Develops Kloosterman refinement for F_q(t) and uses it to establish quantitative arithmetic for rational points on smooth complete intersections of two quadrics in P^{n-1} for n>=9 and q odd.
Birch's theorem in function fields
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abstract
We establish an aysmptotic formula for the number of points with coordinates in $\mb{F}_q[t]$ on a complete intersection of degree $d$ defined over $\mb{F}_q[t]$, with explicit error term, provided that the characteristic of $\mb{F}_q$ is greater than $d$, the codimension of the singular locus of the complete intersection is large enough, and this intersection has a non-singular point at each place of $\mb{F}_q[t]$. In particular, when this complete intersection is non-singular, we show that it satisfies weak approximation.
fields
math.NT 1years
2019 1verdicts
UNVERDICTED 1representative citing papers
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Rational points on complete intersections over $\mathbb{F}_q(t)$
Develops Kloosterman refinement for F_q(t) and uses it to establish quantitative arithmetic for rational points on smooth complete intersections of two quadrics in P^{n-1} for n>=9 and q odd.