By B.H. Gross, B. Mazur
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Extra info for Arithmetic of elliptic curves with complex multiplication
The 2-torus is the quotient of the plane by the fundamental group π1 (T ) which is isomorphic to H1 (T ) as it is abelian. The action is by deck transformations. Given a point x on the plane and a homology class λ we ﬁnd a curve on the torus with this homology class which passes through the projection of x. x to be the end point. This depends only on the homology class, and not on the choice of curve. We note that in particular, the action of [η] ∈ H1 (Q) on an arrow ηn in is the length of one a representative zig-zag ﬂow η, is the arrow ηn+ , where period of η.
If they did, then there is an arrow a ∈ Q which is (without loss of generality) a zig of η and a zag of η . This projects down to an arrow which is both a zig and a zag of η. Thus η intersects itself and the corresponding train track is not a simple closed curve. This contradicts (1), and so η and η do not intersect. This proves part of condition (c). Now we prove that (a) holds. A zig-zag ﬂow can intersect itself in two ways. If an arrow a occurs as a zig and a zag in η, then η projects down to a zig-zag path η which contains the image of a as a zig and a zag.
First we prove that P (σ) evaluates to zero on the two systems of boundary paths. Let η be a representative zig-zag path of γ + or γ − . Since P (σ) is a perfect matching we know that it is non-zero on precisely one arrow in the boundary of every face. By construction, the boundary path of a zig-zag path was pieced together from paths back around faces which have a zig-zag or zag-zig pair in the boundary. 23, P (σ) is non-zero on either the zig or zag of η in the boundary of each of these faces and therefore it is zero on all the arrows in the boundary path.
Arithmetic of elliptic curves with complex multiplication by B.H. Gross, B. Mazur