Quadratic points on intersections of quadrics

Abstract: A projective degree $d$ variety always has a point defined over a degree $d$ field extension. For many degree $d$ varieties, this is the best possible statement, that is, there exist classes of degree $d$ varieties that never have points over extensions of degree less than $d$ (nor even over extensions whose degree is nonzero modulo $d$). However, there are some classes of degree $d$ varieties that obtain points over extensions of smaller degree, for example, degree $9$ surfaces in $\mathbb{P}^9$, and $6$-dimensional intersections of quadrics over local fields. In this talk, we explore this question for intersections of quadrics. In particular, we prove that a smooth complete intersection of two quadrics of dimension at least $2$ over a number field has index dividing $2$, i.e., that it possesses a rational $0$-cycle of degree $2$. This is joint work with Brendan Creutz.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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