We formulate a tropical analogue of Grothendieck’s section conjecture: for all stable graphs G of genus g>2 and for all fields k, a generic curve with reduced form G over k satisfies the section conjecture. . We will prove this conjecture in many cases. In doing so, we generate many examples of curves satisfying the section conjecture over the field of geometric interest, and then over the p-adic and number fields via the Chebotarev argument. Construct two Galois cohomology classes o_1 and o_2 . These prevent the existence of the pi_1 section and thus the existence of rational points. The first is the Abelian disorder, which is closely related to the period of the curve and the cohomology class on the moduli space of the curve M_g studied by Morita. The second is a nilpotent failure of 2 and looks new. We study the degeneracy of these classes using topological methods and create examples of surface bundles on surfaces where these classes interfere with sections. These constructs are then used to generate curves over p-adic and numeric fields where each class disturbs the pi_1 section and thus the rational points. Among the geometric results are a new proof of the section conjecture for general curves of genus g>2 and a proof of the section conjecture for general curves of even genus with rational divisor class of degree 1 (where , the existence of the section is purely non-abelian).
