[Paper Review] On the number of representations by certain octonary quadratic forms with coefficients 1, 2, 3, 4 and 6

B. Ramakrishnan, Brundaban Sahu+1 more|arXiv (Cornell University)|Jul 13, 2016

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper derives explicit formulas for the number of representations of integers by certain diagonal octonary quadratic forms with coefficients 1, 2, 3, 4, and 6 using the theory of modular forms. By constructing explicit bases for the space of modular forms of weight 4 on $̳_0(48)$ with a specific character, the authors complete previously open cases, particularly for mixed-parity forms with coefficients 1, 2, 4, and extend results to include coefficients 3 and 6, yielding new closed-form expressions for 203 cases.

ABSTRACT

In this paper, we find formulas for the number of representations of certain diagonal octonary quadratic forms with coefficients $1,2,3,4$ and $6$. We obtain these formulas by constructing explicit bases of the space of modular forms of weight $4$ on $Γ_0(48)$ with character.

Motivation & Objective

• To complete the classification of representation numbers for diagonal octonary quadratic forms with coefficients 1, 2, 3, 4, and 6.
• To resolve the remaining 12 cases of mixed-parity forms with coefficients 1, 2, 4 not previously covered.
• To extend prior results on forms with coefficients 1, 2, 3, 6 and 2, 3, 6 to include forms with coefficients 1, 2, 3, 4, 6.
• To provide explicit formulas for representation numbers using modular forms of weight 4 on $̳_0(48)$ with a Dirichlet character.

Proposed method

• The authors construct an explicit basis for the space of modular forms of weight 4 on $̳_0(48)$ with a specific Dirichlet character $χ_8$.
• They use the theta series associated with the octonary quadratic forms, which are modular forms of weight 4 on $̳_0(48)$ with character.
• The representation numbers $N(1^i,2^j,3^k,4^l,6^m;n)$ are expressed as linear combinations of arithmetic functions: $σ_{3;\chi_8,\mathbf{1}}(n)$, $σ_{3;\mathbf{1},\chi_8}(n)$, and coefficients $a_{4,8,\chi_8;1}(n)$, $a_{4,8,\chi_8;2}(n)$.
• The modular form space is analyzed via its dimension and basis, and the representation formulas are derived by matching Fourier coefficients.
• The method relies on the theory of modular forms, particularly the decomposition of the space of cusp forms and Eisenstein series.
• The authors verify the formulas by comparing with known results and handling special cases via normalization and reduction techniques.

Experimental results

Research questions

• RQ1 What are the explicit formulas for the number of representations of integers by diagonal octonary quadratic forms with coefficients 1, 2, 3, 4, and 6?
• RQ2 How can the remaining 12 open cases of mixed-parity forms with coefficients 1, 2, 4 be resolved using modular forms?
• RQ3 What is the structure of the space of modular forms of weight 4 on $̳_0(48)$ with character, and how does it support the derivation of representation formulas?
• RQ4 How do the new formulas for coefficients 1, 2, 3, 4, 6 compare with prior results for forms with coefficients 1, 2, 3, 6 or 2, 3, 6?
• RQ5 Can the representation numbers be uniformly expressed using a common basis of modular forms across all 203 cases?

Key findings

• The paper completes the 12 missing cases for octonary forms with coefficients 1, 2, 4 by deriving explicit formulas using modular forms of weight 4 on $̳_0(48)$.
• For the case $N(1^1,2^1,4^6;n)$, the formula is $\frac{2}{11}\sigma_{3;\chi_8,\mathbf{1}}(n) + \frac{2}{11}\sigma_{3;\mathbf{1},\chi_8}(n/2) + \frac{6}{11}a_{4,8,\chi_8;1}(n) + \frac{14}{11}a_{4,8,\chi_8;2}(n) + \frac{48}{11}a_{4,8,\chi_8;1}(n/2) - \frac{28}{11}a_{4,8,\chi_8;2}(n/2)$.
• For $N(1^1,2^3,4^4;n)$, the formula is $\frac{4}{11}\sigma_{3;\chi_8,\mathbf{1}}(n) + \frac{2}{11}\sigma_{3;\mathbf{1},\chi_8}(n/2) + \frac{1}{11}a_{4,8,\chi_8;1}(n) + \frac{17}{11}a_{4,8,\chi_8;2}(n) + \frac{48}{11}a_{4,8,\chi_8;1}(n/2) + \frac{16}{11}a_{4,8,\chi_8;2}(n/2)$.
• The authors derive 203 new formulas for forms with coefficients 1, 2, 3, 4, 6, excluding cases reducible to earlier results with coefficients 2, 3 or 1, 2, 3, 6.
• The representation numbers are expressed as rational linear combinations of arithmetic functions involving divisor sums and coefficients of modular forms.
• The formulas are verified numerically for specific values of $n$, such as $n = 40013$, $40031$, $42011$, $50021$, and $60011$, with exact fractional coefficients provided in tables.