A \(p\)-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties (Q2792352)

Classically, the Birch and Swinnerton-Dyer (BSD) conjecture connects arithmetic invariants of an abelian variety over a number field to the order of zero and the leading coefficient of the Taylor expansion of its \(L\)-function at the central point. The arithmetic invariants include the real period, the regulator, and the order of the Shafarevich-Tate group of the given abelian variety.NEWLINENEWLINENEWLINE\textit{B. Mazur} et al. [Invent. Math. 84, 1--48 (1986; Zbl 0699.14028)] gave a \(p\)-adic analogue of this conjecture for an elliptic curve over the rationals and a prime \(p\) of good ordinary or multiplicative reduction. The authors formulate a generalization of their conjecture to higher dimensional modular abelian varieties in the good ordinary case. This generalization encompasses the classical BSD conjecture in rank \(0\) because of the interpolation property of \(p\)-adic \(L\)-series.NEWLINENEWLINENEWLINELet \(A/\mathbb{Q}\) be a modular abelian variety associated to a new form \(f\). The authors define a \(p\)-adic \(L\)-function \(L_p(A,s)\) to be the product of \(p\)-adic functions \(L_p(f^\sigma,s)\) associated to Galois conjugate of \(f\). Although there is no canonical way of normalizing \(L_p(f^\sigma,s)\), the authors make a careful choice of a set of periods \(\Omega^+_{f^\sigma}\) (called Shimura periods in the paper) whose product is the real period \(\Omega^+_A\), so that the \(p\)-adic series associated to the \(p\)-adic function \(L_p(A,s)\) has the expected interpolation property. This allows them to define one side of the equation in their conjecture, the leading coefficient of the \(p\)-adic \(L\)-series at zero.NEWLINENEWLINENEWLINEOn the other side, assuming that the order of the Shafarevich-Tate group is given by the classical BSD conjecture, the missing piece is the \(p\)-adic regulator. The authors make use of definitions of \(p\)-adic height pairings on abelian varieties in the literature, and give an algorithm -- in the cases they need -- to compute the \(p\)-adic regulator in terms of the height pairing. They provide numerical evidence of their conjecture for the modular abelian varieties of dimension 2 and rank 2 in [\textit{E. V. Flynn} et al., Math. Comput. 70, No. 236, 1675--1697 (2001; Zbl 1020.11043)] and the Jacobian of a twist of \(X_0(31)\) of rank 4.NEWLINENEWLINENEWLINEIt would be interesting to formulate a conjecture that generalizes to the case where the modular abelian variety does not have good ordinary reduction. But this seems difficult for higher dimensions, because even in dimension 1, one needs to modify the BSD conjecture in the exceptional case that an elliptic curve has split multiplicative reduction at \(p\), see [Mazur et al., loc. cit.].