Generalized triple product $p$-adic $L$-functions and rational points on elliptic curves
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909991092355072 |
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| author | Marannino, Luca |
| author_facet | Marannino, Luca |
| contents | We generalize and simplify the constructions of Darmon-Rotger and Hsieh of an unbalanced triple product $p$-adic $L$-function $\mathscr{L}_p^f(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ attached to a triple $(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ of $p$-adic families of modular forms, allowing more flexibility for the choice of $\boldsymbol{g}$ and $\boldsymbol{h}$. Assuming that $\boldsymbol{g}$ and $\boldsymbol{h}$ are families of theta series of infinite $p$-slope, we prove a factorization of (an improvement of) $\mathscr{L}_p^f(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ in terms of two anticyclotomic $p$-adic $L$-functions. As a corollary, when $\boldsymbol{f}$ specializes in weight $2$ to the newform attached to an elliptic curve $E$ over $\mathbb{Q}$ with multiplicative reduction at $p$, we relate certain Heegner points on $E$ to certain $p$-adic partial derivatives of $\mathscr{L}_p^f(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ evaluated at the critical triple of weights $(2,1,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06565 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalized triple product $p$-adic $L$-functions and rational points on elliptic curves Marannino, Luca Number Theory 11F67 (Primary) 11F80, 11G40 (Secondary) We generalize and simplify the constructions of Darmon-Rotger and Hsieh of an unbalanced triple product $p$-adic $L$-function $\mathscr{L}_p^f(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ attached to a triple $(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ of $p$-adic families of modular forms, allowing more flexibility for the choice of $\boldsymbol{g}$ and $\boldsymbol{h}$. Assuming that $\boldsymbol{g}$ and $\boldsymbol{h}$ are families of theta series of infinite $p$-slope, we prove a factorization of (an improvement of) $\mathscr{L}_p^f(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ in terms of two anticyclotomic $p$-adic $L$-functions. As a corollary, when $\boldsymbol{f}$ specializes in weight $2$ to the newform attached to an elliptic curve $E$ over $\mathbb{Q}$ with multiplicative reduction at $p$, we relate certain Heegner points on $E$ to certain $p$-adic partial derivatives of $\mathscr{L}_p^f(\boldsymbol{f},\boldsymbol{g},\boldsymbol{h})$ evaluated at the critical triple of weights $(2,1,1)$. |
| title | Generalized triple product $p$-adic $L$-functions and rational points on elliptic curves |
| topic | Number Theory 11F67 (Primary) 11F80, 11G40 (Secondary) |
| url | https://arxiv.org/abs/2312.06565 |