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

Xavier Caruso (1, 2) , Quentin Gazda (3)

1 IMB - Institut de Mathématiques de Bordeaux
2 CANARI - Analyse cryptographique et arithmétique
3 CMLS - Centre de Mathématiques Laurent Schwartz

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 $v$.

Keywords: Anderson motives; L-series; algorithms

Domains: Number Theory [math.NT]; Symbolic Computation [cs.SC]

Xavier Caruso, Quentin Gazda. Computation of classical and $v$-adic $L$-series of $t$-motives. 2024. ⟨hal-04410981⟩