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Sums of three cubes over a function field

Tim Browning, Jakob Glas, Victor Y. Wang

Abstract

We use a function field version of the circle method to prove that a positive proportion of elements in $\mathbb{F}_q[t]$ are representable as a sum of three cubes of minimal degree from $\mathbb{F}_q[t]$, assuming a suitable form of the Ratios Conjecture and that the characteristic is greater than 3. The analogue of this conjecture for quadratic Dirichlet $L$-functions is known for large fixed $q$, via recent developments in homological stability.

Sums of three cubes over a function field

Abstract

We use a function field version of the circle method to prove that a positive proportion of elements in are representable as a sum of three cubes of minimal degree from , assuming a suitable form of the Ratios Conjecture and that the characteristic is greater than 3. The analogue of this conjecture for quadratic Dirichlet -functions is known for large fixed , via recent developments in homological stability.
Paper Structure (26 sections, 61 theorems, 343 equations)

This paper contains 26 sections, 61 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Ratios Conjecture and applications
  4. Euler product factorisations
  5. Integral estimates
  6. Estimates for exponential sums
  7. Moments of bad exponential sums
  8. Ekedahl sieve
  9. Treatment of Sigma(1)
  10. Treatment of Sigma(2)
  11. Final touches
  12. Endgame outside dual variety
  13. The case hatC<|P|1/2-delta
  14. The case hatC>=|P|1/2-delta
  15. Contribution from the centre
  16. ...and 11 more sections

Key Result

Theorem 1.1

Suppose $\mathop{\mathrm{char}}\nolimits(\mathbb{F}_q) > 3$ and assume the Ratios Conjecture CNJ:(R2o) for $L(s,\bm{c})$, as $\bm{c}\in \mathcal{O}^6$ varies. Then each of the following sets has positive lower density in $\mathcal{O}$.

Theorems & Definitions (120)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 110 more