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Balanced triple product $p$-adic $L$-functions and Stark points

Luca Dall'Ava, Aleksander Horawa

Abstract

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.

Balanced triple product $p$-adic $L$-functions and Stark points

Abstract

Let be an elliptic curve over and be two odd Artin representations. We use -adic methods to investigate the part of the Mordell-Weil group on which the Galois group acts via . When the rank of the group is two, Darmon-Lauder-Rotger used a dominant triple product -adic -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 -adic -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 -adic formula for a point which should conjecturally lie in . 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.
Paper Structure (69 sections, 47 theorems, 320 equations, 3 figures, 9 tables)

This paper contains 69 sections, 47 theorems, 320 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Construction of the balanced $p$-adic $L$-function
  3. The Elliptic Stark Conjecture
  4. Organization of the paper
  5. Local JL correspondence and test vectors
  6. Local $\epsilon$-factors and $L$-factors
  7. Local Jacquet--Langlands correspondence
  8. Local integral for zero, two, and three supercuspidal representations of level $\ell^2$
  9. Case (1): zero supercuspidal representations
  10. Case (2): two supercuspidal representations
  11. Case (3): three supercuspidal representations
  12. Global JL correspondence and test vectors
  13. A remark on the structure of quaternion algebras at ramified primes
  14. Residually inert orders
  15. Lifting characters
  16. ...and 54 more sections

Key Result

Theorem 1

Under the above hypotheses, there exist open admissible neighborhoods $\mathcal{U}_g$ and $\mathcal{U}_h$ of the classical weight $1$ in the weight space, and a (square root) balanced triple product $p$-adic $L$-function $\mathcal{L}_p^{\operatorname{bal}}\colon \mathcal{U}_g \times \mathcal{U}_h \t where $c = (\ell + m)/2$ is the center of the functional equation, and are as in Hsieh2021 and Hsi

Figures (3)

  • Figure 1: We fix $k = 2$, and consider two weights $\ell, m$ with $\ell + m \equiv 0 \pmod 2$. We indicate the four regions for the weights $\ell$, $m$; $\Sigma^F$ is the region where $F$ is dominant, and $\Sigma^{\operatorname{bal}}$ is the region where the weights are balanced and $\ell, m \geq 2$. We also indicate the point $(\ell, m) = (1,1)$ where our Elliptic Stark Conjecture \ref{['conj:ES']} applies.
  • Figure 2: The following diagram shows the diagram of infinity types $(\kappa_1, \kappa_2)$ for the characters in $\Sigma$. We indicate interpolation region $\Sigma_K^{(2)}$ for the Katz $p$-adic $L$-function in blue \ref{['eqn:Katz_interpolation']}, the functional equation \ref{['eqn:Katz_functional_eqn']} with the axis of symmetry given by the dotted line, and the point where the $p$-adic Kroenecker limit formula \ref{['eqn:p-adic_Kronecker']} is valid. The dashed line is the central critical line
  • Figure 3: The following diagram shows the diagram of infinity types $(\kappa_1, \kappa_2)$ for the character in $\Sigma$. We indicate interpolation region $\Sigma_{f, c}^{(2)}$ for the BDP $p$-adic $L$-function in blue \ref{['eqn:Katz_interpolation']}, the expected functional equation with the axis of symmetry given by the dotted line, and the point where the $p$-adic Gross--Zagier formula \ref{['eqn:BDP_Gross_Zagier']} is valid.

Theorems & Definitions (129)

  • Theorem 1: Corollary \ref{['cor:interpolation']}
  • Proposition 2: Proposition \ref{['prop: mult-1 with varpi_d']}
  • Conjecture 3: Rank one Elliptic Stark Conjecture \ref{['conj:ES']}
  • Theorem 4: Theorem \ref{['thm:CM']}
  • Proposition 2.1: Prasad
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • ...and 119 more