As for the second question, I'm not familiar with the particulars of the NC class, but given c you can distribute the multiplication of the matrices $\begin{pmatrix} 1 & a_i \\ & 1 \end{pmatrix}$: use $n/2$ processors to multiply consecutive pairs, then $n/4$ processors to multiple consecutive pairs from the previous stage... until you've multiplied them all. There are $O(\log{n})$ stages, and this produces the $p$ (and $q$, even though you had it as input). Is this algorithm in NC? What I'm worried about is that the sizes of the integers in the matrices grow until it they reach $n$-bits...

There is a relevant paper by Sedjelmaci, The Integer Greatest Common Divisor is NC (2007). In the conclusion section they mention that this implies that modular inversion is also in NC. So I'm guessing running a full extended Euclidean algorithm is also in NC, which would mean continued fractions are there too. Check out the paper and see if it gives you a direction.

Also, when dealing with the classification of representations of a finite reductive group the dual is taken over the same base field. See for example Digne and Michel's book.

When taking the log term out of the second integral, do you incur a dependence on $a$ in the implied constant?

As for the bad primes, I think they are exactly the primes dividing the number $Res_x(Res_t(f,\frac{\partial f}{\partial x}), Res_t(f,\frac{\partial f}{\partial t}))$.

Nice! Thanks for putting lots of details in the computation. Good luck with your question, even if this direction is a dead end :)

Take $a=1+E_{12}+E_{34}+E_{56}$, $b=1+E_{23}+E_{45}+E_{67}\in Mat_{7\times7}(\mathbb{F}_2)$, where $E_{ij}$ has a unique nonzero element at row $i$ and column $j$ equal to 1. Set $c=ab$. Let $V$ be $\overline{\mathbb{F}}_2^7$, and let $D_{16}\cong<x, y|x^2=y^8=xyxy=1>$ act on $V$ by sending $x$ to $a$ and $y$ to $c$. With this action $V$ is an indecomposable module for $D_{16}$. Let $\overline{D_{16}}$ be the Zariski closure of $<a, c>$ in $GL(V)$, which is unipotent. Does it satisfy $\overline{D_{16}}(\mathbb{F}_2)=<a,c>\cong D_{16}$?

I think what you're looking for is called Deligne-Lustzig theory, which, with the work of many people over a few decades, culminates in a classification of the irreducible representations of finite reductive groups. Carter's Finite groups of Lie type: Conjugacy classes and complex characters is an excellent introduction textbook that covers much ground. Digne and Michel's Representations of Finite Groups of Lie Type is also highly recommended. There are also many related question on MO, such as this one: mathoverflow.net/questions/127691/reconciling-lusztigs

Given that this is a duplicate of a question six years earlier, how is this not closed?

The corresponding integral module is isomorphic to copies of the integers, a copy per prime. For each permutation of the involved primes you get a different decomposition of the matrix, and the $U_n$ are simply permutation matrices. These have cyclotomic polynomials as factors of their characteristic polynomials.

If you allow $x,y$ to be rational then the parity conjecture for elliptic curves easily implies a positive proportion of primes are representable (as long as there are no simple local obstructions, like $gcd(A, B)\ne 1$). But this doesn't help much with integer solutions...

Surely that O should be a U.