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Let \mathbb {F}_q denote the finite field of characteristic p and order q. Let \mathbb {Z}_q denote the unramified extension of the p-adic rational integers \mathbb {Z}_p with residue field \mathbb {F}_q. Given two positive integers m,n, define a box \mathcal B_m to be a subset of \mathbb {Z}_q^n with q^{nm} elements such that \mathcal B_m modulo p^m is equal to (\mathbb {Z}_q/p^m \mathbb {Z}_q)^n. For a collection of nonconstant polynomials f_1,\dots,f_s\in \mathbb {Z}_q[x_1,\ldots,x_n] and positive integers m_1,\dots,m_s, define the set of common zeros inside the box \mathcal B_m to be \begin{equation*} V=\{X\in \mathcal B_m:\; f_i(X)\equiv 0\mod {p^{m_i}}\text { for all } 1\leq i\leq s\}. \end{equation*} It is an interesting problem to give the sharp estimates for the p-divisibility of |V|. This problem has been partially solved for the three cases: (i) m=m_1=\cdots =m_s=1, which is just the Ax-Katz theorem, (ii) m=m_1=\cdots =m_s>1, which was solved by Katz [Proc. Amer. Math. Soc. 137 (2009), pp. 4065–4075; Amer. J. Math. 93 (1971), pp. 485–499], Marshal and Ramage [Proc. Amer. Math. Soc. 49 (1975), pp. 35–38], and (iii) m=1, and m_1,\dots,m_s\geq 1, which was recently solved by Cao, Wan [Finite Fields Appl. 91 (2023), p. 25] and Grynkiewicz [ A generalization of the Chevalley-Warning and Ax-Katz theorems with a view towards combinatorial number theory , Preprint, arXiv: 2208.12895 , 2022]. Based on the multi-fold addition and multiplication of the finite Witt rings over \mathbb {F}_q, we investigate the remaining unconsidered case of m>1 and m\neq m_j for some 1\leq j\leq s, and finally provide a complete answer to this problem. [ABSTRACT FROM AUTHOR]

Let p be a prime number and let k be an algebraically closed field of characteristic p. A BT_1 group scheme over k is a finite commutative group scheme which arises as the kernel of p on a p-divisible (Barsotti–Tate) group. Our main result is that every BT_1 group scheme over k occurs as a direct factor of the p-torsion group scheme of the Jacobian of an explicit curve defined over {\mathbb F}_p. We also treat a variant with polarizations. Our main tools are the Kraft classification of BT_1 group schemes, a theorem of Oda, and a combinatorial description of the de Rham cohomology of Fermat curves. [ABSTRACT FROM AUTHOR]

We study the finite field extension estimates for Hamming varieties Hj, j ∈ Fq*, defined by Hj = {x ∈ Fqd: ∏k=1d xk = j}, where Fq^d denotes the d-dimensional vector space over a finite field Fq with q elements. We show that although the maximal Fourier decay bound on Hj away from the origin is not good, the Stein-Tomas L2 → Lr extension estimate for Hjholds. [ABSTRACT FROM AUTHOR]

Let (R,m) be a local Noetherian ring with residue field k. While much is known about the generating sets of reductions of ideals of R if k is infinite, the case in which k is finite is less well understood. We investigate the existence (or lack thereof) of proper reductions of an ideal of R and the number of generators needed for a reduction in the case k is a finite field. When R is one-dimensional, we give a formula for the smallest integer n for which every ideal has an n-generated reduction. It follows that in a one-dimensional local Noetherian ring every ideal has a principal reduction if and only if the number of maximal ideals in the normalization of the reduced quotient of R is at most |k|. In higher dimensions, we show that for any positive integer, there exists an ideal of R that does not have an n-generated reduction and that if n ≥ dimR this ideal can be chosen to be m-primary. In the case where R is a two-dimensional regular local ring, we construct an example of an integrally closed m-primary ideal that does not have a 2-generated reduction and thus answer in the negative a question raised by Heinzer and Shannon. [ABSTRACT FROM AUTHOR]

Consider the Weil sum WF,d(u) = Σx∈F ∈(xd + ux), where F is a finite field of characteristic p, ∈ is the canonical additive character of F, d is coprime to |F*|, and u ∈ F*. We say that WF,d(u) is three-valued when it assumes precisely three distinct values as u runs through F*: this is the minimum number of distinct values in the nondegenerate case, and threevalued WF,d are rare and desirable. When WF,d is three-valued, we give a lower bound on the p-adic valuation of the values. This enables us to prove the characteristic 3 case of a 1976 conjecture of Helleseth: when p = 3 and [F : F3] is a power of 2, we show that WF,d cannot be three-valued. [ABSTRACT FROM AUTHOR]