# Weierstrass Models and Canonical Models

In the book "Algebraic Geometry and Arithmetic Curves," Liu mentions in passing that the minimal Weierstrass model of an elliptic curve plays a similar role to the canonical model of curve of (arithmetic) genus $\geq2$. As far as I can tell the meaning of this comment is their respective roles in desingularization, and my question is how much further the comparison can be drawn?

For example, one defines the Hasse-Weil zeta function as the product of zeta functions of the local minimal Weierstrass equations and, if the elliptic curve has a global minimal Weierstrass equation, this agrees with the (Hasse) zeta function when the model is viewed as a scheme of finite type over $\mathbb{Z}$. This model is related to all others through birational geometry, and so the zeta functions of all models differ by a product of finitely many zeta functions of affine lines. As the Hasse-Weil zeta function is connected to the L-function of the curve through the equation $\zeta_E(s)=\frac{\zeta_k(s)\zeta_{k}(s-1)}{L_E(s)}$, where $k$ is the base field, the passage just described links the L-function to the zeta function of any regular model.

Is a similar passage known to exist for curves $C$ of arbitrary genus? Taking the L-function to be defined as that associated to $H^1$ easily gives us the expression above for the Hasse-Weil'' zeta function of the curve, which I take to be defined as the alternating product of the L-functions of cohomology groups (maybe there is a better definition?), the question is, does the zeta function of the canonical model agree with this Hasse-Weil zeta?

• Let $R$ be a strictly henselian dvr with fraction field $K$ and residue field $k$, and $X$ a proper flat $R$-scheme with generic fiber $X_K$ geometrically connected and smooth of dimension 1 and positive genus. Assume $X$ is the minimal regular proper model of $X_K$. Choose a prime $\ell \in R^{\times}$. For $G_K := {\rm{Gal}}(K_s/K)$, you want $H^1(X_k, \mathbf{Q}_{\ell}) \rightarrow H^1(X_{K_s},\mathbf{Q}_{\ell})^{G_K}$ to be an isomorphism. This problem can be "localized" via vanishing cycles, and in general is subtle (but is tractable when $X_k$ is semistable). Let's see what Q. Liu says. – user28172 Feb 13 '13 at 5:46
• @TomPGR:The canonical model in higher genus is obtained by contracting some $(-2)$-rational curves, and all such configurations of rational curves is classified by M. Artin (see my book, 10.1.53). They all appear in Kodaira-Néron's classification for elliptic curves. So the computation of zeta function is the same as for elliptic curves. – Qing Liu Feb 15 '13 at 17:04
• @nosr: the isomorphism is explained in S. Bloch: "de Rham cohomology and conductors of curves", Duke Math. J. 54 (1987), Lemma 1.2(i) when the model $X$ is regular. If we contract $X$ to the canonical model, this does not change the generic fiber, and the $H^1$ of the closed fiber doesn't change either (by explicit computations, e.g. Dolgacev: "On the purity of the degeneration loci of families of curves". Invent. Math. 8 (1969), Prop. 2.4). This can also be explained (at least when the curve has a rational point) by the fact that the neutral component of the Néron model of $Jac(X_{K})$... – Qing Liu Feb 15 '13 at 17:10
• continued: does not change when contracting $X$ to the canonical model $W$ (the latter has only rational singularities), this can be found in "Néron models", Theorem 9.7/1. – Qing Liu Feb 15 '13 at 17:14
• Thank you both, this is very helpful. I have some reading to do! – TomPGR Feb 21 '13 at 10:13