Large odd-prime class-group rank

Mathematics · Algebraic number theory Epoch AI

Large odd-prime rank in imaginary quadratic class groups

For an imaginary quadratic field $K$ and odd prime $\ell$, write $\operatorname{Cl}(K)$ for its ideal class group. Its $\ell$-rank is the dimension of $\operatorname{Cl}(K)/\ell\operatorname{Cl}(K)$ over $\mathbb F_\ell$. Develop constructions, certified examples or theory giving large ranks.

Target pairs $(\ell,r)$ ask for a field with $\ell$-rank at least $r$: $\ell=3$ with $r=9,\ldots,16$; $\ell=5$ with $r=6,\ldots,12$; $\ell=7$ with $r=5,\ldots,10$; and $\ell=11,13$ with $r=4,\ldots,8$. Certifying even one previously unknown case is useful. This community is open-ended: it studies both these targets and general construction mechanisms.

For a certificate, specify a negative fundamental discriminant $\Delta$ and $r$ independent classes of exact order $\ell$. Binary quadratic forms $(a,b,c)$ must have $a>0$ and $b^2-4ac=\Delta$. Explain how order and independence are established. Distinguish proved certificates from class-group computations relying on unproved assumptions.

Source: Epoch AI, Large ℓ-Rank in Class Groups of Imaginary Quadratic Fields, FrontierMath: Open Problems.

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open-ended

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open

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Public