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math.NT — Number Theory

Thakur's Conjectures in Function Field Arithmetic: A Report On Queries to Frontier Agents

Contributed by Taylor Dupuy

This report was created on 23 October 2026 from six AI research reports generated with the frontier-dump skill on Dinesh Thakur's conjectures in function field arithmetic. Taylor Dupuy initiated those queries during his talk How to Use AI in Mathematical Research Effectively at the University of Rochester Number Theory Seminar on that date. The six reports were synthesized with harvest-dumps and developed into this manuscript with using-latexpowers. We survey the current status of several conjectures of Dinesh Thakur and his coauthors on the arithmetic of $\mathbb{F}_q[t]$ : the degree hypotheses for power sums, the characterisation of zeta-like multizeta values in genus zero and in higher genus, the relations among multizeta values, the distribution of zeros of Goss zeta functions, and the automata-theoretic transcendence questions for special values. For each conjecture we give the precise statement and its status in the literature. The six computational research dumps report several results and refutations. Among them, the recursion predicted by Thakur's second power-sum hypothesis fails for an explicit family of exponents at thirteen non-prime values of $q$ (a computer-assisted result); the shuffle relations, even with Frobenius and Carlitz's Eulerian evaluations, do not generate all relations among multizeta values; a closed formula for a $v$ -adic valuation difference conjectured by Thakur holds only after a sign change (and, for $d\ge4$ , halving), and the corrected formula is proved, with computer assistance, for degrees $d\le 11$ ; and the $v$ -adic Newton polygon of a Goss zeta function over $\mathbb{F}_2[t]$ at a prime of degree two has a slope of multiplicity one, which appears to contradict a result claimed in a 2023 preprint (this conflict needs human verification). The shuffle result and an assembly of published theorems describing all relations are elementary, and we claim no novelty for them. We also write out a proof of a theorem of Wen and Yao on $v$ -adic arithmetic gamma values. Computational evidence is labelled as such throughout, and the open problems are listed. This manuscript is AI-generated by Claude Code (model claude-opus-5-5) from AI-generated research dumps; its proofs received only AI review, and no human has checked them.

Primarily AI-generated textHuman understanding: no partsGoss zeta functionsThakur conjecturesfunction field arithmeticmultizeta values

Oct 27, 2026PDF ↗View work →

hexagon:2610.00525v1

math.NT — Number Theory

A modular obstruction to the $P$ -adic Leopoldt conjecture for the Carlitz module

Contributed by Shawn Krisman

Let $L$ be a finite extension of $K=\mathbb{F}_q(\theta)$ and $P$ a monic prime of $\mathbb{F}_q[\theta]$ . Anglès asked whether the $P$ -adic Leopoldt defect of Taelman's module of units of the Carlitz module over $L$ always vanishes, and Lucas conjectured this, together with the equivalence " $L_P(E/\mathcal{O}_L)\neq 0$ iff $\exp_E$ is injective", for Anderson modules. We show that both statements fail. For every odd prime $p$ , the extension of $\mathbb{F}_p(\theta)$ generated by a root of $z^p-(-\theta)^{p-2}z-(-\theta)^{p-1}$ is real, has Galois group $\mathrm{AGL}(1,p)$ , and has positive $P$ -adic Leopoldt defect and vanishing $P$ -adic zeta value at $s=1$ for every $P$ . The obstruction comes from modular representation theory at the infinite place. We attach to every separable $L/K$ an integer $\delta_{\mathrm{gen}}(L)$ , depending only on the Galois group of the Galois closure, its decomposition group at $\infty$ and the character by which this group acts on the Carlitz period, and prove that the $P$ -adic defect is at least $\delta_{\mathrm{gen}}(L)$ for all $P$ . Using Bosser's $P$ -adic Baker-Brumer theorem, we prove that the defect vanishes for every $P$ for real abelian extensions and for abelian extensions with cyclic Sylow $p$ -subgroup. We also show that a $v$ -adic analogue of Papanikolas's algebraic independence theorem for Carlitz logarithms would imply that the defect equals $\delta_{\mathrm{gen}}(L)$ in general.

Primarily AI-generated textHuman understanding: no partsCarlitz logarithmsCarlitz moduleDrinfeld modulesGoss zeta functionLeopoldt conjectureP-adic L-seriesTaelman unitsmodular representation theory

Oct 26, 2026PDF ↗View work →

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Thakur's Conjectures in Function Field Arithmetic: A Report On Queries to Frontier Agents

A modular obstruction to the $P$ -adic Leopoldt conjecture for the Carlitz module

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Matching works

Thakur's Conjectures in Function Field Arithmetic: A Report On Queries to Frontier Agents

A modular obstruction to the PPP-adic Leopoldt conjecture for the Carlitz module

A modular obstruction to the $P$ -adic Leopoldt conjecture for the Carlitz module