[Saito (1988)](https://projecteuclid.org/euclid.dmj/1077306852) 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](https://mathoverflow.net/questions/23561/nice-definition-of-discriminant-as-alluded-to-in-an-answer-of-qing-liu). And $\textrm{Art}(M/R)$ is the [Artin Conductor](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Conductor_of_an_elliptic_curve) 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](http://jstor.org/stable/2373092) 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. 



How does Saito's treatment of the conductor and discriminant reconcile with an elliptic curve?
This is a great question, but I don't think there is an easy answer. 
[Saito](https://projecteuclid.org/euclid.dmj/1077306852) 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](https://en.wikipedia.org/wiki/Different_ideal), [discriminant](https://en.wikipedia.org/wiki/Discriminant_of_an_algebraic_number_field) and 
[conductor](https://en.wikipedia.org/wiki/Artin_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](https://en.wikipedia.org/wiki/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](http://www.numdam.org/item/CM_1994__94_1_51_0) Prop. 1
(or using [Bloch](https://projecteuclid.org/euclid.dmj/1077305665) 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](https://www.ams.org/books/pspum/046.2/) (I have no access to this) 
and [here](https://projecteuclid.org/euclid.dmj/1077305665) in 1987. Then $\Delta\_{Del}$ is the Deligne discriminant, defined by 
[Deligne in a letter to Quillen](http://publications.ias.edu/sites/default/files/Lettre-a-Quillen.pdf). 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](https://projecteuclid.org/euclid.dmj/1077306852), 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](http://www.numdam.org/item/CM_1994__94_1_51_0) and by [Eriksson](https://arxiv.org/abs/1106.3096)
on conductors and discriminants. 
They treat genus 2 curves and plane curves, respectively, and they are much more accessible.


