Shadow line distributions
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909608158691328 |
|---|---|
| author | Balakrishnan, Jennifer S. Çiperiani, Mirela Mazur, Barry Rubin, Karl |
| author_facet | Balakrishnan, Jennifer S. Çiperiani, Mirela Mazur, Barry Rubin, Karl |
| contents | Let $E$ be an elliptic curve over $\mathbb{Q}$ with Mordell--Weil rank $2$ and $p$ be an odd prime of good ordinary reduction. For every imaginary quadratic field $K$ satisfying the Heegner hypothesis, there is (subject to the Shafarevich--Tate conjecture) a line, i.e., a free $\mathbb{Z}_p$-submodule of rank $1$, in $ E(K)\otimes \mathbb{Z}_p$ given by universal norms coming from the Mordell--Weil groups of subfields of the anticyclotomic $\mathbb{Z}_p$-extension of $K$; we call it the {\it shadow line}. When the twist of $E$ by $K$ has analytic rank $1$, the shadow line is conjectured to lie in $E(\mathbb{Q})\otimes\mathbb{Z}_p$; we verify this computationally in all our examples. We study the distribution of shadow lines in $E(\mathbb{Q})\otimes\mathbb{Z}_p$ as $K$ varies, framing conjectures based on the computations we have made. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shadow line distributions Balakrishnan, Jennifer S. Çiperiani, Mirela Mazur, Barry Rubin, Karl Number Theory Let $E$ be an elliptic curve over $\mathbb{Q}$ with Mordell--Weil rank $2$ and $p$ be an odd prime of good ordinary reduction. For every imaginary quadratic field $K$ satisfying the Heegner hypothesis, there is (subject to the Shafarevich--Tate conjecture) a line, i.e., a free $\mathbb{Z}_p$-submodule of rank $1$, in $ E(K)\otimes \mathbb{Z}_p$ given by universal norms coming from the Mordell--Weil groups of subfields of the anticyclotomic $\mathbb{Z}_p$-extension of $K$; we call it the {\it shadow line}. When the twist of $E$ by $K$ has analytic rank $1$, the shadow line is conjectured to lie in $E(\mathbb{Q})\otimes\mathbb{Z}_p$; we verify this computationally in all our examples. We study the distribution of shadow lines in $E(\mathbb{Q})\otimes\mathbb{Z}_p$ as $K$ varies, framing conjectures based on the computations we have made. |
| title | Shadow line distributions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.00891 |