Small solutions of generic ternary quadratic congruences to general moduli
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912009820307456 |
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| author | Baier, Stephan Chattopadhyay, Aishik |
| author_facet | Baier, Stephan Chattopadhyay, Aishik |
| contents | We study small non-trivial solutions of quadratic congruences of the form $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$, with $q$ being an odd natural number, in an average sense. This extends previous work of the authors in which they considered the case of prime power moduli $q$. Above, $α_2$ is arbitrary but fixed and $α_3$ is variable, and we assume that $(α_2α_3,q)=1$. We show that for all $α_3$ modulo $q$ which are coprime to $q$ except for a small number of $α_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_3,q)=1$ holds if $N\ge q^{11/24+\varepsilon}$ and $q$ is large enough. It is of significance that we break the barrier 1/2 in the above exponent. Key tools in our work are Burgess's estimate for character sums over short intervals and Heath-Brown's estimate for character sums with binary quadratic forms over small regions whose proofs depend on the Riemann hypothesis for curves over finite fields. We also formulate a refined conjecture about the size of the smallest solution of a ternary quadratic congruence, using information about the Diophantine properties of its coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15360 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small solutions of generic ternary quadratic congruences to general moduli Baier, Stephan Chattopadhyay, Aishik Number Theory 11D79, 11E04, 11E25, 11L40, 11T24 We study small non-trivial solutions of quadratic congruences of the form $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$, with $q$ being an odd natural number, in an average sense. This extends previous work of the authors in which they considered the case of prime power moduli $q$. Above, $α_2$ is arbitrary but fixed and $α_3$ is variable, and we assume that $(α_2α_3,q)=1$. We show that for all $α_3$ modulo $q$ which are coprime to $q$ except for a small number of $α_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_3,q)=1$ holds if $N\ge q^{11/24+\varepsilon}$ and $q$ is large enough. It is of significance that we break the barrier 1/2 in the above exponent. Key tools in our work are Burgess's estimate for character sums over short intervals and Heath-Brown's estimate for character sums with binary quadratic forms over small regions whose proofs depend on the Riemann hypothesis for curves over finite fields. We also formulate a refined conjecture about the size of the smallest solution of a ternary quadratic congruence, using information about the Diophantine properties of its coefficients. |
| title | Small solutions of generic ternary quadratic congruences to general moduli |
| topic | Number Theory 11D79, 11E04, 11E25, 11L40, 11T24 |
| url | https://arxiv.org/abs/2408.15360 |