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I read the book of Kazuya Kato——"Number Theory: Iwasawa theory and modular forms". In the proof of Mordell's proof of Ramanujan's conjecture, enter image description here

he constructed "Mordell operator" enter image description here

I've known his proof, but still don't known how the motivation of his construction. The appearance of "Mordell operator" is a little strange, can anybody explain the motivation? Besides, he said the "Mordell operator" can be generalized to "Hecke operator", and it is much more complicated...... So, I just begin with the "simpler" case.

Any help will be appreciated.:)

Thanks for @David Loeffler's answer, and I read serre's book further. He used lattice to define Hecke operator,

enter image description here

and then derive how it looks like. Besides, he calculates the example of Eisenstein series and $\Delta$-function. For $\delta$ the corresponding eigenvalue is $\tau(n)$.

The definition of Hecke operator is much more simple and great. However, I still wonder the motivation of the definition and what's the nature of it?

Thanks for @David Loeffler again~~ :)

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I've never seen the name "Mordell operators" before, but these are very well-known objects under their usual name of "Hecke operators", and googling for that name will bring up references by the hundred. Serre's book "A Course in Arithmetic" gives a very nice account in the special case of level one modular forms (like the $\Delta$ appearing here), which avoids much of the complication of the general case.

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    It's a good suggestion! I read sere's book and found him use lattice to define Hecke operator, and derive how it looks like. Thanks for your advice, but I still can't understand "transforms a lattice to the sum of its sub-lattices of index n" to definite Hecke operator. I will edit it to make my question clear. – 2017-01-13