 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 ...

**5**

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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 ...