A new and improved version of the circle method over function fields

Abstract: Birch's theorem is a classical result about counting rational points on smooth hypersurfaces with many variables; its function field analogue counts rational curves on them. I'll explain a new variant of the circle method that does the same (and works just as well!) for maps from any projective variety $Y$ to a smooth hypersurface $X$. There are many arithmetic and geometric applications: Over finite fields, this gives an higher dimensional analogue of Birch's theorem for $X$ over function fields. Over the complex numbers, this proves that the moduli spaces of maps $Y$ to $X$ are irreducible of the expected dimension. The key input is to stratify the "circle" differently by an unusual invariant called the multiplication rank and a degeneration result from the theory of Gröbner bases. Many parts of this are joint with Jakob Glas.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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