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Saito (1988) gives a proof that $$\textrm{Art}(M/R) = \nu(\Delta)$$ Here, $M$ is the minimal regular projective model of a projective smooth and geometrically connected curve $C$ of positive genus over the field of fractions of $R$, a ring with perfect residue field. The $\Delta$ here is the "discriminant" of $M$ which measures defects in a functorial isomorphism described in this answer. And $\textrm{Art}(M/R)$ is the Artin Conductor.

I know at some point, somewhere in the literature, someone took this result and applied it to elliptic curves and realized that when $C$ is an elliptic curve, $\Delta$ above represents the discriminant of the minimal Weierstrass equation of the elliptic curve and that $\textrm{Art}(M/R)$ is the standard global conductor of the elliptic curve.

So my question is: how exactly that is done? How does one take the definition of the discriminant and conductor in Saito's paper and in the linked answer and prove that when looking at an elliptic curve, they are the regular discriminants and conductors?

Or am I really confused?

EDIT: Watson Ladd has provided an amazing paper by Ogg that attempts a case-by-case proof of the result specialized to an elliptic curve. And upon researching, I've found that many people claim Saito (1988)'s general result proved Ogg's result "fully". So I guess my question can also be restated as how you get from Saito's result to Ogg's formula.

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  • $\begingroup$ I don't have access to the Saito text, so in all probability, Saito himself does this. But I would appreciate an answer that doesn't just point me to another article as our high school only gives us jstor. $\endgroup$
    – Nico A
    Feb 9, 2018 at 12:28
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    $\begingroup$ Somewhat tangentially, the issue of people not having free access to research is a major current news topic. This article discusses it in detail theverge.com/2018/2/8/16985666/… $\endgroup$ Feb 9, 2018 at 22:03
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    $\begingroup$ You're in luck: jstor.org/stable/2373092 is the result specialized to an elliptic curve. $\endgroup$ Feb 9, 2018 at 22:20
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    $\begingroup$ @FelipeVoloch Oh man, tell me about it. As a high school student, trying to get access to a few papers related to a cool problem can cost upwards of $300 USD. And often, you don't know what the paper is about until you pay for it. It's like finding a needle in a haystack when you have to pay for every piece of hay you disturb. $\endgroup$
    – Nico A
    Feb 10, 2018 at 3:06
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    $\begingroup$ It is also not too hard to get most papers you want by "alternative avenues". Saito's paper is available, for instance. $\endgroup$
    – Asvin
    Feb 12, 2018 at 3:18

1 Answer 1

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This is a great question, but I don't think there is an easy answer. Saito himself proves on p.156 (Cor. 2) that his results imply Ogg's formula, including the missing case of 2-adic fields. However, the proofs are quite condensed and the underlying technology very advanced.

Relative dimension 0. Saito's approach is motivated by the classical relation between the different, discriminant and conductor for number fields or local fields. Say $K/{\mathbb Q}_p$ is finite, and $$ f: X=\textrm{Spec }{\mathcal O}_K \longrightarrow \textrm{Spec }{\mathbb Z}_p=S. $$ Classically, there is the different (ideal upstairs) $$ \delta = \{\,x\in K\>|\>\textrm{Tr}(x{\mathcal O}_K)\subset{\mathbb Z}_p\,\}^{-1} \>\>\subset {\mathcal O}_K, $$ the discriminant (ideal downstairs) $$ \Delta = (\det \textrm{Tr} (x_i x_j)_{ij})\subset {\mathbb Z}_p, \quad\qquad x_1,...,x_n\textrm{ any }{\mathbb Z}_p\textrm{-basis of }{\mathcal O}_K $$ and the Artin conductor ${\mathfrak f}({\mathcal O}_K/{\mathbb Z}_p)$ (also an ideal downstairs). The conductor-discriminant formula in this case says $$ \textrm{order }(\Delta) = \textrm{order }\textrm{Norm}_{K/{\mathbb Q}_p}(\delta) = \textrm{order }{\mathfrak f}({\mathcal O}_K/{\mathbb Z}_p), $$ where order is just the valuation: order$(p^n{\mathbb Z}_p)=n$.

Saito interprets these sheaf-theoretically as follows: $\Delta$ is a homomorphism of invertible ${\mathcal O}_S$-modules $$ \Delta: (\textrm{det }f_*{\mathcal O}_X)^{\otimes 2} \rightarrow {\mathcal O}_S, \qquad (x_1\wedge...\wedge x_n)\otimes (y_1\wedge...\wedge y_n) \mapsto \det(\mathrm{Tr}_{X/S}(x_iy_j)), $$ which is an isomorphism on the generic fibre; the classical discriminant is its order = length of the cokernel on the special fibre. Then, $-\textrm{order }{\mathfrak f}({\mathcal O}_K/{\mathbb Z}_p)=\textrm{Art}(X/S)$; this can be defined for any relative scheme over $S$ in terms of l-adic etale cohomology, $$ \textrm{Art}(X/S) = \chi_{et}(\textrm{generic fibre}) - \chi_{et}(\textrm{special fibre}) - (\textrm{Swan conductor}). $$ Finally, the different happens to be the localised first Chern class $$ \delta = c_1(\Omega_{X/S}) = c_1({\mathcal O}_X\to \omega_{X/S}). $$

Relative dimension 1. Now we move to one dimension higher, so that $S$ is the same, but $X$ is now a regular model of, say, an elliptic curve $E/{\mathbb Q}_p$. The conductor $\textrm{Art}(X/S)$ is still defined, and it is essentially the conductor of $E$, except that $\chi_{et}(\textrm{special fibre})$ has an $H^2$-contribution from the irreducible components of the special fibre. To be precise, as explained in Liu Prop. 1 (or using Bloch Lemma 1.2(i)), $$ -\textrm{Art}(X/S) = n + f - 1, $$ where $f$ is the the classical conductor exponent of an elliptic curve, and $n$ is the number of components of the special fibre of the regular model $X$. So, Ogg's formula in this language reads $$ -\textrm{Art}(X/S) = \textrm{ord }\Delta_{min}, $$ where $\Delta_{min}$ is the discriminant of the minimal Weierstrass model. So Ogg's formula is like "conductor=discriminant" formula in the number field setting, and Saito proves it through "conductor=different=discriminant". To be precise, there are three equalities $$ -\textrm{Art}(X/S) = -\deg c_1(\Omega^1_{X/S}) = \textrm{ord }\Delta_{Del} = \textrm{ord }\Delta_{min} $$ The first one, "conductor=different" was done by Bloch here (I have no access to this) and here in 1987. Then $\Delta_{Del}$ is the Deligne discriminant, defined by Deligne in a letter to Quillen. Analogously to the sheaf-theoretic interpretation of the discriminant in the relative dimension 0 case, Deligne constructs a canonical map $$ \Delta: \det(Rf_*(\omega_{X/S}^{\otimes 2})) \rightarrow \det(Rf_*(\omega_{X/S}))^{\otimes 13}). $$ It is again an isomorphism on the generic fibre, and Saito calls the (Deligne) discriminant the order of this map on the special fibre. The second equality, "different=discriminant", is the main result of Saito's paper, and it is very technical. And finally, Saito on pp.155-156 proves that for elliptic curves, $\textrm{ord }\Delta_{Del} = \textrm{ord }\Delta_{min}$ (third equality), using properties of minimal Weierstrass models (at most one singular point), Neron models and existence of a section for models of elliptic curves.

Personal note. It would be amazing if someone deciphered Saito's proof and wrote it down in elementary terms. I don't think such a treatment exists, though there are very nice papers by Liu and by Eriksson on conductors and discriminants. They treat genus 2 curves and plane curves, respectively, and they are much more accessible.

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  • $\begingroup$ This clears up soooo much confusion! I'm surprised most literature doesn't touch on this background machinery - it might be a little involved but now that I understand it, it gives a ton of intuition for the objects I've been studying. Seriously, thank you so much. I'd give you way more than 50 bounty if I could afford it. $\endgroup$
    – Nico A
    Feb 13, 2018 at 15:51
  • $\begingroup$ I can follow your answer, but I'm not all that familiar with the technicalities of sheaf theory, are there any resources you reccomend? $\endgroup$
    – Nico A
    Feb 13, 2018 at 15:59
  • $\begingroup$ A good starting place to see the analogy of number field language and function field language (which is essentially the dim. 0 part of the answer) is chapter 3 of Neukirch's book "Algebraic number theory". It seems to me like going from dim. 0 to dim. 1 is not at all obvious tho, regardless I don't know a source for it. $\endgroup$ Feb 13, 2018 at 16:23
  • $\begingroup$ @TreFox: You are very welcome! This question interested me for a long time, and still does. Neukirch is indeed a good place to start for the analogy. But regarding sheaf theory, and the 1-dim case, I don't know. Hartshorne's Algebraic Geometry, Ch. 2 (2.1-2.5) sets up schemes, sheaves and sheaves of modules, and that's how Saito phrases everything. But to decode the actual results... I have a feeling that one should start with Bloch's paper, rather than Saito's paper, maybe I'll add that to the answer. $\endgroup$ Feb 18, 2018 at 9:30

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