Equivariant Birch and Swinnerton-Dyer conjecture and Bloch-Kato finiteness for elliptic curves twisted by Artin representations $\rho_1\otimes\rho_2$ in the diagonal-cycle setting beyond Heegner points

For an elliptic curve $E/\mathbb{Q}$ and an Artin representation $\rho$, the equivariant Birch and Swinnerton-Dyer conjecture predicts that the order of vanishing of the Hasse-Weil-Artin $L$-function $L(E,\rho,s)$ at $s=1$ equals the dimension of the $\rho$-isotypic component of the Mordell-Weil group of $E$ over the field cut out by $\rho$. Darmon and Rotger (J. Amer. Math. Soc. 30, 2017) proved the rank-zero case for $\rho=\rho_1\otimes\rho_2$ a tensor product of two odd irreducible two-dimensional Artin representations with inverse determinants — a setting that includes ring class characters of real quadratic fields and $A_5$-type representations and lies beyond the reach of Heegner points — and gave conditional rank-two lower bounds depending on non-vanishing of certain non-classical Garrett-Hida $p$-adic $L$-values (values of a three-variable $p$-adic $L$-function outside its range of classical interpolation). A three-variable $\Lambda$-adic family of diagonal-cycle classes has since been constructed by the authors, and anticyclotomic Euler systems of diagonal cycles now yield Bloch-Kato results on the imaginary-dihedral side; but the central questions remain open, namely finiteness of the $\rho$-isotypic Selmer and Tate-Shafarevich groups in analytic rank zero outside the imaginary-dihedral case, the unconditional rank-two statement including the upper bound, and systematic non-vanishing of the non-classical $p$-adic $L$-values.

Background

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $\rho$ be an Artin representation, i.e. a continuous representation of the absolute Galois group of $\mathbb{Q}$, with finite image, over a finite extension $L/\mathbb{Q}$. The Hasse-Weil-Artin $L$-function $L(E,\rho,s)$ is the $L$-function of the twist of $E$ by $\rho$. If $H/\mathbb{Q}$ is the finite Galois extension cut out by $\rho$ and $E(H)^\rho$ denotes the $\rho$-isotypic component of the Mordell-Weil group $E(H)$ of $H$-rational points, the equivariant Birch and Swinnerton-Dyer (BSD) conjecture predicts

$$r_{\mathrm{an}}(E,\rho) \stackrel{?}{=} r(E,\rho),$$

where $r_{\mathrm{an}}(E,\rho)=\mathrm{ord}_{s=1}L(E,\rho,s)$ is the analytic rank and $r(E,\rho)=\dim_L E(H)^\rho$ is the algebraic rank; the classical BSD conjecture is the case $\rho=\mathbf{1}$. The Tate-Shafarevich group $\Sha(E/H)$, the group of everywhere-locally-solvable principal homogeneous spaces of $E$ modulo the trivial ones, measures the failure of the local-to-global principle and is conjectured to be finite; its $\rho$-isotypic part is controlled by the Bloch-Kato Selmer group $\mathrm{Sel}_p(E,\rho)=H^1_{\mathrm{fin}}(\mathbb{Q},V_p(E)\otimes V_\rho)$, where $V_p(E)$ is the $p$-adic Tate module of $E$ and $V_\rho=\rho\otimes_{\mathbb{Q}}\mathbb{Q}_p$: the subgroup of the Galois cohomology $H^1(\mathbb{Q},V_p(E)\otimes V_\rho)$ of classes unramified away from the level and crystalline at $p$ (the standard $p$-adic Hodge-theoretic local condition at $p$).

Darmon and Rotger studied this conjecture for $\rho=\rho_1\otimes\rho_2$, where $\rho_1,\rho_2$ are odd (i.e. complex conjugation acts with determinant $-1$), irreducible, two-dimensional Artin representations with $\det(\rho_1)=\det(\rho_2)^{-1}$, so that $\rho$ is self-dual. By modularity, $E$, $\rho_1$, and $\rho_2$ are attached to newforms $f,g,h$ (normalized Hecke-eigen cusp forms) of weights $2,1,1$, and $L(E,\rho,s)$ identifies with the Garrett-Rankin triple-product $L$-function $L(f,g,h,s)$ — the degree-eight $L$-function of the triple product of the Galois representations attached to $f$, $g$, and $h$ — whose analytic continuation and functional equation are known. When the level $N_f$ of $f$ is coprime to $N_gN_h$, all local root numbers of $L(f,g,h,s)$ — the local constants whose product is the sign of the functional equation — are $+1$, so $r_{\mathrm{an}}(E,\rho)$ is even. If one factor, say $\rho_2$, is induced from a ring class character of an imaginary quadratic field (a character of the Galois group of a ring class field, the natural abelian extension attached to an order of the quadratic field), this is the classical Heegner setting: Heegner points, i.e. special points on modular curves arising from the theory of complex multiplication, combined with the Gross-Zagier formula (which expresses heights of Heegner points as first derivatives of $L$-functions) and Kolyvagin's method of Euler systems (norm-compatible families of global cohomology classes converted by descent into bounds on Selmer groups), give the rank-zero statement. These tools fail when the quadratic field is real — the conjectural replacement, the analytically defined Stark-Heegner points, are not even known to be algebraic — and for exotic projective images such as $A_5$ (the alternating group on five letters).

The method of the source rests on the Gross-Kudla-Schoen diagonal cycle, the modified diagonal on the triple product of three modular curves, a homologically trivial algebraic cycle whose arithmetic reflects the triple product $L$-function; on its $p$-adic Abel-Jacobi images; and on their variation in Hida families, which are $p$-adic analytic families of ordinary modular forms interpolating a fixed eigenform at all classical weights. The authors construct a one-variable $\Lambda$-adic cohomology class $\kappa(\mathbf{f},\mathbf{g}\mathbf{h})$ with the weight-two form $f$ held fixed; its specializations at the unbalanced point of weights $(2,1,1)$ produce four generalized Kato classes in $H^1(\mathbb{Q},V_p(E)\otimes V_\rho)$, so named because Kato's Euler system of Beilinson elements is the case where $g,h$ are weight-one Eisenstein series. An explicit reciprocity law relates the Bloch-Kato dual exponential of these classes (the $p$-adic counterpart of the classical exponential map, evaluating Galois cohomology classes in the de Rham cohomology of $V_p(E)\otimes V_\rho$) to the Garrett-Hida $p$-adic $L$-function, the $p$-adic analytic function on the triple product of weight spaces interpolating central critical values $L(f_x,g_y,h_z,c)$ built by pairing Hida families against $p$-adic test vectors. The main theorems are: Theorem A, if $L(E,\rho,1)\neq 0$ then $E(H)^\rho=0$; and Theorems B and D, if $L(E,\rho,1)=0$ and one of four non-classical $p$-adic $L$-values — values of the Garrett-Hida $p$-adic $L$-function at points outside its range of classical interpolation — is non-zero, then $\dim_{L_p}\mathrm{Sel}_p(E,\rho)\ge 2$, where $L_p=L\otimes_{\mathbb{Q}}\mathbb{Q}_p$ is the coefficient field of the Selmer group, with two linearly independent classes constructed in the Selmer group. The proof of Theorem A does not bound the Selmer or Tate-Shafarevich group, and the paper states that extending Kato's and Lei-Loeffler-Zerbes' finiteness refinements to the diagonal-cycle setting 'would be of great interest'.

Problem Statement

Let $E/\mathbb{Q}$ be an elliptic curve and let $\rho_1,\rho_2:G_{\mathbb{Q}}\to\mathrm{GL}_2(L)$ be odd, irreducible, two-dimensional Artin representations over a finite extension $L/\mathbb{Q}$ with $\det(\rho_1)=\det(\rho_2)^{-1}$; set $\rho=\rho_1\otimes\rho_2$, and let $H/\mathbb{Q}$ be the finite Galois extension cut out by $\rho$. Let $f,g,h$ be the newforms of weights $2,1,1$ attached to $E,\rho_1,\rho_2$, assume $(N_f,N_gN_h)=1$ so that all local root numbers of $L(E,\rho,s)=L(f,g,h,s)$ are $+1$, fix an odd prime $p\nmid N_fN_gN_h$ at which $E$ is ordinary (i.e. the trace of Frobenius $a_p(E)=p+1-|E(\mathbb{F}_p)|$ is a $p$-adic unit), and write $\mathrm{Sel}_p(E,\rho)=H^1_{\mathrm{fin}}(\mathbb{Q},V_p(E)\otimes V_\rho)$ for the Bloch-Kato Selmer group. Establish the remaining cases of the equivariant Birch and Swinnerton-Dyer conjecture for these Hasse-Weil-Artin $L$-functions, namely:

1. In analytic rank zero, prove that $L(E,\rho,1)\neq 0$ implies not only $E(H)^\rho=0$ (Darmon-Rotger's Theorem A) but also the vanishing — and in particular the finiteness — of $\mathrm{Sel}_p(E,\rho)$, equivalently the finiteness of the $\rho$-isotypic Tate-Shafarevich group $\Sha(E/H)^\rho[p^\infty]$, for every $\rho=\rho_1\otimes\rho_2$ outside the imaginary-dihedral case, i.e. in the settings where no Heegner-point Euler system exists; this entire class is the required scope, and the cases named below are examples of it rather than an exhaustive list: when $\rho_1$ or $\rho_2$ is induced from a ring class character of a real quadratic field, or when $\rho$ has exotic (e.g. $A_5$) projective image.

2. In analytic rank two, prove unconditionally that $r_{\mathrm{an}}(E,\rho)=r(E,\rho)$ when $L(E,\rho,s)$ vanishes to exact order two at $s=1$: remove the non-vanishing hypothesis on the four non-classical Garrett-Hida $p$-adic $L$-values in Theorems B and D of the source, and prove the upper bound $\dim_{L_p}\mathrm{Sel}_p(E,\rho)=2$, where $L_p=L\otimes_{\mathbb{Q}}\mathbb{Q}_p$ is the coefficient field of the Selmer group.

The single objective underlying both deliverables is the equivariant BSD conjecture $r_{\mathrm{an}}(E,\rho)=r(E,\rho)$, together with the finiteness of the associated Selmer and Tate-Shafarevich groups predicted by the Shafarevich-Tate conjecture, for the class of Artin representations $\rho=\rho_1\otimes\rho_2$ as above — the setting introduced by Darmon-Rotger's diagonal-cycle method.

References

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