1 Hecke Operators 1.1 Classical Setting First, we will recall the classical theory of Hecke operators. Let be a congruence subgroup of GL+ 2 (R) and f2M k(N). Recall the operation of on M k(N) is given by: (fj k)(z) = f az+ b cz+ d p det cz+ d! k The space of Hecke Operators H is defined to be the free abelian group generated by the double ...

2025年2月16日 · A family of operators mapping each space M_k of modular forms onto itself. For a fixed integer k and any positive integer n, the Hecke operator T_n is defined on the set M_k of entire modular forms of weight k by (T_nf)(tau)=n^(k-1)sum_(d|n)d^(-k)sum_(b=0)^(d-1)f((ntau+bd)/(d^2)).

Hecke Operators We will motivate Hecke Operators following [3] by introducing double coset operators. 2.1 Double Coset De nition. Let 1 and 2 be congruence subgroups and let 2 GL+ 2 (Q), de ne 1 2 = f 1 2: 1 2 1; 2 2 2g to be the double coset in GL+ 2 (Q). The group 1 acts on 1 2 by left multiplication, partitioning it into orbits. It can be ...

2024年2月15日 · Let $M ( k )$ be the vector space of (entire) modular forms of weight $k$, see Modular form or [a1]. Then the Hecke operator $T _ { n }$ is defined for $f \in M ( k )$ by. \begin {equation} \tag {a1} (T_n f) (\tau) = n^ {k-1} \sum_ {d|n} d^ {-k} \sum_ {b=0}^ {d-1} f \left ( \frac {n\tau+bd} {d^2} \right), \end {equation}

In this paper, we aim to discuss Hecke operators in the theory of modular forms. We will show how the Hecke ring H(GL 2(Q p);GL 2(Z p)), the ring of all locally constant, compactly supported GL 2(Z p)-bi-invariant functions f : GL 2(Q p) ! C, is associated with the Hecke operator attached to p. To do so, we will rst

Hecke Operators We recall that an entire modular form of weight k is an analytic function on the upper half-plane H that satis es the transformation property f az+ b cz+ d = (cz+ d)kf(z) for all matrices a b c d in the modular group , and has a Fourier expansion f(z) = P 1 m=0 c(m)e2ˇimz:

1. Hecke operators: First definition We now return to the multiplicative properties of ˝(n). There is a family of operators, called Hecke operators: T m: M k!M k; m= 2;3;:::: We write the action of T m on a form f as either T m(f) or fjT m. The action of Hecke operators can be de ned via the action on Fourier expansions: X n a nq n! T m= X n b ...

Theorem 2. (Hecke) If f is a normalized eigenform on SL2(Z) and if L(f;s) is its associ-ated L-function then for <e(s) > k=2+1 L(f;s) = Y p (1 app s +pk 12s) proof : Relations among the Tn’s and Tpr’s imply that if f is a normalized eigenform : a(mn) = a(m)a(n) if (m;n) = 1 and a(pr) = a(p)a(pr 1) pk 1a(pr 2) thus L(f;s) = X1 n=1 ann s = Y ...

The Hecke operator Ta on L2(Γ\G(R)) is defined as follows: for any f ∈ L2(Γ\G(R)), where |Tax| denotes the cardinality of the set Tax. Since Γ and a−1Γa are commensurable with each other and |Γ\ΓaΓ| = [Γ : Γ∩a−1Γa], the set Γ\ΓaΓ is finite for a ∈ G(Q). The cardinality of this set will be denoted by deg(a).

the Hecke operators act on cusp forms of a given weight and level. The first goal of these two lectures is to pick a very special function f whose convolution action “is” precisely the action of a Hecke operator on the space of cusp forms: that is, we want R(f) to act as the Hecke operator on cusp forms (embedded as usual into L2