Computing isogeny classes of typical principally polarized abelian surfaces over the rationals
We describe an efficient algorithm which, given a principally polarized (p.p.) abelian surface $A$ over $\mathbb{Q}$ with geometric endomorphism ring equal to $\mathbb{Z}$, computes all the other p.p. abelian surfaces over $\mathbb{Q}$ that are isogenous to $A$. This algorithm relies on explicit ope...
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| creator | van Bommel, Raymond Chidambaram, Shiva Costa, Edgar Kieffer, Jean |
| description | We describe an efficient algorithm which, given a principally polarized
(p.p.) abelian surface $A$ over $\mathbb{Q}$ with geometric endomorphism ring
equal to $\mathbb{Z}$, computes all the other p.p. abelian surfaces over
$\mathbb{Q}$ that are isogenous to $A$. This algorithm relies on explicit open
image techniques for Galois representations, and we employ a combination of
analytic and algebraic methods to efficiently prove or disprove the existence
of isogenies. We illustrate the practicality of our algorithm by applying it to
1 440 894 isogeny classes of Jacobians of genus 2 curves. |
| doi_str_mv | 10.48550/arxiv.2301.10118 |
| format | Article |
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(p.p.) abelian surface $A$ over $\mathbb{Q}$ with geometric endomorphism ring
equal to $\mathbb{Z}$, computes all the other p.p. abelian surfaces over
$\mathbb{Q}$ that are isogenous to $A$. This algorithm relies on explicit open
image techniques for Galois representations, and we employ a combination of
analytic and algebraic methods to efficiently prove or disprove the existence
of isogenies. We illustrate the practicality of our algorithm by applying it to
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(p.p.) abelian surface $A$ over $\mathbb{Q}$ with geometric endomorphism ring
equal to $\mathbb{Z}$, computes all the other p.p. abelian surfaces over
$\mathbb{Q}$ that are isogenous to $A$. This algorithm relies on explicit open
image techniques for Galois representations, and we employ a combination of
analytic and algebraic methods to efficiently prove or disprove the existence
of isogenies. We illustrate the practicality of our algorithm by applying it to
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(p.p.) abelian surface $A$ over $\mathbb{Q}$ with geometric endomorphism ring
equal to $\mathbb{Z}$, computes all the other p.p. abelian surfaces over
$\mathbb{Q}$ that are isogenous to $A$. This algorithm relies on explicit open
image techniques for Galois representations, and we employ a combination of
analytic and algebraic methods to efficiently prove or disprove the existence
of isogenies. We illustrate the practicality of our algorithm by applying it to
1 440 894 isogeny classes of Jacobians of genus 2 curves.</abstract><doi>10.48550/arxiv.2301.10118</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Algebraic Geometry Mathematics - Number Theory |
| title | Computing isogeny classes of typical principally polarized abelian surfaces over the rationals |
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