A finiteness theorem for positive definite strictly n-regular quadratic forms

Chan, Wai Kiu; Marino, Alicia

doi:10.1007/s40993-018-0108-z

Cited by 3 publications

(3 citation statements)

References 16 publications

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“…Recall that a Z-lattice L is called primitive if npLq " Z. For general properties of Λ p -transformation, see [2] and [3].…”

Section: Watson's Transformations On the Set Of Spinor Generamentioning

confidence: 99%

Genus-correspondences respecting spinor genus

2017

Journal of Number Theory

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Abstract. For two positive definite integral ternary quadratic forms f and g and a positive integer n, if n¨g is represented by f and n¨dg " df , then the pair pf, gq is called a representable pair by scaling n. The set of all representable pairs in genpf qˆgenpgq is called a genus-correspondence. In [7], Jagy conjectured that if n is square free and the number of spinor genera in the genus of f equals to the number of spinor genera in the genus of g, then such a genus-correspondence respects spinor genus in the sense that for any representable pairs pf, gq, pf 1 , g 1 q by scaling n, f 1 P spnpf q if and only if g 1 P spnpgq. In this article, we show that by giving a counter example, Jagy's conjecture does not hold. Furthermore, we provide a necessary and sufficient condition for a genus-correspondence to respect spinor genus.

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“…Recall that a Z-lattice L is called primitive if npLq " Z. For general properties of Λ p -transformation, see [2] and [3].…”

Section: Watson's Transformations On the Set Of Spinor Generamentioning

confidence: 99%

Genus-correspondences respecting spinor genus

2017

Journal of Number Theory

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“…The readers are referred to [3] for more properties of the operators Λ p . Let L be a ternary Z-lattice and let p be a fixed prime.…”

Section: A Generalization Of Watson Transformationmentioning

confidence: 99%

A generalization of Watson transformation and representations of ternary quadratic forms

Lee

2016

Journal of Number Theory

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Abstract. Let L be a positive definite (non-classic) ternary Z-lattice and let p be a prime such that a 1 2Zp-modular component of Lp is nonzero isotropic and 4¨dL is not divisible by p. For a nonnegative integer m, let G L,p pmq be the genus with discriminant p m¨d L on the quadratic spaceZp-modular component of Tp is nonzero isotropic, and Tq is isometric to pL p m qq for any prime q different from p. Let rpn, M q be the number of representations of an integer n by a Z-lattice M . In this article, we show that if m ď 2 and n is divisible by p only when m " 2, then for any T P G L,p pmq, rpn, T q can be written as a linear summation of rppn, S i q and rpp 3 n, S i q for S i P G L,p pm`1q with an extra term in some special case. We provide a simple criterion on when the extra term is necessary, and we compute the extra term explicitly. We also give a recursive relation to compute rpn, T q, for any T P G L,p pmq, by using the number of representations of some integers by lattices in G L,p pm`1q for an arbitrary integer m.

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“…A quadratic form is strictly regular if it primitively represents all elements that are primitively represented by the genus of f . The work of Earnest-Kim-Meyer was later generalized by Chan and Marino [2], who showed that there exist only finitely many inequivalent integral positive definite normalized strictly n-regular quadratic forms over Q in n + 4 variables.…”

Section: Acknowledgementsmentioning

confidence: 99%

Finiteness Of Strictly N-regular And Almost N-regular Hermitian Lattices

Li¹

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Let E be an imaginary quadratic field. A Hermitian lattice L is said to be regular if L globally represents all elements that are locally represented by L. It is nregular if L globally represents all Hermitian lattices of rank n that are locally represented by L. In 2005 Rokicki [23] proved that, for a fixed imaginary quadratic field, there exist only finitely many isometry classes of nondegenerate normalized positive definite n-regular Hermitian lattices of rank n + 1. The notion of n-regularity can be strengthened to strict n-regularity. A sublattice, N , of a Hermitian lattice, L, is primitive in L if N is a direct summand of L. We say that L is strictly n-regular if L primitively represents all Hermitian lattices of rank n that are locally primitively represented by L. In this thesis, we prove an analogous result to Rokicki's for strictly n-regular Hermitian lattices, for n ≥ 2. Another notion of regularity is almost n-regularity. A Hermitian lattice L is almost n-regular if L globally represents all but finitely many Hermitian lattices of rank n that are locally represented by L. In this thesis, we also consider almost n-regular lattices, and prove a finiteness result for almost n-regular lattices of rank n + 1.

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A finiteness theorem for positive definite strictly n-regular quadratic forms

Cited by 3 publications

References 16 publications

Genus-correspondences respecting spinor genus

Genus-correspondences respecting spinor genus

A generalization of Watson transformation and representations of ternary quadratic forms

Finiteness Of Strictly N-regular And Almost N-regular Hermitian Lattices

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