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Computation of classical and $v$-adic $L$-series of $t$-motives

Xavier Caruso, Quentin Gazda

Abstract

We design an algorithm for computing the $L$-series associated to an Anderson $t$-motives, exhibiting quasilinear complexity with respect to the target precision. Based on experiments, we conjecture that the order of vanishing at $T=1$ of the $v$-adic $L$-series of a given Anderson $t$-motive with good reduction does not depend on the finite place $v$.

Computation of classical and $v$-adic $L$-series of $t$-motives

Abstract

We design an algorithm for computing the -series associated to an Anderson -motives, exhibiting quasilinear complexity with respect to the target precision. Based on experiments, we conjecture that the order of vanishing at of the -adic -series of a given Anderson -motive with good reduction does not depend on the finite place .
Paper Structure (27 sections, 25 theorems, 72 equations, 4 figures)

This paper contains 27 sections, 25 theorems, 72 equations, 4 figures.

Table of Contents

  1. $t$-motives and their $L$-series
  2. $t$-motives
  3. A formula for the local $L$-factor
  4. Maximal models
  5. Frobenius spaces
  6. Local $L$-factors of Frobenius spaces
  7. Proof of Theorem \ref{['thm:another-formula-local']}
  8. Anderson's trace formula
  9. Global $L$-series
  10. Algorithm for computing the $L$-series
  11. Determination of the maximal model
  12. Construction of the maximal model
  13. Step 1.
  14. Step 2.
  15. Step 3.
  16. ...and 12 more sections

Key Result

Theorem 1

There exists an algorithm taking as input and outputs the $L$-series $L_v(\underline M, T)$ at precision $v^{\operatorname{prec}}$ for a cost of operations in $\mathbb{F}\xspace$ where $d_\theta$ is the maximal $\theta$-degree of an entry of the matrix of $\tau_M$ and $\Omega$ denotes a feasible exponent for the computation of the characteristic polynomial over an abstract ring.

Figures (4)

  • Figure 1: $t$-adic valuation of the ten first coefficients of $L_t(M; T)$ ($q=3$)
  • Figure 2: Orders of vanishing for the $t$-motive defined by Eq. \ref{['eq:example']} ($q=2$)
  • Figure 3: Algorithm for computing $L$-series of a $t$-motive
  • Figure 4: Timings for the computation of the $L$-series with $q = d_\theta = 9$.

Theorems & Definitions (53)

  • Remark
  • Theorem 1: cf Theorem \ref{['thm:complexity']}
  • Theorem 2
  • Conjecture 3
  • Definition 1.1.1
  • Definition 1.1.2
  • Remark 1.1.3
  • Definition 1.2.1
  • Theorem 1.2.2
  • Remark 1.2.3
  • ...and 43 more