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arXiv · 2412.08469

Semi-topological Galois cohomology and Weierstrass realizability

Jyh-Haur Teh
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Abstract

Semi-topological Galois theory associates a canonical finite splitting covering to a monic Weierstrass polynomial. The inverse limit of the corresponding deck groups defines the absolute semi-topological Galois group, $\PiST(X,x)$. This paper develops a cohomology theory for $\PiST(X,x)$ with discrete torsion coefficients, establishing its fundamental properties and canonical comparison maps to singular cohomology. A Lyndon-Hochschild-Serre spectral sequence is used to yield an obstruction theory for semi-topological embedding problems. We prove several structural and vanishing results, including ST-fullness for free fundamental groups and triviality for finite fundamental groups. As applications, we provide a criterion for lifting finite projective monodromy to linear monodromy, formulate the $π_1$-detectable Weierstrass realizability conjecture for divisor classes and show that this conjecture is true for abelian varieties, smooth complex projective curves and ruled surfaces over positive-genus curves.

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Jyh-Haur Teh. 2026-03-04. Semi-topological Galois cohomology and Weierstrass realizability. https://arxiv.org/abs/2412.08469

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