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Computation of the unipotent Albanese map on elliptic and hyperelliptic curves. (English. French summary) Zbl 1472.11173

Let \(X/K\) be an elliptic or hyperelliptic curve over a number field, let \(C = X \smallsetminus \{\infty\}\) be an odd-degree affine model, and let \(v\) be a place of \(K\). The article in question provides a number of algorithms for computing with the universal unipotent connection \(\mathcal U\) on \(C\). Let \(\mathcal U_{n}\) be the maximal quotient of \(\mathcal U\) with unipotency index \(n\). Given a choice of either a rational basepoint or a tangential basepoint, and for all \(n\), these algorithms explicitly compute:
1.
the \(n\)-{th} \(p\)-adic de Rham period map \(j_{n}^{\mathrm{dr}}: X(K_{v}) \to F^{0}U_n^{\mathrm{dr}}\backslash U_n^{\mathrm{dr}}\) mapping the \(K_{v}\) points of \(C\) to a certain unipotent quotient of the deRham fundamental group (This is also called the unipotent Albanese map.);
2.
an extension of \(\mathcal U_n\) to a unipotent connection with logarithmic singularities at \(\infty\) on \(X\); and
3.
the Hodge filtration on \(\mathcal U_{n}\), in terms of iterated \(p\)-adic Coleman integrals.
The algorithms are generally defined inductively, first computing structures on \(\mathcal U_{n-1}\) and then extending to \(\mathcal U_{n}\). Assuming a certain lifting condition, the author shows that \(j_{n}^{\mathrm{dr}}\) admits a particularly simple and compact description in terms of \(j_{n-1}^{\mathrm{dr}}\) The author uses their algorithm to compute several of these maps/structures beyond what was previously known, including explicitly determining \(j_{4}^{\mathrm{dr}}\) for an elliptic curve with a rational basepoint, \(j_{3}^{\mathrm{dr}}\) for elliptic curves with either a rational basepoint or a tangential basepoint at infinity and \(j_{2}^{\mathrm{dr}}\) for odd degree hyperelliptic curves with either a rational basepoint or a tangential basepoint at infinity. The article also contains a clear background section providing an extended exposition of unipotent connections and their logarithmic extensions which are an excellent reference for readers less familiar with the field.
A primary motivation for studying unipotent Albanese maps \(j_{n}^{\mathrm{dr}}\), the Hodge filtration on \(\mathcal U_{n}\), and Frobenius-invariant paths, comes from Minhyong Kim’s “nonabelian Chabauty” program (initiated in [M. Kim, Invent. Math. 161, No. 3, 629–656 (2005; Zbl 1090.14006); Publ. Res. Inst. Math. Sci. 45, No. 1, 89–133 (2009; Zbl 1165.14020)]) to compute \(K\)-rational points on curves whose genus is not strictly greater than the rank of their Jacobian. (The ‘abelian’ Chabauty-Coleman method typically requires the Jacobian to have small rank.) To carry out Kim’s method, given a curve \(X\), one identifies \(X(K)\) inside of the intersection of the image \(j_{n}^{\mathrm{dr}}(X(K_{v}))\) and the ‘global Selmer variety’ of level \(n\). Under a certain dimension hypothesis, this intersection is finite. This hypothesis is satisfied in several cases for sufficiently large \(n\), including for \(X = \mathbb P^1\smallsetminus \{0,1,\infty\}\) and for \(X\) a solvable cover of \(\mathbb P^1\). See, for example [J. Coates and M. Kim, Kyoto J. Math. 50, No. 4, 827–852 (2010; Zbl 1283.11092)], [J. S. Ellenberg, and D. R. Hast, “Rational points on solvable curves over \(\mathbb Q\) via non-abelian Chabauty.” Int. Math. Res. Not. IMRN 2021 (to appear)].
Kim’s strategy is also amenable to explicit computation. To date, most work on carrying out Kim’s strategy explicitly has focused on the case \(n = 2\). In this case, the global Selmer variety can be described in terms of \(p\)-adic height functions. J. S. Balakrishnan and N. Dogra have also carried out a detailed study of such objects and \(j_{2}^{\mathrm{dr}}\) to prove explicit bounds on the number of rational points on hyperelliptic curves beyond the ‘abelian’ Chabauty-Coleman rank versus dimension bounds [Duke Math. J. 167, No. 11, 1981–2038 (2018; Zbl 1401.14123)]. Extending such results to more general curves and in combination with explicit computations of \(j_{2}^{\mathrm{dr}}\), Balakrishnan-Dogra-Muller-Tuitman-Vonk [J. Balakrishnan et al., Ann. Math. (2) 189, No. 3, 885–944 (2019; Zbl 1469.14050)] computed the set of \(\mathbb Q\)-points on the ‘cursed’ split Cartan modular curve \(X_{s}(13)\) of level \(13\). Since then, the methods have been used to compute rational points on several other modular curves which were previously inaccessible. The work of the article under review is a step towards one side of the effort needed to extend such algorithms to even more general classes of curves.

MSC:

11G05 Elliptic curves over global fields
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
11Y40 Algebraic number theory computations

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References:

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