Poster presentation at Ramification in geometric Langlands and non-abelian Hodge theory

Here is the list of families of pairs $(d,e)$ potentially giving finite monodromy for $\mathcal{F}_{x^d(x-1)^e}$, that is, such that $d$, $e$ and $d+e$ are FM-exponents.

Theorem. Let $d,e$ be positive integers such that $d$ is prime to $p$ and all of $d$, $e$ and $d+e$ are FM-exponents. Then $(d,e)$ is one of the following pairs (or their reversed pairs):

1. $(\frac{p^a+1}{p^b+1},p^b\frac{p^a+1}{p^b+1})$ for $a\geq1$ an odd multiple of $b$.
2. $(\frac{p^{a+2b}+1}{p^b+1},p^b\frac{p^a+1}{p^b+1})$ for $a\geq1$ an odd multiple of $b$.
3. $p\geq3$, $(\frac{p^a+1}{2},\frac{p^a+1}{2})$ for $a\geq0$.
4. $p=2$, $(3,10)$, $(5,6)$, $(5,8)$, $(5,17)$, $(5,52)$, $(9,2)$, $(9,4)$, $(9,11)$, $(9,13)$, $(9,17)$, $(9,34)$, $(9,43)$, $(9,48)$, $(11,2)$, $(11,13)$, $(11,57)$, $(13,44)$, $(13,228)$, $(17,26)$, $(17,40)$, $(33,10)$, $(33,24)$, $(33,172)$, $(33,208)$, $(65,176)$, $(171,34)$ or $(205,36)$.
5. $p=2$, $(2^a+1,2^a+1)$ for $a\geq1$.
6. $p=2$, $(\frac{2^a+1}{2^b+1},\frac{2^a+1}{2^b+1})$ for $b\geq1$, $a$ an odd multiple of $b$.
7. $p=2$, $(1,2^a+1)$ for $a\geq1$.
8. $p=2$, $(2^a+1,2^a)$ for $a\geq1$.
9. $p=2$, $(3,2^a+1)$ for $a\geq1$.
10. $p=2$, $(2^a+1,3\cdot2^a)$ for $a\geq1$.
11. $p=2$, $(2^a+1,\frac{2^{3a}+1}{2^a+1})$ for $a\geq1$.
12. $p=2$, $(\frac{2^{3a}+1}{2^a+1},2^a(2^a+1))$ for $a\geq1$.
13. $p=2$, $(2^a+1,\frac{2^{a}+1}{3})$ for odd $a\geq1$.
14. $p=2$, $(1,\frac{2^{a}+1}{3})$ for odd $a\geq1$.
15. $p=2$, $(\frac{2^{a}+1}{3},2^a)$ for odd $a\geq1$.
16. $p=2$, $(3,\frac{2^{2a}+1}{5})$ for odd $a\geq1$.
17. $p=2$, $(\frac{2^{2a}+1}{5},2^{2a}\cdot3)$ for odd $a\geq1$.
18. $p=2$, $(5,\frac{2^{a}+1}{3})$ for odd $a\geq1$.
19. $p=2$, $(\frac{2^{a}+1}{3},5\cdot2^a)$ for odd $a\geq1$.
20. $p=2$, $(\frac{2^{3a}+1}{9},\frac{2^{3a}+1}{3})$ for odd $a\geq1$.
21. $p=2$, $(\frac{2^{3a}+1}{9},2\frac{2^{3a}+1}{9})$ for odd $a\geq1$.
22. $p=3$, $(2,5)$, $(4,3)$, $(4,10)$, $(5,7)$, $(10,63)$, $(28,45)$ or $(61,12)$.
23. $p=3$, $(2,3^a+1)$ for $a\geq0$.
24. $p=3$, $(3^a+1,2\cdot3^a)$ for $a\geq0$.
25. $p=3$, $(3^a+1,\frac{3^a+1}{2})$ for $a\geq0$.
26. $p=3$, $(1,\frac{3^a+1}{2})$ for $a\geq0$.
27. $p=3$, $(\frac{3^a+1}{2},3^a)$ for $a\geq0$.
28. $p=3$, $(4,\frac{3^a+1}{2})$ for $a\geq0$.
29. $p=3$, $(\frac{3^a+1}{2},4\cdot3^a)$ for $a\geq0$.
30. $p=3$, $(\frac{3^a+1}{2},\frac{3^a+1}{4})$ for odd $a\geq1$.
31. $p=3$, $(\frac{3^a+1}{4},\frac{3^a+1}{4})$ for odd $a\geq1$.
32. $p=3$, $(2,\frac{3^a+1}{4})$ for odd $a\geq1$.
33. $p=3$, $(\frac{3^a+1}{4},2\cdot3^a)$ for odd $a\geq1$.
34. $p=5$, $(1,6)$, $(2,5)$, $(3,7)$, $(6,7)$ or $(6,15)$.
35. $p=5$, $(2,\frac{5^a+1}{2})$ for $a\geq0$.
36. $p=5$, $(\frac{5^a+1}{2},2\cdot5^a)$ for $a\geq0$.
37. $p=7$, $(2,2)$.