Diagonal Hypersurfaces Research Articles

Artin L-functions of diagonal hypersurfaces and generalized hypergeometric functions over finite fields

Artin L-functions of diagonal hypersurfaces and generalized hypergeometric functions over finite fields

Number of 𝔽q-points on diagonal hypersurfaces and hypergeometric function

Let [Formula: see text] denote the family of monomial deformations of diagonal hypersurface over a finite field [Formula: see text] given by [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. The Dwork hypersurface is the case when [Formula: see text], that is, [Formula: see text]. Formulas for the number of [Formula: see text]-points on the Dwork hypersurfaces in terms of McCarthy’s [Formula: see text]-adic hypergeometric functions are known. In this paper, we provide a formula for the number of [Formula: see text]-points on [Formula: see text] in terms of McCarthy’s [Formula: see text]-adic hypergeometric function which holds for [Formula: see text].

Rational lines on diagonal hypersurfaces and subconvexity via the circle method

Fix k , s , n ∈ N k,s,n\in \mathbb {N} , and consider non-zero integers c 1 , … , c s c_1,\ldots ,c_s , not all of the same sign. Provided that s ⩾ k ( k + 1 ) s\geqslant k(k+1) , we establish a Hasse principle for the existence of lines having integral coordinates lying on the affine diagonal hypersurface defined by the equation c 1 x 1 k + … + c s x s k = n c_1x_1^k+\ldots +c_sx_s^k=n . This conclusion surmounts the conventional convexity barrier tantamount to the square-root cancellation limit for this problem.

Counting rational points of quartic diagonal hypersurfaces over finite fields

<abstract><p>Let $ \mathbb{F}_q $ be the finite field of order $ q $ where $ q = p^{k} $, $ k $ is a positive integer and $ p $ is an odd prime. Let $ \mathbb{F}_q^* $ represent the nonzero elements of $ \mathbb{F}_{q} $. For $ f(x_1, \cdots, x_n)\in\mathbb{F}_q[x_1, \cdots, x_n] $, we used $ N\big(f(x_1, \cdots, x_n) = 0\big) $ to denote the number of $ \mathbb{F}_q $-rational points of the affine hypersurface $ f(x_1, \cdots, x_n) = 0 $. In 2020, Zhao et al. obtained the explicit formulae for $ N(x_1^4+x_2^4 = c) $, $ N(x_1^4+x_2^4+x_3^4 = c) $ and $ N(x_1^4+x_2^4+x_3^4+x_4^4 = c) $ over $ \mathbb{F}_q $, with $ c\in\mathbb{F}_q^* $. In this paper, by using Jacobi sums and an analog of the Hasse-Davenport theorem, we arrived at explicit formulae for $ N(a_1x_1^4+a_2x_2^4 = c) $ and $ N(b_1x_1^4+b_2x_2^4+b_3x_3^4 = c) $ with $ a_i, b_j\in \mathbb{F}_q^* (1\leq i \leq 2, 1 \leq j \leq 3) $ and $ c\in \mathbb{F}_q $. Furthermore, by using the reduction formula for Jacobi sums, the number of rational points of the quartic diagonal hypersurface $ a_1x_1^4+a_2x_2^4+\cdots+a_nx_n^4 = c $ of $ n\geq 4 $ variables with $ a_i\in\mathbb{F}_q^* $ $ (1\leq i\leq n) $, $ c\in\mathbb{F}_q $ and $ p\equiv1({\rm{mod}} \ 4) $, can also be deduced. These extended and improved earlier results.</p></abstract>

Tight closure of powers of parameter ideals in hypersurface rings and their tight Hilbert polynomials

In this paper, we find the tight closure of powers of parameter ideals of certain diagonal hypersurface rings. In many cases, the associated graded ring with respect to tight closure filtration turns out to be Cohen–Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. We determine the tight Hilbert polynomial in the following cases: (1) F-pure diagonal hypersurfaces where number of variables is equal to the degree of defining equation, (2) diagonal hypersurface rings where characteristic of the ring is one less than the degree of defining equation and (3) quartic diagonal hypersurface in four variables.

How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces

In this paper, we establish a strategy for the calculation of the proportion of everywhere locally soluble diagonal hypersurfaces of [Formula: see text] of fixed degree. Our strategy is based on the product formula established by Bright, Browning and Loughran. Their formula reduces the problem into the calculation of the proportions of [Formula: see text]-soluble diagonal hypersurfaces for all places [Formula: see text]. As worked examples, we carry out our strategy in the cases of quadratic and cubic hypersurfaces. As a consequence, we prove that around [Formula: see text] of diagonal cubic fourfolds have [Formula: see text]-rational points under a hypothesis on the Brauer–Manin obstruction.

The number of rational points of certain quartic diagonal hypersurfaces over finite fields

Let $p$ be an odd prime and let $\mathbb{F}_q$ be a finite field of characteristic $p$ with order $q = p^s$. For $f(x_1, \cdots, x_n)\in\mathbb{F}_q[x_1, ..., x_n]$, we denote by $N(f(x_1, \cdots, x_n) = 0)$ the number of $\mathbb{F}_q$-rational points on the affine hypersurface $f(x_1, \cdots, x_n) = 0$. In 1981, Myerson gave a formula for $N(x_1^4+\cdots+x_n^4 = 0)$. Recently, Zhao and Zhao obtained an explicit formula for $N(x_1^4+x_2^4 = c)$ with $c\in\mathbb{F}_q^*: = \mathbb{F}_q\setminus \{0\}$. In this paper, by using the Gauss sum and Jacobi sum, we arrive at explicit formulas for $N(x_1^4+x_2^4+x_3^4 = c)$ and $N(x_1^4+x_2^4+x_3^4+x_4^4 = c)$ with $c\in\mathbb{F}_q^*$.

On Artin’s conjecture: Linear slices of diagonal hypersurfaces

Artin’s conjecture is established for all forms that can be realised as a diagonal form on a hyperplane.

Maximal functions and ergodic averages related to Waring’s problem

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, l2-maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of lp spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.

A note on diagonal and Hermitian hypersurfaces

Aspects of the properties, enumeration and constructionof points on diagonal and Hermitian hypersurfaceshave been considered extensively in the literature and arefurther considered here. The zeta function of diagonal hypersurfacesis given as a direct result of the work of Wolfmann.Recursive construction techniques for the set ofrational points of Hermitian hypersurfaces are of interest.The relationship of these techniques here tothe construction of codes on hypersurfaces is briefly noted.

Rational points on linear slices of diagonal hypersurfaces

AbstractAn asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

Rational points on linear slices of diagonal hypersurfaces

AbstractAn asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

Diagonal hypersurfaces and the Bloch-Kato conjecture, I

Diagonal hypersurfaces and the Bloch-Kato conjecture, I

$F$-invariants of diagonal hypersurfaces

In this note, we derive a formula for the F-pure threshold of diagonal hypersurfaces over a perfect field of prime characteristic. We also calculate the associated test ideal at the F-pure threshold, and give formulas for higher jumping numbers of Fermat hypersurfaces.

A direct limit for limit Hilbert–Kunz multiplicity for smooth projective curves

This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert–Kunz multiplicity, a possible candidate for a characteristic zero Hilbert–Kunz multiplicity. The main goal is to establish an affirmative answer for one of the main cases for which the limit Hilbert–Kunz multiplicity is even known to exist, namely that of graded ideals in the homogeneous coordinate ring of smooth projective curves. The proof involves more careful estimates of bounds found independently by Brenner and Trivedi on the dimensions of the cohomologies of twists of the syzygy bundle as the characteristic p goes to infinity and uses asymptotic results of Trivedi on the slopes of Harder–Narasimham filtrations of Frobenius pullbacks of bundles. In view of unpublished results of Gessel and Monsky, the case of maximal ideals in diagonal hypersurfaces is also discussed in depth.

Abelian points on algebraic varieties

We attempt to determine which classes of algebraic varieties over $\Q$ must have points in some abelian extension of $\Q$. We give: (i) for every odd $d>1$, an explicit family of degree $d$, dimension $d-2$ diagonal hypersurfaces without $\Q^{\ab}$-points, (ii) for every number field $K$, a genus one curve $C_{/\Q}$ with no $K^{\ab}$-points, and (iii) for every $g \geq 4$ an algebraic curve $C_{/\Q}$ of genus $g$ with no $\Q^{\ab}$-points. In an appendix, we discuss varieties over $\Q((t))$, obtaining in particular a curve of genus $3$ without $\Q((t))^{\ab}$-points.

On crepant resolution of some hypersurface singularities and a criterion for UFD

In this article, we find some diagonal hypersurfaces that admit crepant resolutions. We also give a criterion for unique factorization domains.

A Computation of Tight Closure in Diagonal Hypersurfaces

The aim of this paper is to settle a question about the tight closure of Ž 2 2 2 . w x Ž 3 3 3. the ideal x , y , z in the ring R s K X, Y, Z r X q Y q Z where Ž K is a field of prime characteristic p / 3. Lower case letters denote the . images of the corresponding variables. M. McDermott has studied the tight closure of various irreducible ideals in R and has established that Ž 2 2 2 . w x xyz g x , y , z * when p 200, see Mc . The general case however existed as a classic example of the difficulty involved in tight closure w x Ž 2 2 2 . computations, see also Hu, Example 1.2 . We show that xyz g x , y , z * in arbitrary prime characteristic p, and furthermore establish that xyz g Ž 2 2 2 .F x , y , z whenever R is not F-pure, i.e., when p ' 2 mod 3. We move on to generalize these results to the diagonal hypersurfaces R s w x Ž n n. K X , . . . , X r X q ??? qX . 1 n 1 n These issues relate to the question whether the tight closure I* of an ideal I agrees with its plus closure, Iqs IRql R, where R is a domain over a field of characteristic p and Rq is the integral closure of R in an algebraic closure of its fraction field. In this setting, we may think of the Frobenius closure of I as I F s IR l R where R is the extension of R obtained by adjoining pth roots of all nonzero elements of R for e g N. It is not difficult to see that Iq: I*, and equality in general is a formidable open question. It should be mentioned that in the case when I is an ideal generated by part of a system of parameters, the equality is a result