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FINAL EDIT (9/5/15)

1. By combining my answer to the initial recursion with the results of my arXiv note 1508.07523 I can show the following: Let K consist of those mod 2 modular forms of level Gamma_0 (3) that are killed by the maps f-->U_2(f) and f-->f+U_3(f). When one completes the shallow Hecke algebra acting on K at the maximal ideal generated by the T_p with p> 3, one gets a power series ring in T_7 and T_13.

2. My answer to the variation on a theme question should lead to a similar result in level Gamma_0 (5), but I'd first need level 5 analogs to the results of the arXiv note, and this may not be entirely straightforward.

3. Despite the lack of response to my answers (I don't think I'd have found the answers if I hadn't posted the questions here) I find them elegant. Since no elementary answers have been found, and in view of 1. , I'll accept my first.

4. I think I can answer the characteristic 3 questions too. But to avoid accusations of self-abuse I'll refrain. And though I find the question on Jordan blocks very interesting (and similar questions attached to the other recursions should also be interesting), the thread has become long and so I've deleted that question.

FINAL EDIT (12/7/16)

1. I've found elementary but apparently mysterious answers to the first 2 questions--see my 2 answers. The arguments are similar, and I'll randomly accept the level 3 answer.

2. I think I can answer the characteristic 3 questions too. But to avoid accusations of self-abuse I'll refrain. And though I find the question on Jordan blocks very interesting (and similar questions attached to the other recursions should also be interesting), the thread has become long and so I've deleted that question.

EDIT-- I have further questions related to the initial recursion:

Let T: Z/2[x]-->Z/2[x] be the Z/2-linear map taking x^n to the c(n) of my initial recursion. T is degree lowering; in particular if q is a power of 2, T acts nilpotently on the space V(q) of polynomials of degree less than or equal to (q^2+2)/3, and its matrix decomposes into Jordan blocks. When q is at most 16, it appears that the largest block has size q+1.

Question: Is this true for all q?

1. Note that if my initial conjecture holds, then (when q>1) the number of Jordan blocks is (q^2+8)/12. It would interest me to know what the sizes of all these Jordan blocks are--does the computer give a plausible suggestion?

2. The map T can be interpreted as U_3+I acting on the space of odd mod 2 modular forms of level Gamma_0 (3), so the underlying question concerns the nilpotence order of such forms under the action of U_3+I. This is the basic motivation.

FINAL EDIT (9/5/15)

1. By combining my answer to the initial recursion with the results of my arXiv note 1508.07523 I can show the following: Let K consist of those mod 2 modular forms of level Gamma_0 (3) that are killed by the maps f-->U_2(f) and f-->f+U_3(f). When one completes the shallow Hecke algebra acting on K at the maximal ideal generated by the T_p with p> 3, one gets a power series ring in T_7 and T_13.

2. My answer to the variation on a theme question should lead to a similar result in level Gamma_0 (5), but I'd first need level 5 analogs to the results of the arXiv note, and this may not be entirely straightforward.

3. Despite the lack of response to my answers (I don't think I'd have found the answers if I hadn't posted the questions here) I find them elegant. Since no elementary answers have been found, and in view of 1. , I'll accept my first.

4. I think I can answer the characteristic 3 questions too. But to avoid accusations of self-abuse I'll refrain. And though I find the question on Jordan blocks very interesting (and similar questions attached to the other recursions should also be interesting), the thread has become long and so I've deleted that question.

EDIT-- I have further questions related to the initial recursion:

Let T: Z/2[x]-->Z/2[x] be the Z/2-linear map taking x^n to the c(n) of my initial recursion. T is degree lowering; in particular if q is a power of 2, T acts nilpotently on the space V(q) of polynomials of degree less than or equal to (q^2+2)/3, and its matrix decomposes into Jordan blocks. When q is at most 16, it appears that the largest block has size q+1.

Question: Is this true for all q?

1. Note that if my initial conjecture holds, then (when q>1) the number of Jordan blocks is (q^2+8)/12. It would interest me to know what the sizes of all these Jordan blocks are--does the computer give a plausible suggestion?

2. The map T can be interpreted as U_3+I acting on the space of odd mod 2 modular forms of level Gamma_0 (3), so the underlying question concerns the nilpotence order of such forms under the action of U_3+I. This is the basic motivation.