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3
votes
1answer
289 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 $M_k(\Gamma_0(N))$. ...
1
vote
1answer
166 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{...
12
votes
1answer
342 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 $\...
5
votes
1answer
805 views

Origin of Hecke operators

What is the original paper in which Erich Hecke had first introduced the Hecke operators?
7
votes
3answers
433 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 $\newcommand{\End}{\operatorname{End}}\End_{\mathbb Q}(J_0(p))$ ...
2
votes
0answers
42 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 \...
3
votes
1answer
112 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
votes
0answers
118 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
votes
0answers
231 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 ...
8
votes
1answer
172 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 ...
5
votes
0answers
112 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
votes
0answers
244 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
votes
0answers
121 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 ...
22
votes
2answers
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
1answer
115 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
2answers
416 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
votes
0answers
178 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
votes
0answers
514 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
votes
1answer
365 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
votes
1answer
380 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 ...
15
votes
1answer
622 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
votes
1answer
233 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?
2
votes
0answers
237 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(\...
1
vote
0answers
115 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
1answer
123 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 ...
6
votes
0answers
112 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 ...
6
votes
2answers
463 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
votes
1answer
533 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 ...