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Let K be a global field, and let C/K be a smooth, geometrically integral projective curve. Let X = {x1 , . . , xm } ⊂ C(K) be a finite set of points stable under Aut(K/K), and let E = v Ev ⊂ v Cv (Cv ) be a K-rational adelic set compatible with X, so each Ev is stable under Autc (Cv /Kv ) and all but finitely many Ev are X-trivial. Let S ⊂ MK be a finite set of places v, / S. containing all archimedean v, such that Ev is X-trivial for each v ∈ Assume that γ(E, X) > 1. Assume also that for each v ∈ S, there is a (possibly empty) Autc (Cv /Kv )-stable Borel subset ev ⊂ Cv (Cv ) of inner capacity 0 such that (A) If v is archimedean and Kv ∼ = C, then each point of cl(Ev )\ev is analytically accessible from the Cv (C)-interior of Ev .

47) z−ζ z−ζ . If z and ζ are not in the same component, then G(z, ζ; E) = 0. 49) γζ (E) = lim − log z→ζ z−ζ z−ζ + log(|z − ζ|) = log(2| Im(ζ)|) , 1 . 2| Im(ζ)| The Disc with Opposite Radial Arms. Take L1 , L2 ≥ 0, and let E be the union of D(0, R) with the segment [−L1 − R, R + L2 ]; thus E is a disc with opposite radial arms of length L1 , L2 . 50) γ∞ (E) = 1 4 2R + R2 + RL2 + L22 R2 + RL1 + L21 + R + L1 R + L2 . To see this, first take R = 1. Put a1 = 1 + L1 , a2 = 1 + L2 ; then E = D(0, 1)∪[−a1 , a2 ].

We will now define the notion of a compact Berkovich adelic set compatible with X. For each place v of K, let Ev ⊂ Cvan be a compact, nonpolar set disjoint from X. 2 has good reduction, the points of X specialize to distinct points in the special fibre rv (Cv ), and Ev consists of all points z ∈ Cvan whose specialization rv (z) ∈ rv (Cv ) is distinct from {rv (x1 ), . . , rv (xm )}. Equivalently, − Ev is X-trivial if it is the closure of the X-trivial set Ev = Cv (Cv )\( m i=1 B(xi , 1) ) in Cv (Cv ).

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An Explicit Approach To Elementary Number Theory by stein

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