Exceptionality and Serre's Open Image Theorem: Davenport and
Lewis were deciding when the error estimate for the number of points on
a curve over a finite field would accumulate significantly. So, they
took the curve <span style="font-style: italic;">w</span><sup>2</sup>=<span
style="font-style: italic;">f</span>(<span style="font-style: italic;">u</span>)-<span
style="font-style: italic;">x</span> – call its non-singular
projective model <span style="font-style: italic;">X<sub>x</sub> – <small></small></span>with
<span style="font-style: italic;">x</span> a parameter, and considered
the sum of squares of the number of points minus<span
style="font-style: italic;"> p</span>+1 on <span
style="font-style: italic;">X<sub>x</sub></span>, summed over <span
style="font-style: italic;">x</span>∈<span style="font-weight: bold;">Z</span>/<span
style="font-style: italic;">p</span>.<br>
<br>
"Significant error" meant that sum would be bounded away from 0 by a
nonzero constant times <i>p</i> (for infinitely many <i>p</i>). Katz's result
connects it to the monodromy statement (one irreducible component to
the completely reducible action). Mine was to characterize for which <i>f</i>
that happened. My paper above (also said in my math reviews) says the
connections with other topics deepens when you extend "polynomial" <i>f</i>(<i>u</i>)
to rational function. The connection is to alternative forms of Serre's
Open Image Theorem, a topic that first appeared in [Fr78, &sect;3]<br>