The large-characteristic Newton polygon conjecture for twisted binomial L-functions

Let $\chi$ be a multiplicative character of order $c$ over a finite field, let $f(x)=x^d+\lambda x^e$ with $(d,e)=1$ , and let $p$ be the characteristic. Write $\operatorname{NP}_{u,m}(f)$ and $\operatorname{NP}_{u,T}(f)$ for the twisted classical and $T$ -adic Newton polygons, respectively, and let $P_{u,e,d}$ be the associated lower-bound polygon. Large-characteristic Newton polygon conjecture. If $p$ is large enough with respect to $c$ and $d$ , then

\operatorname{NP}_{u,m}(f)=\operatorname{NP}_{u,T}(f)=P_{u,e,d}.

This conjecture generalizes the conjecture of Zhang and Niu and is motivated by the criterion that equality holds when the associated integral constant is not divisible by $p$ . The paper proves the equality for $p>c(d^2-d+1)$ when $e=d-1$ , but the general large-characteristic assertion remains open.

The conjecture asserts that, for sufficiently large characteristic, the twisted classical and $T$ -adic Newton polygons both equal the predicted lower-bound polygon. Shenxing Zhang stated it as Conjecture 1.5 in a preprint posted on September 30, 2021, for binomials $f(x)=x^d+\lambda x^e$ .

Known results

Zhang, 2021: if $p>(d-e)(2d-1)$ , both Newton polygons lie above $P_{u,e,d}$ .
Zhang, 2021: equality holds exactly when the associated integral constant $H_{\mu,c,\mathbf p,e,d}$ is not divisible by $p$ .
Zhang, 2021: if $e=d-1$ and $p>c(d^2-d+1)$ , then $\operatorname{NP}_{u,m}(f)=\operatorname{NP}_{u,T}(f)=P_{u,e,d}$ .

September 30, 2021 preprint

Zhang’s preprint establishes the lower bound and the case $e=d-1$ , but explicitly leaves the general sufficiently-large- $p$ assertion as a conjecture. The retrieved record contains no later proof, counterexample, or claimed settlement.

Current status (as of September 2026): The conjecture is unproved in general; the equality is settled for $e=d-1$ under $p>c(d^2-d+1)$ and related special cases, while the general large-characteristic assertion remains open.

The large-characteristic Newton polygon conjecture for twisted binomial L-functions

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