24
votes
What is the matter with Hecke operators?
To keep things simple, let $G$ be a finite group and $K$ a subgroup of it. The simplest definition of the Hecke algebra associated to this pair $(G, K)$ is that it is the algebra of $G$-endomorphisms $...
- 118k
16
votes
Accepted
Origin of Hecke operators
Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I, Math. Ann. 114 (1937), 1-28; II, ibid., 316-351. These two papers are available here and here.
- 105k
16
votes
Accepted
What is the matter with Hecke operators?
If $K$ is a local field and $\mathcal O$ is its ring of integers, we say an irreducible representation of $G(K)$ is unramified if it contains a vector invariant under $G(\mathcal O)$. It is known that ...
- 148k
10
votes
Endomorphism ring of $J_0(p)$ and Hecke operators
Mazur proved that $\mathbb{T}' = \text{End}_{\mathbf{Q}}(J_0(p))$ where $\mathbb{T}'$ is generated by $\textit{all}$ the Hecke operators $T_n$ for $n\geq 1$ (including $n$ divisible by $p$). Note that ...
- 1,109
10
votes
Accepted
Distribution relation in the Euler system of Heegner points
What I don't understand is why the exactly the same terms should appear in both sums.
The Galois action on CM points is described in adelic terms via the fundamental theorem of complex multiplication ...
- 10.9k
9
votes
Accepted
Reaching Hecke eigenvalues from a trace formula
Yes, this is a standard thing to do. If you want to look at traces of Hecke operators on a definite quaternion algebra, this is the same as what are known as "traces of Brandt matrices." These have ...
- 6,039
7
votes
Accepted
Is every eigenvector sequence for the Hecke operators a eigenform?
The answer is no. Your form $f$ would have to be of weight $k$, and of level $SL_2(\mathbb Z)$, so should be in a finite dimensional vector space of dimension $d$. That would mean that the Hecke ...
- 26k
7
votes
Endomorphism ring of $J_0(p)$ and Hecke operators
EDIT: This is an answer to a different question, namely whether removing operators other than $U_p$ can result in a strict sub-algebra. In particular, the example given shows that $\mathbb T^{(2)}$ is ...
- 10.9k
7
votes
Accepted
Origin of definitions of ramified Hecke operators
These operators certainly appeared in the 1970 paper by Atkin and Lehner:
Atkin, A. O. L.; Lehner, J. Hecke operators on $\Gamma_0(m)$. Math. Ann. 185
(1970), 134–160.
I don't know for sure that ...
- 37k
7
votes
Accepted
Explicit computation of the effect of the Atkin-Lehner operator/Fricke involution's effect on $q$-expansion
In short, there is no simple formula for the $q$-expansion of the transform of a modular form under an Atkin-Lehner operator $W_Q$, in terms of the $q$-expansion of the original modular form.
The ...
- 20.8k
5
votes
Endomorphism ring of $J_0(p)$ and Hecke operators
The very recent paper of Noah Taylor (Section 5 of https://arxiv.org/abs/2001.01814) proves that the Hecke algebra $\mathbb{T}$ generated by $T_q$ for $q \nmid p$ over $\mathbf{Z}_2$ already includes ...
- 206
4
votes
Accepted
What is the image of the Hecke operator $U_p$?
The statement, as claimed, is false.
Let $p = 2, N = 11$, and let $f_0$ be the unique normalised eigenform in $S_2(\Gamma_0(11))$; and set $f(\tau) = f_0(8\tau)$. Then $f \in M_2(\Gamma_0(Np^3))$, but ...
- 37k
4
votes
Accepted
Hecke operators on universal elliptic curves
The universal elliptic curve can be written $y'^2 = 4x'^3- g_2 x - g_3$ where $g_2$ and $g_3$ are Eisenstein series. Given any differential on $E$, simply change coordinates to this family, then ...
- 148k
4
votes
Accepted
Integrality of Atkin-Lehner operator for $\Gamma_1(N)$
Theorem. Let $\ell$ be prime, and $Q, R \ge 1$ such that $(\ell, Q, R)$ are pairwise coprime. Let $N = QR$ and for simplicity assume $N \ge 4$. Then $W_Q$ preserves $M_k(\Gamma_1(N), \mathbf{Z}[1/N, \...
- 37k
4
votes
Accepted
Algebra of Hecke operators on $M_k(\mathrm{SL}_2\mathbb{Z})$ is an integral domain?
You mean the (commutative, normal for the Petersson inner product thus diagonalizable) complex algebra $\Bbb{T(C)}$ of endomorphisms of the complex vector space $M_k(SL_2(Z))$ ($k$ even) generated by ...
- 3,403
3
votes
Distribution relation in the Euler system of Heegner points
Thank you Olivier for this great answer, I wish I could be one of your students ;-) !
Though I still not quite fully understand the adelic setting, here is some "classical" explanation I found, using ...
- 329
3
votes
Lower bound of Hecke eigenvalues of Maass form
Following up on @Idoneal's comment, one can prove something a bit more precise. In particular, $S(x)\asymp\frac{x}{\log x}$, provided that $\log x\gg\log(\lambda N)$, where $f$ has level $N$ and ...
- 5,089
3
votes
Accepted
How does one compute the Hecke algebra acting on modular forms?
This isn't quite an answer, but since I cant comment, I'll do it here.
In MAGMA you can ask for HeckeAlgebra of a space of ModularSymbols (or maybe it only works for the cuspidal subspace of such), ...
- 302
1
vote
Computing double coset operators in a computer algebra system
From the MAGMA documentation:
Since V2.8, Magma has included packages for modular forms and modular symbols. These were originally developed by William Stein, and are continually being developed ...
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1
vote
Hecke operator which changes character
This was essentially done by Wohlfhart in his paper Über Operatoren Heckescher Art bei Modulformen reeller Dimension, and also explained in Stromberg's paper Hecke Operators for Maass Waveforms on $\...
- 2,215
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