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 Abstract: A hyperelliptic curve over $\mathbb Q$ is called "locally soluble" if it has a point over every completion of $\mathbb Q$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\mathbb Q$ of genus $g\geq 1$ are locally soluble but have no points over any odd degree extension of $\mathbb Q$. We also obtain a number of related results. For example, we prove that for any fixed odd integer $k > 0$, the proportion of locally soluble hyperelliptic curves over $\mathbb Q$ of genus $g$ having no points over any odd degree extension of $\mathbb Q$ of degree at most $k$ tends to 1 as $g$ tends to infinity. We also show that the failures of the Hasse principle in these cases are explained by the Brauer-Manin obstruction. Our methods involve a detailed study of the geometry of pencils of quadrics over a general field of characteristic not equal to 2, together with suitable arguments from the geometry of numbers. Publication Date: Apr-2017 Electronic Publication Date: 27-Jul-2016 Citation: Bhargava, Manjul, Gross, Benedict H, Wang, Xiaoheng. A positive proportion of locally soluble hyperelliptic curves over $\mathbb Q$ have no point over any odd degree extension, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 30 (2017), no. 2, 451-493 , DOI 10.1090/jams/863 DOI: 10.1090/jams/863 Pages: 451-493 Type of Material: Journal Article Journal/Proceeding Title: JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Version: Author's manuscript