Diagonal cycles and Euler systems. II: The Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin \(L\)-functions. (English) Zbl 1397.11090

Let \(E\) be an elliptic curve over \(\mathbb{Q}\). Let \(\rho\) be a \(G_{\mathbb{Q}}:=\mathrm{Gal}({\overline{\mathbb{Q}}/\mathbb{Q}})\)-representation with values in \(\mathrm{Aut}(V_\rho)\), where \(V_\rho\) is a \(r\)-dimension vector space over a number field \(L\), factoring through the Galois group of a finite extension of \(H\) of \(\mathbb{Q}\) with coefficients in a number field \(L\).
To this data one can associate the Hasse-Weil-Artin \(L\)-series \(L(E,\rho,s)\) of \(E\) twisted by \(\rho\), defined on the right half-plane \(\mathrm{Re}(s) > 3/2\) by an absolutely convergent Euler product of degree \(2n\). Let \(E(H)^{\rho} := \mathrm{Hom}_{G_{\mathrm{Q}}} (V_\rho, E(H) \otimes L)\) denote the \(\rho\)-isotypical component of the Mordell-Weil group \(E(H)\), and define the analytic and algebraic rank of the twist of E by \(\rho\) to be the order of vanishing of \(L(E,\rho,s)\) in \(s=1\).
An equivariant refinement of the Birch-Swinnerton-Dyer conjecture predicts that the analytic rank equals the rank of \(E(H)^{\rho}\).
In this paper, this conjecture is studied in the case that \(\rho=\rho_1\otimes \rho_2\), where the \(\rho_i\) are odd, irreducible two-dimensional Artin representations satisfying \(\det(\rho_1)=\det(\rho_2)^{-1}\).
It is now known that the elliptic curve \(E\) and the Artin representations \(\rho_1\) and \(\rho_2\) have associated cuspidal newforms \(f,g,h\) of weights 2, 1, 1, and level \(N_f,N_g,N_h\), respectively. The nebentype character \(\chi=\det(\rho_1)\) of \(g\) is an odd Dirichlet character whose conductor divides both \(N_g\) and \(N_h\). The \(L\)-function \(L(E,\rho,s)\) can be identified with the triple-product convolution \(L(f,g,h,s)\), whose analytic continuation and functional equation are known. It is assumed throughout the article that the level \(N_f\) of f is relatively prime to \(N_gN_h\). This implies that the analytic rank is even. The first main result is that subject to these hypotheses one has that if \(L(E,\rho,1)\neq 0\), then \(E(H)^\rho=0\).
A second result considers the case when the analytic rank is positive (and since it is even it is at least 2). Under some non-vanishing condition on the so-called Garret-Hida \(p\)-adic \(L\)-function associated with \(f,g,h\) the authors show that \[ \dim_{L_p} \mathrm{Sel}_p(E,\rho)\geq 2 \] in this case.
The other two main theorems of the paper are stronger, but much more technical results each of which implies one of the two above mentioned results.
For Part I, see [the authors, Ann. Sci. Éc. Norm. Supér. (4) 47, No. 4, 779–832 (2014; Zbl 1356.11039)].


11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture


Zbl 1356.11039
Full Text: DOI


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