Balanced triple product $p$-adic $L$-functions and Stark points
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910358403284992 |
|---|---|
| author | Dall'Ava, Luca Horawa, Aleksander |
| author_facet | Dall'Ava, Luca Horawa, Aleksander |
| contents | Let $E$ be an elliptic curve over $\mathbb{Q}$ and $\varrho_1, \varrho_2 \colon \mathrm{Gal}(H/\mathbb{Q}) \to \mathrm{GL}_2(L)$ be two odd Artin representations. We use $p$-adic methods to investigate the part of the Mordell-Weil group $E(H) \otimes L$ on which the Galois group acts via $\varrho_1 \otimes \varrho_2$. When the rank of the group is two, Darmon-Lauder-Rotger used a dominant triple product $p$-adic $L$-function to study this group, and gave an Elliptic Stark Conjecture which relates its value outside of the interpolation range to two Stark points and one Stark unit. Our paper achieves a similar goal in the rank one setting. We first generalize Hsieh's construction of a 3-variable balanced triple product $p$-adic $L$-function in order to allow Hida families with classical weight one specializations. We then give an Elliptic Stark Conjecture relating its value outside of the interpolation range to a Stark point and two Stark units. As a consequence, we give an explicit $p$-adic formula for a point which should conjecturally lie in $E(H) \otimes L$. We prove our conjecture for dihedral representations associated with the same imaginary quadratic field. This requires a generalization of the results of Bertolini-Darmon-Prasanna which we prove in the appendix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05183 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Balanced triple product $p$-adic $L$-functions and Stark points Dall'Ava, Luca Horawa, Aleksander Number Theory 11G05, 11G40, 11F11, 11F12, 11R23 Let $E$ be an elliptic curve over $\mathbb{Q}$ and $\varrho_1, \varrho_2 \colon \mathrm{Gal}(H/\mathbb{Q}) \to \mathrm{GL}_2(L)$ be two odd Artin representations. We use $p$-adic methods to investigate the part of the Mordell-Weil group $E(H) \otimes L$ on which the Galois group acts via $\varrho_1 \otimes \varrho_2$. When the rank of the group is two, Darmon-Lauder-Rotger used a dominant triple product $p$-adic $L$-function to study this group, and gave an Elliptic Stark Conjecture which relates its value outside of the interpolation range to two Stark points and one Stark unit. Our paper achieves a similar goal in the rank one setting. We first generalize Hsieh's construction of a 3-variable balanced triple product $p$-adic $L$-function in order to allow Hida families with classical weight one specializations. We then give an Elliptic Stark Conjecture relating its value outside of the interpolation range to a Stark point and two Stark units. As a consequence, we give an explicit $p$-adic formula for a point which should conjecturally lie in $E(H) \otimes L$. We prove our conjecture for dihedral representations associated with the same imaginary quadratic field. This requires a generalization of the results of Bertolini-Darmon-Prasanna which we prove in the appendix. |
| title | Balanced triple product $p$-adic $L$-functions and Stark points |
| topic | Number Theory 11G05, 11G40, 11F11, 11F12, 11R23 |
| url | https://arxiv.org/abs/2403.05183 |