The number of representations of squares by integral ternary quadratic forms

Abstract

Let f be a positive definite (non-classic) integral quaternary quadratic form. We say f is strongly s-regular if it satisfies a regularity property on the number of representations of squares of integers. In this article, we prove that there are only finitely many strongly s-regular quaternary quadratic forms up to isometry if the minimum of the nonzero squares that are represented by the quadratic form is fixed. Furthermore, we show that there are exactly 34 strongly s-regular diagonal quaternary quadratic forms representing one (see Table 1). In particular, we use eta-quotients to prove the strongly s-regularity of the quaternary quadratic form x 2`2 y 2`3 z 2`1 0w 2 , which is, in fact, of class number 2 (see Lemma 5.5 and Proposition 5.6).2010 Mathematics Subject Classification. Primary 11E12, 11E20, 11E45.

The number of representations of squares by integral ternary quadratic forms (II)

“…For any ternary form f with class number one, since rpn 2 , f q " rpn 2 , genpf qq for any integer n, f is strongly s-regular. In the previous article [5], we proved that every ternary quadratic form in the genus of f is strongly s-regular if and only if the genus of f is indistinguishable by squares, that is rpn 2 , f 1 q " rpn 2 , f q for any f 1 P genpf q and any integer n. Also, we completely resolved the conjecture given by Cooper and Lam in [2].…”

“…If x 2`4 y 2`4 yz`p2 3 3 t`1 qz 2 " n 2 , then either x or z is odd, but not both. Hence we have rpn 2 , K 1,t q " rpn 2 , x1, 4, 2 5 3 t yq`rˆn 2 , x4y Kˆ4 2 2 2 3 3 t`1˙˙.…”

“…For any ternary form f with class number one, since rpn 2 , f q " rpn 2 , genpf qq for any integer n, f is strongly s-regular. In the previous article [5], we proved that every ternary quadratic form in the genus of f is strongly s-regular if and only if the genus of f is indistinguishable by squares, that is rpn 2 , f 1 q " rpn 2 , f q for any f 1 P genpf q and any integer n. Also, we completely resolved the conjecture given by Cooper and Lam in [2]. In this article, we prove that every strongly s-regular form represents all squares that are represented by its genus, and there are only finitely many strongly s-regular ternary forms up to equivalence if m s pf q " min nPZ`t n : rpn 2 , f q ‰ 0u is fixed.…”

“…where rgs is the equivalence class containing g in the genus genpf q of f and opf q is the order of the isometry group Opf q. Then the Minkowski-Siegel formula (see also Theorem 3.2 of [5]) says that rpn 2 1 n 2 2 , genpf qq " rpn 2 1 , genpf qq ź p∤8df h p pdf, λ p q.…”

Quadratic forms with a strong regularity property on the representations of squares

“…Related to this question, we [15] proved that there are only finitely many isometry classes of strongly s-regular ternary quadratic forms if the minimum of the nonzero squares that are represented by the form is fixed, and we classified all strongly s-regular ternary quadratic forms that represent 1. Recently, the first author [13] proved that there are only finitely many isometry classes of strongly s-regular quaternary quadratic forms if the minimum of the nonzero squares that are represented by them is fixed, and classifies all strongly s-regular diagonal quaternary quadratic forms that represent 1.…”