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 $...

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.

15
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 ...

15
votes

Accepted

### Why is there a factor $p$ in the definition of $T_p$ via Hecke correspondences on modular curves?

This question is somehow a "characteristic 0" question, so let me treat $Y = Y_1(N)$ and $X = X_1(N)$ as $\mathbf{Q}$-varieties rather than doing anything complicated with integral models.
There's an ...

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 ...

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 ...

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 ...

8
votes

Accepted

### Restriction to the diagonal of Hilbert eigenforms

It is extremely unusual for the restriction of a Hilbert modular form to the diagonal to be an elliptic modular eigenform. It happens occasionally in some small cases (by coincidence, essentially), ...

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 ...

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 ...

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 ...

6
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 ...

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 ...

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 ...

4
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 ...

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, \...

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
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 ...

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 ...

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), ...

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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