Representations of finite number of quadratic forms with same rank
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Representations of finite number of quadratic forms with same rank Daejun Kim1
· Byeong-Kweon Oh2
Received: 25 January 2020 / Accepted: 25 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let m, n be positive integers with m ≤ n. Let κ(m, n) be the largest integer k such that for any (positive definite and integral) quadratic forms f 1 , . . . , f k of rank m, there exists a quadratic form of rank n that represents f i for any i with 1 ≤ i ≤ k. In this article, we determine the number κ(m, n) for any integer m with 1 ≤ m ≤ 8, except for the cases when (m, n) = (3, 5) and (4, 6). In the exceptional cases, it will be proved that 1 ≤ κ(3, 5), κ(4, 6) ≤ 2. We also discuss some related topics. Keywords Quadratic forms · Representations Mathematics Subject Classification 11E12 · 11E20 · 11E25
1 Introduction For a positive integer m, let φm (X , Y ) be the classical modular polynomial (for the definition of this, see [6]). For three positive integers m 1 ,m 2 , and m 3 , Gross and Keating [6] showed that the quotient ring R(m 1 , m 2 , m 3 ) = Z[X , Y ]/(φm 1 , φm 2 , φm 3 ) is finite if and only if there is no positive definite binary quadratic form Q(x, y) = ax 2 + bx y + cy 2 over Z which represents the three integers m 1 , m 2 , m 3 . Moreover, when m 1 , m 2 , m 3 satisfy this condition, they found an explicit formula for the cardinality of R(m 1 , m 2 , m 3 ). Later, Görtz proved in [5] that there is no positive definite binary
This work was supported by the National Research Foundation of Korea (NRF-2019R1A2C1086347).
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Daejun Kim [email protected] Byeong-Kweon Oh [email protected]
1
Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea
2
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea
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D. Kim, B.-K. Oh
quadratic form Q(x, y) = ax 2 + bx y + cy 2 over Z which represents m 1 , m 2 , m 3 if and only if every positive semi-definite half-integral quadratic form with diagonal (m 1 , m 2 , m 3 ) is non-degenerate. Motivated by the above, we consider the following problem: For positive integers m and n, find the largest integer k such that for any positive definite quadratic forms f 1 , f 2 , . . . , f k with rank m, there is a quadratic form of rank n that represents f i for any i with 1 ≤ i ≤ k. In this article, the largest integer k satisfying the above property will be denoted by κ(m, n). As a sample, if m = 1 and n = 2, which is exactly the above case, then one may easily show that there does not exist a binary quadratic form that represents 1, 2, and 15. Since there is always a binary quadratic form representing any two positive integers given in advance, we have κ(1, 2) = 2. It seems to be very difficult problem to determine the number κ(m, n) for arbitrary positive integers m and n. The aim of this article is to determine the number κ(m, n) for any positive integer m with m ≤ 8 except for the cases when (m, n) = (3, 5) and (4, 6). In the exceptional cases, we onl
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