Conjecture on minimal a-number for a binomial family of Artin–Schreier curves with ramification break n p^2 − 1

Prove that for every odd prime p and every integer n ≥ 0, the a-number of the Artin–Schreier curve over an algebraically closed field of characteristic p defined by y^p − y = −x^{n p^2 − 1} − x^{(n p^2 + (n − 1)p − 1)/2} equals L(n p^2 − 1), where for a single branch point with ramification break d the Booher–Cais lower bound is L(d) := max_{1 ≤ j ≤ p − 1} ∑_{i=j}^{p−1} (⌊i d / p⌋ − ⌊i d / p − (1 − 1/p) j d / p⌋).

Background

The paper studies whether the Booher–Cais lower bound for the a-number of Z/pZ-Galois Artin–Schreier covers is optimal. It proves optimality in several infinite families (notably for ramification breaks congruent to −1 mod p^2) by constructing explicit curves and using formal patching.

In Remark (experiment), the author reports computational evidence: for 0 ≤ n ≤ 7 and odd primes p ≤ 13, the specific binomial Artin–Schreier curves with equation y^p − y = −x^{n p^2 − 1} − x^{(n p^2 + (n − 1)p − 1)/2} have a-number equal to the lower bound L(n p^2 − 1). Motivated by this, the remark formulates a conjecture asserting that this equality holds for all n and all odd primes p, which would further support the optimality of the lower bound across a broad family.