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P. Bruin and F. Najman have determined the exceptional quadratic points on $X_0(33)$.
See Table 8 of https://arxiv.org/pdf/1406.0655.pdf
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The set of solutions to the $S$-unit equation for $k(X)$ is finite. Let me explain why. (You can "theoretically" find all solutions, as the finiteness eventually boils down to the "effective" finiteness result of de Franchis-Severi on maps of curves.)
Let $k$ be a number field, let $X$ be a smooth projective geometrically connected curve over $k$, let $S$ ...
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A flippant response is that people had the idea of using perfectoid theory in Iwasawa theory long before perfectoid theory even existed. What I'm referring to here is the work of Fontaine--Wintenberger, who studied the "field of norms" of a tower of p-adic fields, or the "tilt" as youngsters like you would call it, way back in the 1970's. Scholze's tilting ...
2
Kahn has quite a panoramic view of L-functions in arithmetic geometry. He has written a small book in french, which has been translated to english.
Bruno Kahn makes a clear distinction between zeta-functions which count points in fibers of arithmetic varieties, and L-functions which are associated to varieties "indirectly" via their motives and depend only ...
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you really shouldn't crosspost. Anyway, you've slightly misstated Hensel's lemma, you left out the assumption that $f(x)$ has a simple root in $R_v/\mathcal{M}_v$. That's where the $e$ is coming from. In general, if $f(x)$ has a root of higher multiplicity in $R_v/\mathcal{M}_v$, then you need to work in $R_v/\mathcal{M}_v^e$. So if the curve is non-singular ...
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This more of comment. If $A$ is an abelian surface with Picard rank $4$, then the associated Kummer surface $X$ has Picard rank $20$. So the answer to your question is presumably yes. To be a bit more explicit, the etale cohomology $$H^2(\bar X_{et}, \mathbb{Q}_\ell)= H^2(\bar A_{et}, \mathbb{Q}_\ell)\oplus \mathbb{Q}_\ell(-1)^{16}$$
So you'll see the same ...
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The Galois group of $L/{\mathbb Q_p}$ acts on the component group over $L$ and the component group over $\mathbb Q_p$ should be the subgroup of elements fixed by the Galois group.
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