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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 of double cosets $$\Gamma_0(N) \begin{pmatrix} a&b \\ c&d \end{pmatrix} \Gamma_0(N),$$ where $a, b, c, d \in \mathbb Z$ with $ad-bc = p^m$ and $c \equiv 0$ mod $N$. The usual definition for a ramified Hecke operator $T_{p^m}$ ($p | N$), the union of double cosets as above, now with the additional restriction that $p \nmid a$.

On the other hand, Eichler in his work on the basis problem worked with ramified Hecke operators without this restriction $p \nmid a$.

To me, the usual definition seems rather ad hoc, and the one Eichler uses seems more natural from the point of view of orders and ideals. (Note these definitions give different operators.) Of course, the usual definition is nice because it acts nicely on Fourier coefficients.

Question: Where were these ramified Hecke operators first defined?

For bonus credit: how did these definitions actually come about?

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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 this is the first time these operators appeared in print, but the Mathematical Reviews entry for this paper, by Rankin, seems to suggest so; the concluding paragraph of the review is

These results show that it is possible to obtain a satisfactory theory of Hecke operators on $\Gamma_0(m)$ not only for primes $p \nmid m$, but also for primes that do divide the level m.

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  • $\begingroup$ Thanks. FYI I believe Eichler considered the other kind of ramified Hecke operators back in the 1950's, though I haven't read his earlier papers (in German, which I sadly do not read). $\endgroup$ – Kimball Feb 5 '18 at 13:25

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