Rational torsion on hyperelliptic jacobian varieties
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915727374548992 |
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| author | Kuru, Hamide Sadek, Mohammad |
| author_facet | Kuru, Hamide Sadek, Mohammad |
| contents | It was conjectured by Flynn that there exists a constant $κ$ such that, for any integer $g \ge 2$, any $m \le κg$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its Jacobian. Leprévost proved this conjecture with $κ=3$. In this work we prove that given an integer $N$ in the interval $[3g,4g+1]$, $g\ge 3$, satisfying certain partition conditions, there exist parametric families of hyperelliptic Jacobian varieties with a rational torsion point of order $N$. In particular, we establish the existence of such varieties for $N=4g+1$ when $g$ is odd and for $N=4g-1$ when $g$ is even. A few explicit applications of this result produce the first known infinite examples of torsion $13$ when $g=3$, torsion $15$ when $g=4$, and torsion $17,18,21$ when $g=5$. In fact, we show that infinitely many of the latter abelian varieties are absolutely simple. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_14454 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rational torsion on hyperelliptic jacobian varieties Kuru, Hamide Sadek, Mohammad Number Theory Algebraic Geometry It was conjectured by Flynn that there exists a constant $κ$ such that, for any integer $g \ge 2$, any $m \le κg$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its Jacobian. Leprévost proved this conjecture with $κ=3$. In this work we prove that given an integer $N$ in the interval $[3g,4g+1]$, $g\ge 3$, satisfying certain partition conditions, there exist parametric families of hyperelliptic Jacobian varieties with a rational torsion point of order $N$. In particular, we establish the existence of such varieties for $N=4g+1$ when $g$ is odd and for $N=4g-1$ when $g$ is even. A few explicit applications of this result produce the first known infinite examples of torsion $13$ when $g=3$, torsion $15$ when $g=4$, and torsion $17,18,21$ when $g=5$. In fact, we show that infinitely many of the latter abelian varieties are absolutely simple. |
| title | Rational torsion on hyperelliptic jacobian varieties |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2410.14454 |