Would you please explain the importance of Hecke operators on modular forms? I am studying modular forms mostly on my own and I have a pretty good understanding up to Hecke operators. So, I just wonder why we care about them.
Number Theory – Hecke Operators on Modular Forms
modular-formsnumber theory
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It might help to go back to the definition of Hecke operators in level $1$ in Serre's Course in arithmetic. For a prime $p$ and a lattice $\Lambda$, the $p$the Hecke corresondence (I forget if Serre uses exactly this terminology) takes $\Lambda$ to $\sum \Lambda'$, where $\Lambda'$ runs over all index $p$ sublattices of $\Lambda$.
This is a multi-valued function from lattices to lattices (it is $1$-to-$p+1$-valued).
Now lattices (mod scaling) are just elliptic curves: $\Lambda \mapsto \mathbb C/\lambda$. And so we can also think of this as a multi-valued map from the moduli space of ellitic curves (i.e. the $j$-line, or $Y_0(1)$ if you like) to itself.
How to describe a multi-valued map more geometrically? Think about its graph inside $Y_0(1) \times Y_0(1)$. The graph of a function has the property that its projection onto the first factor is an isomorphism. The graph of a $p+1$-valued function has the property that its projection onto the first factor is of degree $p+1$.
This graph has an explicit description: it is just $Y_0(p)$ (the modular curve of level $\Gamma_0(p)$). Remember that $Y_0(p)$ parameterizes pairs $(E,E')$ of $p$-isogenous curves. We embed it into $Y_0(1) \times Y_0(1)$ in the obvious way, by mapping the pair $(E,E')$ (thought of as an element of $Y_0(p)$) to $(E,E')$ (thought of as an element of the product).
In terms of the upper half-plane variable $\tau$, one can think of this map as being $\tau \bmod \Gamma_0(p)$ maps to $\bigl(\tau \bmod SL_2(\mathbb Z), p\tau \bmod SL_2(\mathbb Z) \bigr).$
So we have recast Serre's description of the $p$th Hecke operator in terms of a correspondence on lattices in the geometric language of correspondences on curves: i.e. the $p$th Hecke operator is given by a mutli-valued morphism from $Y_0(1)$ to itself, rigorously encoded by its graph thought of as a curve in the product surface $Y_0(1) \times Y_0(1)$, which is in fact isomorphic to $Y_0(p)$.
We can easily compactify the situation, to get $X_0(p)$ embedding as the graph of a correspondence on $X_0(1) \times X_0(1)$.
[Caveat: Actually the map $Y_0(p) \to Y_0(1) \times Y_0(1)$ need not be an embedding; it is a birational map onto its image, but the image can be singular (and the same applied with $X$'s instead of $Y$'s). This is because the point on $Y_0(p)$ is not just the pair $(E,E')$, but the additional data of the $p$-isogeny $E\to E'$, which is not uniquely determined up to isomorphism in some exceptional cases. But this is a technical point which is not worth fussing about at the beginning.]
The advantage of having a geometric correspondence in sight is that whenever we apply any kind of linearization functor to our curve, the correspondence will turn into a genuine single valued operator.
The point is that if we have a multi-valued function from one abelian group to another, we can just add up the values to get a single-valued function.
So the correspondence $T_p$ induces genuine maps from the Jacobian of $X_0(1)$ to itself, or from the cohomology of $X_0(1)$ to itself, or from the space of holomorphic differentials on $X_0(1)$ to itself.
Now actually in the case of $X_0(1)$, which has genus zero, the Jacobian and the space of holomorphic differentials are trivial. But we can do everything with $X_0(N)$ or $X_1(N)$ in place of $X_0(1)$ for any $N$, and all the same remarks apply.
Remembering that the holomorphic differentials on $X_0(N)$ are the weight two cuspforms of level $N$, one can compute that the $p$th Hecke correspondence gives rise to the usual $p$th Hecke operator on cuspforms in this way.
What's the point of considering the correspondence? There are many; here's one:
if we reduce everything mod $p$, we get a mod $p$ correspondence on the mod $p$ reduction of $X_0(N)$, whose graph is the mod $p$ reduction of $X_0(Np)$. But this latter reduction is well-known to be singular, and in fact reducible; it is the union of two copies of $X_0(N)$. Thus the $p$th Hecke correspondence mod $p$ decomposes as the sum of two simpler correspondences, which one checks to be the Frobenius morphism from $X_0(N)$ Mod $p$ to iself, and its dual.
This is the Eichler--Shimura congruence relation (in some form it actually goes back to Kronecker), and it underlies the relationship between $T_p$-eigenvalues and the trace of Frobenius in the $2$-dimensional Galois reps. attached to Hecke eigenforms.
Some MO posts which are vaguely relevant:
If $M$ divides $N$, and $d$ divides $N/M$, then for any modular form $f(\tau)$ on $\Gamma_0(M)$, the function $f(d\tau)$ is a modular form on $\Gamma_0(N)$.
Let's write $V_N$ for the vector space of cuspforms on $\Gamma_0(N)$ (of some fixed weight). Then the above map, i.e. $f(\tau) \mapsto f(d\tau)$, gives an embedding of $V_M$ into $V_N$; write $V_{M,d}$ its image.
Then the space of old forms in $V$ is the subspace of $V$ obtained by taking the sum over all proper divisors $M$ of $N$ and all possible corresponding choices of $d$ of the subspaces $V_{M,d}$. We use the word old for them because these forms, while they have level $N$, actually come from a smaller level (namely $M$), and so are "old" from the point of view of the level $N$. The space of new forms is the orthogonal complement in $V$ of the space of old forms; elements of $V$ are genuinely new to level $N$, hence their name.
Best Answer
This is a layman answer. Hecke operators commute and are self-adjoint, hence the modular forms which are eigenvectors wrt. all Hecke operators form a basis of the space of all modular forms (and the same for cusp forms). If $f$ is such an eigenvector then the L-series corresponding to $f$ has multiplicative coefficients, i.e. it can be represented by an Euler product (over all primes). So AFAIK, Hecke operators are important for connections of modular forms with number theory.