Explicit arithmetic intersection theory and computation of Néron-Tate heights. (English) Zbl 1426.14006
Let \(A\) be an abelian variety defined over a global field \(K\) and \(c\) an ample symmetric divisor class on \(A\). The aim of the paper is the computation of the Néron-Tate height induced by \(c\) (a quadratic form on \(A(K)\)) \(\hat{h}_c(P)\) or, what is the same, the bilinear pairing \(\hat{h}_c(P,Q)\), see [A. Néron, Ann. Math. (2) 82, 249–331 (1965; Zbl 0163.15205)] and their application to the computation of the regulator of \(A\) and to give numerical evidence for the conjecture of Birch and Swinnerton-Dyer for the Jacobian of the Cartan modular curve of level 13.
The paper restricts to the case \(A\) the Jacobian of a smooth projective geometrically connected curve \(C\), \(c\) twice the class of a symmetric theta divisor \(\theta\) and \(K=\mathbb{Q}\) (although “our algorithm can be generalized easily to work over general global fields”).
The algorithm takes advantage of an equality due to Faltings and Hriljac (Theorem 4.1): \(\hat{h}_{2\theta}([D],[E])=-\sum_{v\in M_K}\langle D,E\rangle_v\), where \(D,E\) are divisors of degree 0 on \(C\) without common component, \(M_K\) the set of places of \(K\) and \(\langle D,E\rangle_v\) the local Néron pairing at \(v\) (since \(K=Q\) \(v\) is a \(p\)-adic valuation or the archimedean one). In the case \(C\) a hyperelliptic curve that computation was already done by D. Holmes [J. Number Theory 132, No. 6, 1295–1305 (2012; Zbl 1239.14019)] and J. S. Müller [Math. Comput. 83, No. 285, 311–336 (2014; Zbl 1322.11074)].
Section 2 gives an algorithm to compute the non-archimedean local Néron pairings (in the authors words “our main contribution”) and Section 3 deals with the archimedean case.
Section 4 discusses the computation of \(\hat{h}_{2\theta}([D],[E])\) using Theorem 4.1 and in particular identifies a set of places outside which the local Néron pairing of \(D\) and \(E\) vanished.
Finally Section 5 provides examples of the implementation, using the package Magma, of the algorithm for several curves including the split Cartan modular curve of level 13; it finds the regulator up to an integral square factor giving numerical evidence that the BSD conjecture holds up to an integral square.
The paper restricts to the case \(A\) the Jacobian of a smooth projective geometrically connected curve \(C\), \(c\) twice the class of a symmetric theta divisor \(\theta\) and \(K=\mathbb{Q}\) (although “our algorithm can be generalized easily to work over general global fields”).
The algorithm takes advantage of an equality due to Faltings and Hriljac (Theorem 4.1): \(\hat{h}_{2\theta}([D],[E])=-\sum_{v\in M_K}\langle D,E\rangle_v\), where \(D,E\) are divisors of degree 0 on \(C\) without common component, \(M_K\) the set of places of \(K\) and \(\langle D,E\rangle_v\) the local Néron pairing at \(v\) (since \(K=Q\) \(v\) is a \(p\)-adic valuation or the archimedean one). In the case \(C\) a hyperelliptic curve that computation was already done by D. Holmes [J. Number Theory 132, No. 6, 1295–1305 (2012; Zbl 1239.14019)] and J. S. Müller [Math. Comput. 83, No. 285, 311–336 (2014; Zbl 1322.11074)].
Section 2 gives an algorithm to compute the non-archimedean local Néron pairings (in the authors words “our main contribution”) and Section 3 deals with the archimedean case.
Section 4 discusses the computation of \(\hat{h}_{2\theta}([D],[E])\) using Theorem 4.1 and in particular identifies a set of places outside which the local Néron pairing of \(D\) and \(E\) vanished.
Finally Section 5 provides examples of the implementation, using the package Magma, of the algorithm for several curves including the split Cartan modular curve of level 13; it finds the regulator up to an integral square factor giving numerical evidence that the BSD conjecture holds up to an integral square.
Reviewer: Juan Tena Ayuso (Valladolid)
MSC:
| 14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |
| 11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |
| 11G50 | Heights |
| 37P30 | Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems |
| 11G10 | Abelian varieties of dimension \(> 1\) |
Keywords:
abelian variety; Jacobian; Néron-Tate height; local Néron pairing; conjecture of Birch and Swinnerton-Dyer; Cartan modular curve of level 13Software:
ArbReferences:
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