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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.19268 |
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| _version_ | 1866916413968482304 |
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| author | Arai, Keisuke Hattori, Shin Kondo, Satoshi Papikian, Mihran |
| author_facet | Arai, Keisuke Hattori, Shin Kondo, Satoshi Papikian, Mihran |
| contents | Let $p$ be a rational prime, $q>1$ a power of $p$ and $F=\mathbb{F}_q(t)$. For an integer $d\geq 2$, let $D$ be a central division algebra over $F$ of dimension $d^2$ which is split at $\infty$ and has invariant $\mathrm{inv}_x(D)=1/d$ at any place $x$ of $F$ at which $D$ ramifies. Let $X^D$ be the Drinfeld--Stuhler variety, the coarse moduli scheme of the algebraic stack over $F$ classifying $\mathscr{D}$-elliptic sheaves. In this paper, we establish various arithmetic properties of $\mathscr{D}$-elliptic sheaves to give an explicit criterion for the non-existence of rational points of $X^D$ over a finite extension of $F$ of degree $d$. As an application, for $d=2$, we present explicit infinite families of quadratic extensions of $F$ over which the curve $X^D$ violates the Hasse principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19268 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathscr{D}$-elliptic sheaves and the Hasse principle Arai, Keisuke Hattori, Shin Kondo, Satoshi Papikian, Mihran Number Theory Let $p$ be a rational prime, $q>1$ a power of $p$ and $F=\mathbb{F}_q(t)$. For an integer $d\geq 2$, let $D$ be a central division algebra over $F$ of dimension $d^2$ which is split at $\infty$ and has invariant $\mathrm{inv}_x(D)=1/d$ at any place $x$ of $F$ at which $D$ ramifies. Let $X^D$ be the Drinfeld--Stuhler variety, the coarse moduli scheme of the algebraic stack over $F$ classifying $\mathscr{D}$-elliptic sheaves. In this paper, we establish various arithmetic properties of $\mathscr{D}$-elliptic sheaves to give an explicit criterion for the non-existence of rational points of $X^D$ over a finite extension of $F$ of degree $d$. As an application, for $d=2$, we present explicit infinite families of quadratic extensions of $F$ over which the curve $X^D$ violates the Hasse principle. |
| title | $\mathscr{D}$-elliptic sheaves and the Hasse principle |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.19268 |