It is my understanding that if I twist a hyperelliptic curve of genus 2 whose Jacobian has conductor $N$ by a prime $p$ with $p\nmid N$, that the conductor of the Jacobian of the twist is expected to be $Np^4$, but I am unable to find a reference for this. Is anyone aware of a proof of this "fact"?
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2$\begingroup$ If the definition of the conductor is equal to the conductor of the Jacobian of the curve, then this is true because you can compute the conductor of the ell-adic Galois representation instead, and you have four tame characters each of which contributes $p^1$. $\endgroup$– ericOct 20, 2015 at 19:41
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$\begingroup$ That makes sense, but I'd still like a reference. $\endgroup$– Johnson-LeungOct 21, 2015 at 1:20
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2$\begingroup$ @eric There are curves of genus 2 (for example) that have bad reduction somewhere but whose Jacobian has good reduction, so the conductor of the curve is not the same as of its Jacobian, depending on the definition. $\endgroup$– Felipe VolochOct 21, 2015 at 1:48
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1$\begingroup$ In which case I don't know the definition we're using in this question. $\endgroup$– ericOct 21, 2015 at 20:32
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1$\begingroup$ (Edited) If the curve has good reduction then the Jacobian has good reduction. The standard reference for this is SGA7. Now you need that the ell-adic representation is unramified; this is one direction of Neron-Ogg-Shaferevich, and a reference is the Serre-Tate Annals paper. A quadratic twist is tamely ramified (for $p$ odd -- when $p=2$ the result you want is probably false in fact), and for this you can refer to the Artin-Tate notes on class field theory. $\endgroup$– ericOct 22, 2015 at 10:48
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(This answer is Community Wiki and is extracted from comments by user eric, who has declined to post them as an answer. The CW is to invite others to contribute, especially to provide suitable references for these facts.)
If the curve has good reduction, then the Jacobian has good reduction. The standard reference for this is SGA 7. Now you need that the $l$-adic representation is unramified; this is one direction of Neron-Ogg-Shaferevich. (One reference for that off the top of my head is the Serre-Tate Annals paper, although note that it's only the easy direction of NOS we use here.) Meanwhile, a quadratic twist is tamely ramified (this is for $p$ odd -- when $p=2$ the result you want is probably false in fact). If you need a reference for this, you can try the Artin-Tate notes on class field theory, but much like the previous reference, I'd regard this as overkill for what you are asking for.
In brief: if you have an unramified representation of dimension $d$ and you twist it by a tamely ramified character, the resulting representation has conductor $p^d$. I don't really know what would be a great reference for this specific observation, but it's not a difficult calculation.
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$\begingroup$ (Apologies: I forgot to make this CW until just now.) $\endgroup$– Todd Trimble ♦Oct 23, 2015 at 12:58