On the algebraicity of polyquadratic plectic points

Since $E/F$ is a CM extension, Assumption 1.1 implies that the sign of the functional equation of the $L$-function $L(A/E,s)$ of the base-change of $A_{/F}$ to $E$ is equal to $\varepsilon(A/E)=(-1)^{r}$.

We introduce the following useful notations. First, if $H$ is any commutative prodiscrete group, we denote by $\widehat{H}$ the torsion-free part of its pro-$p$ completion. Second, if $M$ is an abelian group and $\Omega$ a field of characteristic zero, we use $M_{\Omega}$ and $\mathop{\mathchoice{\raisebox{0.0pt}{$\textstyle\bigwedge$}^{\mspace{-2.0mu}r}\kern-0.5pt}{\raisebox{0.0pt}{$\textstyle\bigwedge$}^{\mspace{-2.0mu}r}\kern-0.5pt}{\raisebox{0.0pt}{$\scriptstyle\bigwedge$}^{\mspace{-2.0mu}r}\kern-0.5pt}{\raisebox{0.0pt}{$\scriptscriptstyle\bigwedge$}^{\mspace{-2.0mu}r}\kern-0.5pt}}(M_{\Omega})$ to respectively denote the tensor product $M\otimes_{\mathbb{Z}}\Omega$, and the $r$-th exterior power of the $\Omega$-vector space $M_{\Omega}$.

Consider the tensor product

Under the running assumptions, the plectic Stark–Heegner point (for the trivial character) is

as constructed in ([FG21], Section 4.4), where it was denoted $\mathrm{P}^{\mathbbm{1}}_{A}$. The notion of algebraicity for plectic Stark–Heegner points is formulated in terms of a determinant map. Writing $\iota_{\mathfrak{p}}\colon E\hookrightarrow E_{\mathfrak{p}}$ for the canonical embedding for every $\mathfrak{p}\in S$, we can consider the homomorphism

given by

As usual, $r_{\mathrm{alg}}(A/E)$ denotes the rank of the finitely generated abelian group $A(E)$, and $r_{\mathrm{an}}(A/E)$ the order of vanishing of the $L$-function $L(A/E,s)$ at $s=1$.

There are also plectic Stark–Heegner points associated to non-trivial anticyclotomic characters of $E/F$. Aiming for clarity in the introduction, we discuss them only in the body of the paper.

Assumption 1.5 implies that the elliptic curve $A_{/F}$ is modular by quadratic base change for Hilbert modular forms. Moreover, we deduce that the cardinality $r=\lvert S\rvert$ equals $2^{t}$, and that the prime $\wp$ is inert in $E_{\circ}/F_{\circ}$. By a small abuse of notation, we denote by $\wp$ the unique prime of $E_{\circ}$ lying above $\wp$. Since $\wp$ is completely split in $F/F_{\circ}$, the elliptic curve $A_{\circ/F_{\circ}}$ has multiplicative reduction at $\wp$. Furthermore, if we set $a_{\wp}=1$ (resp. $a_{\wp}=-1$) in case $A_{\circ/F_{\circ}}$ has split (resp. non-split) multiplicative reduction, we have

Now, write the conductor $\mathfrak{f}_{\circ}$ of $A_{\circ/F_{\circ}}$ as

where $\mathfrak{n}_{\circ}^{\sharp}$ is divisible by every prime divisor of $\mathfrak{f}_{\circ}$ split in $E_{\circ}/F_{\circ}$.

Under Assumption 1.7, the sign of the functional equation for $A_{\circ/E_{\circ}}$ equals $\varepsilon(A_{\circ}/E_{\circ})=-1$. Hence, the BSD–conjecture predicts that $A_{\circ}(E_{\circ})$ is non-torsion. Moreover, for any character $\eta\colon\mathfrak{G}\to\{\pm 1\}$, the twist $A^{\eta}_{\circ/F_{\circ}}$ also fulfils the generalized Heegner hypothesis and we have

because every prime ramified in $F/F_{\circ}$ splits in $E_{\circ}/F_{\circ}$ by Assumption 1.5. Thus, we expect that

with equality if and only if $r_{\mathrm{alg}}(A^{\eta}_{\circ}/E_{\circ})=1$ for every character $\eta\colon\mathfrak{G}\to\{\pm 1\}$.

Under our running assumptions (1.1, 1.5, 1.7), we deduce the following theorems about plectic points from the known properties of Heegner points, the $p$-adic uniformization of Shimura curves, and a factorization of anticyclotomic $p$-adic $L$-functions (Corollary 4.4).

Now, set $A^{+}_{/F}=A_{/F}$ and denote by $A^{-}_{/F}$ the quadratic twist of $A_{/F}$ with respect to the extension $E/F$. We partition $S=S^{+}\cup S^{-}$ by declaring that the subset $S^{+}\subseteq S$ contains all the primes in $S$ of split multiplicative reduction for $A^{+}_{/F}$, and define $\varrho_{A}(S):=\max\big{\{}r_{\mathrm{alg}}(A^{\pm}/F)+\lvert S^{\pm}\rvert\big{\}}.$

We note that in the main body of this article, we prove generalizations of Theorems A, B for plectic Stark–Heegner points associated to anticyclotomic characters of $E/F$ that are restrictions of anticyclotomic characters of $E_{\circ}/F_{\circ}$.

The following lemma can be easily deduced from (2) and (3).

Now, both the augmentation ideal $I(G)$ and the relative versions $I_{Q}(G)$ are clearly stable under $(-)^{\vee}$. Equation (4) implies that $(-)^{\vee}$ induces multiplication with $-1$ on $I(G)/I(G)^{2}$ and, thus, it induces multiplication with $(-1)^{n}$ on $I(G)^{n}/I(G)^{n+1}$. This observation readily implies the following relative statement: under the assumptions of Lemma 2.3 the following diagram commutes

We fix an $\mathcal{O}_{F}$-ideal $\mathfrak{c}$ coprime to the conductor of $A_{/F}$. Let $E_{\mathfrak{c}}/E$ denote the anticyclotomic extension of conductor $\mathfrak{c}$ defined in ([FG21], Section 4.2.1) with Galois group $\mathcal{G}_{\mathfrak{c}}=\operatorname{Gal}(E_{\mathfrak{c}}/E)$. Recall that for any character $\chi\colon\mathcal{G}_{\mathfrak{c}}\to\mathbb{C}^{\times}$ its conductor is the maximal divisor $\mathfrak{c}_{\chi}$ of $\mathfrak{c}$ such that $\chi$ factors through $\mathcal{G}_{\mathfrak{c}}\twoheadrightarrow\mathcal{G}_{\mathfrak{c}_{\chi}}$. We write $\mathbb{Q}_{\chi}$ for the extension of $\mathbb{Q}$ generated by the values of $\chi$. With a small change of notation compared to ([FG21], Section 4.4), we denote the plectic Stark–Heegner point associated to an anticyclotomic character $\chi$ by

It follows easily from the construction of plectic Stark–Heegner points that there is an element

such that the equality $\chi(\mathrm{P}_{A,S}^{\mathcal{G}_{\mathfrak{c}}})=\mathrm{P}_{A,S}^{\chi}$ holds for any character $\chi\colon\mathcal{G}_{\mathfrak{c}}\to\mathbb{C}^{\times}$ of conductor $\mathfrak{c}$. A similar statement also holds for plectic $p$-adic invariants ([FG21], Section 4.2): there is an element

such that the equality $\chi(\mathrm{Q}_{A,S}^{\mathcal{G}_{\mathfrak{c}}})=\mathrm{Q}_{A,S}^{\chi}$ holds for any character $\chi\colon\mathcal{G}_{\mathfrak{c}}\to\mathbb{C}^{\times}$ of conductor $\mathfrak{c}$. Moreover, the two elements are related by the following equation

We fix embeddings $\iota_{\mathfrak{p}}\colon\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_{p}$ and $\iota_{\nu}\colon\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$ respectively inducing the $p$-adic prime $\mathfrak{p}$ and an Archimedean place $\nu$ of $F$. Since $\mathfrak{p}$ is inert in $E/F$, it splits completely in the anticyclotomic extension $E_{\mathfrak{c}}/E$. Thus, the embedding $\iota_{\mathfrak{p}}$ restricts to an embedding $\iota_{\wp}\colon E_{\mathfrak{c}}\hookrightarrow E_{\mathfrak{p}}$ inducing an injective homomorphism

We write $r_{\mathrm{alg}}(A/E,\chi)$ for the $\mathbb{Q}_{\chi}$-dimension of the $\chi$-component $A(E_{\mathfrak{c}})^{\chi}_{\mathbb{Q}_{\chi}}$ and define $r_{\mathrm{an}}(A/E,\chi)$ to be the order of vanishing of the $L$-function $L(A/E,\chi,s)$ at $s=1$. As explained in ([GMM20], Appendix A.1.2), there exists a Heegner point $P_{A,\mathfrak{c}}\in A(E_{\mathfrak{c}})$ of conductor $\mathfrak{c}$, arising from a Shimura curve associated to the $F$-quaternion algebra ramified exactly at $\mathfrak{p}\mathfrak{n}^{\flat}$ and all Archimedean places different from $\nu$, such that the image of

under $\iota_{A,\mathfrak{p}}$ is a non-zero rational multiple of the plectic Stark–Heegner point associated to the triple $(A,E,\{\mathfrak{p}\})$, i.e., there exists $k_{A,\mathfrak{p}}^{\chi}\in\mathbb{Q}^{\times}$ such that:

Interestingly, ([BG18], Proposition 5.6) allows us to describe the behaviour of $\mathscr{L}_{S}(A/E)_{\mathfrak{c}}$ under the involution $(-)^{\vee}$ in terms of the global root number $\varepsilon(A/F)$ of $A_{/F}$ and a product of local root numbers $\varepsilon_{S}(A/F):=\prod_{\mathfrak{p}\in S}\varepsilon_{\mathfrak{p}}(A/F)$.