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I am reading the article by Lawrence and Venkatesh on diophantine problems and $p-$adic period mappings. At page $35$ they say that the dimension of the Prym variety of an (unramified) cover of curves $C_1' \to C_2$ of degree $q$ is

$$(2g-1) \cdot \frac{q-1}{2},$$

where $g$ is the genus of $C_2$. The Prym variety is defined as $$\text{coker}(\text{Pic}^0(C_2) \to \text{Pic}^0(C_1')).$$

Riemann-Hurwitz tells us that the genus of $C_1'$ is

$$g' = q(g-1)+1.$$ Since the cover is surjective, I expected the map of Jacobians to be injective (is this true?). If that is the case, the dimension of the cokernel should be just the difference of the dimensions of the Jacobians, which are just the genera of the curves:

$$ g'-g=(q-1)(g-1).$$

This is off by $\frac{q-1}{2}$ with respect to the correct dimension. What am I doing wrong?

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  • $\begingroup$ Check the ramification index $e_P$ at the (only) point of ramification $\endgroup$
    – Stanley Yao Xiao
    Commented Jun 28, 2019 at 16:39
  • $\begingroup$ @StanleyYaoXiao Thanks for the remark! But why can I only have one element in the ramified fiber? Is it because $q$ is prime? But the cover $C_1' \to C_2$ is not Galois, even away from the ramification... $\endgroup$
    – 57Jimmy
    Commented Jun 28, 2019 at 16:51
  • $\begingroup$ I don't think that's the reason... they constructed the covers to have only one point of ramification. This can always be done, by the work of Fulton (see: jstor.org/stable/1970748?seq=1#page_scan_tab_contents) $\endgroup$
    – Stanley Yao Xiao
    Commented Jun 28, 2019 at 16:54
  • $\begingroup$ @StanleyYaoXiao Thanks for the explanation, but it's still not completely clear to me: by Riemann-Hurwitz, the number of points in the ramified fiber determines the genus, so what do you mean by "this can always be done"? Here we consider all Aff(q)-covers branched at a single point, so also those that have more than one point in the fiber, and then take a non-Galois subcover of order $q$ by quotienting $\mathbb{F}_q^*$, right? $\endgroup$
    – 57Jimmy
    Commented Jun 28, 2019 at 17:03

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It's rather a comment that an answer, which requires me more reputation than I own. See the discussion about the fundamental group above Theorem 5.4, we see that a loop around $y_0$ is in the commutator subgroup, so is its image in $Aff(q)$. But $[Aff(q),Aff(q)]=F_q$. See the three possibilities above Lemma 8.2 and we are in the second case. Now read the second paragraph of Section 8.2.2. The monodromy around $y$ is a $q$-cycle acting on a fiber $Z_{y'}$ for $y'\in Y-y$ (the base point of a loop around $y$), so the ramification locus of the brached cover $Z\to Y$ is a singleton. Now apply Riemann-Hurwitz formula to get the genus of $Z$, $gq-\frac{q-1}{2}$ and use your idea to compute the dimension of the cokernel, i.e. the Prym variety. (Told by Mme.Cadoret)

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