p-adic elliptic polylogarithms and cubic Chabauty
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| Format: | Preprint |
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2026
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| _version_ | 1866917429548941312 |
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| author | Balakrishnan, Jennifer S. Bianchi, Francesca Dogra, Netan |
| author_facet | Balakrishnan, Jennifer S. Bianchi, Francesca Dogra, Netan |
| contents | The Chabauty--Coleman--Kim method, under favourable circumstances, describes the set of integral points of a hyperelliptic curve inside the $p$-adic zeroes of certain transcendental functions. For an elliptic curve of Mordell--Weil rank one, the Chabauty--Coleman--Kim set in depth 2 is given by the zeroes of a (finite union of) quadratic polynomial(s) in the $p$-adic logarithm of the elliptic curve and the local $p$-adic height at $p$. Here, we give an explicit formula for a finite set containing the Chabauty--Coleman--Kim set in depth 3 for an elliptic curve of rank at most 2 under an assumption on non-vanishing of a special value of a $p$-adic $L$-function. The finite set is given by the zeroes of a polynomial in $p$-adic elliptic polylogarithms. We use these formulas to verify new instances of Kim's conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_20662 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | p-adic elliptic polylogarithms and cubic Chabauty Balakrishnan, Jennifer S. Bianchi, Francesca Dogra, Netan Number Theory Algebraic Geometry The Chabauty--Coleman--Kim method, under favourable circumstances, describes the set of integral points of a hyperelliptic curve inside the $p$-adic zeroes of certain transcendental functions. For an elliptic curve of Mordell--Weil rank one, the Chabauty--Coleman--Kim set in depth 2 is given by the zeroes of a (finite union of) quadratic polynomial(s) in the $p$-adic logarithm of the elliptic curve and the local $p$-adic height at $p$. Here, we give an explicit formula for a finite set containing the Chabauty--Coleman--Kim set in depth 3 for an elliptic curve of rank at most 2 under an assumption on non-vanishing of a special value of a $p$-adic $L$-function. The finite set is given by the zeroes of a polynomial in $p$-adic elliptic polylogarithms. We use these formulas to verify new instances of Kim's conjecture. |
| title | p-adic elliptic polylogarithms and cubic Chabauty |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2604.20662 |