By E. Ramirez De Arellano
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Additional info for Algebraic Geometry and Complex Analysis
T/. d/k dominating Ei . d/ C Eio ! Eio is a Galois cover of degree di . Eio / j where the first equality follows from the additivity of the Euler characteristic and the second from Hurwitz’ theorem. t/. t/ 7! 5. 1). 2 We say that Ei is a principal component of Ck if the genus of Ei is non-zero or Ei n Eiı contains at least three points. We denote by Iprin Â I the subset corresponding to principal components of Ck . C/ the stabilization index of C. C/ D 1. C/ is well-defined. C / depends on the model C and not only on C.
This is a quadratic Artin-Schreier extension of K. 3/ be the unique tame extension of K 0 in K s of degree 3. Then it is easy to check that C K L has good reduction and that L is the minimal extension of K with this property, so that ŒL W K D 6. t/ if C is tamely ramified. We will now show that this recipe is also valid in the wild case. 6. 8 Assume that g ¤ 1 or that C is an elliptic curve. t/. Proof Assume that C is a relatively minimal sncd-model of C. t/. Eo / < 0 is principal. Thus it suffices to show that N divides e if N > 1 is the multiplicity of a principal component in Ck .
4] the following question: if Cmin is a relatively minimal sncd-model of C and E is a principal component of its special fiber, is it true that the p-part of the multiplicity of E divides ŒLWK? 2], but, to our best knowledge, the question is still wide open. C/ in the wildly ramified case. k/. Let C be the elliptic K-curve with Weierstrass equation y2 D x3 C 2: It is easily computed, using Tate’s algorithm, that this equation is minimal, and that C has reduction type p II over R and acquires good reduction over the wild Kummer extension L D K.
Algebraic Geometry and Complex Analysis by E. Ramirez De Arellano