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Rank one elliptic curves and rank stability
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abstract
For any quadratic extension $L/K$ of number fields, we prove that there are infinitely many elliptic curves $E$ over $K$ so that the abelian groups $E(K)$ and $E(L)$ both have rank $1$. In particular, there are infinitely many elliptic curves of rank $1$ over any number field. This result generalizes theorems of Koymans-Pagano and Alp\"oge-Bhargava-Ho-Shnidman which were used to independently show that Hilbert's tenth problem over the ring of integers of any number field has a negative answer. Our approach differs since we are obtaining our elliptic curves by specializing a nonisotrivial rank $1$ family of elliptic curves and we compute all the ranks involved.
Forward citations
Cited by 2 Pith papers
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On Cohesive Products of Fields
Cohesive products of fields are constructed and their infinite Galois groups plus hyper-automorphism groups are characterized for large classes of computable Galois extensions.
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Infinitely many hyperelliptic curves of small genus and small fixed rank, and of any genus and rank two
For any number field K and genus g ≥ 2, there are infinitely many non-isomorphic hyperelliptic curves over K with Jacobian rank 0, 1, or 2 over K; explicit higher-rank ranges are given for small genera over Q.
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