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Explicit arithmetic of Jacobians of generalized Legendre curves over global function fields. (English) Zbl 1465.11002

Memoirs of the American Mathematical Society 1295. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4219-4/pbk; 978-1-4704-6253-6/ebook). v, 131 p. (2020).
In this text, the authors study the Jacobian \(J\) of the smooth projective curve \(C\) of genus \(r-1\) with affine model \[y^r=x^{r-1}(x+1)(x+t),\] over the function field \(\mathbb{F}_p(t)\), when \(p\) is prime and \(r\geq2\) is an integer prime to \(p\). When \(q\) is a power of \(p\) and \(d\) is a positive integer, they compute the \(L\)-function of \(J\) over \(\mathbb{F}_q(t^{1/d})\) and show that the Birch and Swinnerton-Dyer conjecture holds for \(J\) over \(\mathbb{F}_q(t^{1/d})\). When \(d\) is divisible by \(r\) and of the form \(p^\nu+1\), and \(K_d:=\mathbb{F}_p(\mu_d,t^{1/d})\), they write down explicit points in \(J(K_d)\), show that they generate a subgroup \(V\) of rank \((r-1)(d-2)\) whose index in \(J(K_d)\) is finite and a power of \(p\), and show that the order of the Tate-Shafarevich group of \(J\) over \(K_d\) is \([J(K_d):V]^2\). When \(r>2\), the authors prove that the new part of \(J\) is isogenous over \(\overline{F_p(t)}\) to the square of a simple abelian variety of dimension \(\phi(r)/2\) with endomorphism algebra \(\mathbb{Z}[\mu_r] ^+\). For a prime \(l\) with \(l\nmid pr\), they prove that \(J[l](L)=\{0\}\) for any abelian extension \(L\) of \(\overline{F}_p(t)\). This monograph is organized as follows. Chapter 1, deals with the curve, explicit divisors, and relations. The authors give basic information about the curve \(C\) and Jacobian \(J\) they are studying. They write down explicit divisors in the case \(d=p^\nu+1\), and we find relations satisfied by the classes of these divisors in J. These relations turn out to be the only ones, but that is not proved in general until much later in the paper. Chapter 2, deals with descent calculations. In this chapter, the authors assume that \(r\) is prime and use descent arguments to bound the rank of \(J\) from below in the case when \(d=p^\nu+1\). Chapter 3, deals with minimal regular model, local invariants, and domination by a product of curves. The authors construct the minimal, regular, proper model \(\aleph\longrightarrow\mathbb{P}^1\) of \(C/\mathbb{F}_q(u)\) for any values of \(d\) and \(r\). In particular, they compute the singular fibers of \(\aleph\longrightarrow\mathbb{P}^1\). This allows them to compute the component groups of the Néron model of \(J\). They also give a precise connection between the model \(X\) and a product of curves. Chapter 4, deals with heights and the visible subgroup. The authors consider the case where \(d=p^\nu+1\) and \(r|d\), and they compute the heights of the explicit divisors introduced in Chapter 1. This allows them to compute the rank of the explicit subgroup \(V\) and its structure over the group ring \(\mathbb{Z}[\mu_r\times\mu_d]\). Chapter 5, deals with the \(L\)-function and the \(BSD\) conjecture. The authors give an elementary calculation of the \(L\)-function of \(J\) over \(\mathbb{F}_q(u)\) (for any \(d\) and \(r\)) in terms of Jacobi sums. They also show that the \(BSD\) conjecture holds for \(J\), and we give an elementary calculation of the rank of \(J(\mathbb{F}_q(u))\) for any \(d\) and \(r\) and all sufficiently large \(q\). Chapter 6, deals with analysis of \(J[p]\) and \(NS(\aleph_d)tor\) and Chapter 7, with index of the visible subgroup and the Tate-Shafarevich group. In these technical chapters, the authors prove several results about the surface \(X\) that allow them to deduce that the index of \(V\) in \(J(K_d)\) is a power of \(p\) when \(d=p^\nu+1\) and \(r\) divides \(d\). They also use the \(BSD\) formula to relate this index to the order of the Tate-Shafarevich group. Chapter 8, deals with monodromy of \(l\)-torsion and decomposition of the Jacobian. The authors prove strong results on the monodromy of the \(l\)-torsion of \(J\) for prime to \(pr\). This gives precise statements about torsion points on \(J\) over abelian or solvable extensions of \(\mathbb{F}_p(t)\) and about the decomposition of \(J\) up to isogeny into simple abelian varieties. The paper is supported by an appendix on an additional hyperelliptic family.

MSC:

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H05 Algebraic functions and function fields in algebraic geometry
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