It will depend on what I really cover. See below. It will certainly have to do with my thesis and some of the resulting papers (which I would not really recommend as first reading): G. Wiese. On the faithfulness of parabolic cohomology as a Hecke module over a finite field. arXiv:math.NT/0511115 G. Wiese. On modular symbols and the cohomology of Hecke triangle surfaces. arXiv:math.NT/0511113 G. Wiese. Modular Forms of Weight One Over Finite Fields (PhD thesis), in particular, Chapter IV (computational details). My web page. Useful are also Sections 4,5,6 (the appendices are a bit out of date) of S.J. Edixhoven. Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one (arXiv:math.NT/0312019), Journal of the Inst. of Math. Jussieu (2006) 5(1), 1-34. If I mention Serre's conjecture, I recommend Ribet/Stein and Serre's original Duke paper, as well as S.J. Edixhoven. Serre's conjecture. Cornell, Silverman, Stevens: Modular Forms and Fermat's Last Theorem. Springer.

It is a bit difficult to have detailed comments, only knowing the titles of the lectures and not their contents. I very essentially need the notion of a Hecke algebra: subring in some endomorphism ring generated by the Hecke operators. So I need that they commute, that T_p^n can be got from T_p (and dimanond if applicable) etc. So, I could give a very short lecture on that on the first day (unless that is part of "classical modular forms I"). That lecture would also prove the structure theorem of Artin algebras: they are the direct product of their localisations - that is much nicer than just talking about eigenforms, in particular, if there is no basis of eigenforms which is usually the case over F_p (or, of course, if there are old forms). I could treat the following: 1) Modular symbols over (rather) arbitrary rings. William, I guess you want to do that, which is fine, of course. But I need the definition in particular over a finite field. 2) Manin symbols over (rather) arbitrary rings. I can give a full proof that they coincide with modular symbols (at least a complete sketch) if I'm allowed to use some group cohomology. An ad hoc definition can be given of group cohomology which already leads very far. One would have to believe some of the standard facts of group cohomology (but we also have exercises; maybe, I can find a way of leading the students to an ad hoc proof of the needed facts, in exercises.) 3) I essentially need the Eichler-Shimura isomorphism, or the Shokurov pairing (which I think is much more difficult). I could give a proof of Eichler-Shimura, at least a rather complete sketch. That would give a full proof of the modular symbols algorithm for computing (classical) modular forms, if we identify the modular symbols space with the cohomology group used in Eichler-Shimura (or Shokurov). William, if you want to stick to the presentation in your book (which in the version I read some time ago refers to Merel who refers to Shokurov), then that's fine for me, of course, I just need some result of that kind. 4) Mod p modular forms. One already gets very far by seeing them as reductions of classical modular forms, at least, when the weight is at least 2. Maybe, I present the Katz definition in passing. Really working with it would be too difficult (maybe, also for me...). Deligne-Rapoport etc. would have to be supposed known. I will nevertheless mention that there are big differences between the two definitions in weight one. So, I'll present that area in a black box way. 5) Computing mod p eigenforms. I'll deduce from the modular symbols algorithm over the rationals that one gets all mod p eigenforms by using mod p modular symbols. That is actually very easy. 6) Computing mod p Hecke algebras. In the mod p setting it is desirable not only to know eigenforms, but the whole Hecke algebra, as it encodes much more information, e.g. on congruences. I'll present p- adic Hodge theory in a black box way, as it gives a Z_p-analogue of the Eichler-Shimura isomorphism, so we get a Z_p-analog of the algorithm for computing Hecke algebras, hence, also an F_p-analog which is the one that I'll use. There is, however, the restriction to weights 2 <= k < p. My result that one can also use k=p on the ordinary locus will be mentioned, but I think it is too difficult to prove it, since it makes use of Gross paper in Duke on the tameness criterion (also making use of Deligne-Rapoport). 7) Computing mod p modular forms of weight 1. That part could present Edixhoven's algorithm realised on mod p modular symbols. It works by relating the weight one forms with weight p forms by using the Frobenius and the Hasse invariant. By the results of 6) one can compute the Hecke algebra in weight p using mod p modular symbols. The algorithm then tells one how to use that Hecke algebra to get the weight one forms. 8) Serre's conjecture. The principal motivations (from my point of view) of looking at mod p modular forms are congruences and more importantly Serre's conjecture. I could present it in a nutshell. Now that Serre's conjecture is on the edge of being completely proved, one should mention how it can be used from a computational point of view: _All 2-dimensional representations (mod p, irreducible, odd) of Gal(Qbar/Q) can be computed by computing mod p modular forms._ That sentence is still a bit optimistic (as Edixhoven's project on that is not yet completely finished). But we can alread now rather efficiently compute the traces of all Frobenius elements of all these representations just by computing the eigenvalues of Hecke operators on mod p modular symbols. Dont' worry: That the numbers run till 8 does not mean that I want 8 lectures! 1) and 5) are very short. 7) could be dropped. In that case there would be no mention of Katz' definition of modular forms over a finite field.