Counting points on curves using a map to \(\mathbf P^1\). II.

DOI10.1016/j.ffa.2016.12.008zbMath1402.11097arXiv1412.7217OpenAlexW2572077490MaRDI QIDQ2396763

Jan Tuitman

Publication date: 24 May 2017

Published in: Finite Fields and their Applications (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1412.7217

zbMATH Keywords

\(p\)-adic cohomologyalgebraic curves over finite fieldszeta functionscomputational number theory

Mathematics Subject Classification ID

Number-theoretic algorithms; complexity (11Y16) Curves over finite and local fields (11G20) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) (14G10) Zeta and (L)-functions in characteristic (p) (11M38)

Related Items (16)

Computing zeta functions of arithmetic schemes ⋮ Explicit Vologodsky integration for hyperelliptic curves ⋮ Quadratic Chabauty for modular curves: algorithms and examples ⋮ Improved complexity bounds for counting points on hyperelliptic curves ⋮ Rational points on \(X_0^+(125)\) ⋮ On the enumeration of irreducible polynomials over \(\mathrm{GF}(q)\) with prescribed coefficients ⋮ Computing zeta functions of cyclic covers in large characteristic ⋮ Explicit Coleman integration in larger characteristic ⋮ Zeta functions of nondegenerate hypersurfaces in toric varieties via controlled reduction in \(p\)-adic cohomology ⋮ Explicit Coleman integration for curves ⋮ Computing zeta functions of generic projective hypersurfaces in larger characteristic ⋮ Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus ⋮ Maps between curves and arithmetic obstructions ⋮ Explicit Chabauty-Kim for the split Cartan modular curve of level 13 ⋮ Computing zeta functions of algebraic curves using Harvey's trace formula ⋮ Two Recent p-adic Approaches Towards the (Effective) Mordell Conjecture

Uses Software

• Magma

Cites Work

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• Computation of integral bases
• Gonality of the modular curve \(X_1(N)\)
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• The Magma algebra system. I: The user language
• Computing Riemann-Roch spaces in algebraic function fields and related topics.
• Counting points on hyperelliptic curves in average polynomial time
• Defining equations of modular curves
• An extension of Kedlaya's algorithm to hyperelliptic curves in characteristic \(2\)
• Hurwitz schemes and irreducibility of moduli of algebraic curves
• Counting points on curves using a map to $\mathbf {P}^1$
• Computing zeta functions of nondegenerate curves
• Point counting on curves using a gonality preserving lift
• Explicit Coleman Integration for Hyperelliptic Curves
• Constructing elliptic curves over finite fields with prescribed torsion
• Multiplying matrices faster than coppersmith-winograd
• Kedlaya's Algorithm in Larger Characteristic
• Effective 𝑝-adic cohomology for cyclic cubic threefolds

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