##
**Implementing 2-descent for Jacobians of hyperelliptic curves.**
*(English)*
Zbl 0972.11058

Let \(C\) be a curve over the field \(\mathbb Q\) of rational numbers. Let \(J\) denote the Jacobian of \(C\). Galois cohomology provides an injection of \(J({\mathbb Q})/2J({\mathbb Q})\) into a finite group \({\text{Sel}}^{(2)}({\mathbb Q},J)\) called the \(2\)-Selmer group of \(J\). In theory, the \(2\)-Selmer group is computable.

This paper describes an explicit algorithm to compute the size of the \(2\)-Selmer group in the case where \(C\) is a hyperelliptic curve that either has a rational Weierstrass point or has even genus. In more concrete terms, the hypothesis on \(C\) is equivalent to the requirement that \(C\) be birational to an affine curve defined by \(y^2=f(x)\), where \(f(x) \in {\mathbb Q}[x]\) is a squarefree polynomial such that either \(f(x)\) has a rational root, or the degree of \(f\) is greater than \(4\) and not divisible by \(4\). This is satisfied, for instance, when \(C\) is a curve of genus \(2\) over \(\mathbb Q\).

Let us explain why it is important to be able to compute the \(2\)-Selmer group. The following three problems are known to be algorithmically equivalent, even though we do not yet know an algorithm that is guaranteed to solve them:

Compute the size of \(J({\mathbb Q})/2J({\mathbb Q})\).

Compute the rank of the finitely generated abelian group \(J({\mathbb Q})\).

Compute generators of \(J({\mathbb Q})\) and the relations between them.

The size of the \(2\)-Selmer group is an upper bound for the size of \(J({\mathbb Q})/2J({\mathbb Q})\), and in favorable cases one can find a matching lower bound by searching for points. Hence one obtains an upper bound for the rank of \(J({\mathbb Q})\), and in favorable cases, one obtains generators and relations for \(J({\mathbb Q})\). Such knowledge, in turn, is needed by certain algorithms for determining the set of rational points on \(C\), a set that is known to be finite when the genus of \(C\) is at least \(2\). Thus the computation of the \(2\)-Selmer group can be used to solve many concrete diophantine problems.

The method to compute the \(2\)-Selmer group has its origins in the “descent” methods used in the 1920s by Mordell and Weil, who clarified and generalized techniques used earlier by Fermat and Poincaré. For Jacobians of curves of genus \(2\) and higher, the method was adapted in a series of articles making it more and more explicit, or applicable to more and more curves, beginning with J. W. S. Cassels, and continuing with D. M. Gordon, D. Grant, E. V. Flynn, E. F. Schaefer, this reviewer, and M. Stoll. See B. Poonen and E. F. Schaefer, J. Reine Angew. Math. 488, 141-188 (1997; Zbl 0888.11023) and E. F. Schaefer, Math. Ann. 310, 447-471 (1998; Zbl 0889.11021) and the references listed there. The current paper can be seen as the culmination of this program in the case of hyperelliptic curves: the method has been refined to the point where it can be completely automated. In fact, Stoll has implemented his algorithm as part of the Magma algebra system. It is this algorithm that discovered the first examples of Jacobians whose Shafarevich-Tate groups have nonsquare order [B. Poonen and M. Stoll, Ann. Math. (2) 150, 1109-1149 (1999; Zbl 1024.11040)]. Among the innovations that distinguish this well-written paper from those of previous authors are:

A better way to compute the kernel of the explicit homomorphism used as a substitute for the injection \(J({\mathbb Q})/2J({\mathbb Q}) \rightarrow \text{Sel}^{(2)}({\mathbb Q},J)\) (pp. 258-259 of Section 5).

A better way to compute the local images of \(J({\mathbb Q}_p)/2J({\mathbb Q}_p)\) (Section 6).

Use of the parity of the dimension of the \(2\)-torsion of the Shafarevich-Tate group (Section 7).

This paper describes an explicit algorithm to compute the size of the \(2\)-Selmer group in the case where \(C\) is a hyperelliptic curve that either has a rational Weierstrass point or has even genus. In more concrete terms, the hypothesis on \(C\) is equivalent to the requirement that \(C\) be birational to an affine curve defined by \(y^2=f(x)\), where \(f(x) \in {\mathbb Q}[x]\) is a squarefree polynomial such that either \(f(x)\) has a rational root, or the degree of \(f\) is greater than \(4\) and not divisible by \(4\). This is satisfied, for instance, when \(C\) is a curve of genus \(2\) over \(\mathbb Q\).

Let us explain why it is important to be able to compute the \(2\)-Selmer group. The following three problems are known to be algorithmically equivalent, even though we do not yet know an algorithm that is guaranteed to solve them:

Compute the size of \(J({\mathbb Q})/2J({\mathbb Q})\).

Compute the rank of the finitely generated abelian group \(J({\mathbb Q})\).

Compute generators of \(J({\mathbb Q})\) and the relations between them.

The size of the \(2\)-Selmer group is an upper bound for the size of \(J({\mathbb Q})/2J({\mathbb Q})\), and in favorable cases one can find a matching lower bound by searching for points. Hence one obtains an upper bound for the rank of \(J({\mathbb Q})\), and in favorable cases, one obtains generators and relations for \(J({\mathbb Q})\). Such knowledge, in turn, is needed by certain algorithms for determining the set of rational points on \(C\), a set that is known to be finite when the genus of \(C\) is at least \(2\). Thus the computation of the \(2\)-Selmer group can be used to solve many concrete diophantine problems.

The method to compute the \(2\)-Selmer group has its origins in the “descent” methods used in the 1920s by Mordell and Weil, who clarified and generalized techniques used earlier by Fermat and Poincaré. For Jacobians of curves of genus \(2\) and higher, the method was adapted in a series of articles making it more and more explicit, or applicable to more and more curves, beginning with J. W. S. Cassels, and continuing with D. M. Gordon, D. Grant, E. V. Flynn, E. F. Schaefer, this reviewer, and M. Stoll. See B. Poonen and E. F. Schaefer, J. Reine Angew. Math. 488, 141-188 (1997; Zbl 0888.11023) and E. F. Schaefer, Math. Ann. 310, 447-471 (1998; Zbl 0889.11021) and the references listed there. The current paper can be seen as the culmination of this program in the case of hyperelliptic curves: the method has been refined to the point where it can be completely automated. In fact, Stoll has implemented his algorithm as part of the Magma algebra system. It is this algorithm that discovered the first examples of Jacobians whose Shafarevich-Tate groups have nonsquare order [B. Poonen and M. Stoll, Ann. Math. (2) 150, 1109-1149 (1999; Zbl 1024.11040)]. Among the innovations that distinguish this well-written paper from those of previous authors are:

A better way to compute the kernel of the explicit homomorphism used as a substitute for the injection \(J({\mathbb Q})/2J({\mathbb Q}) \rightarrow \text{Sel}^{(2)}({\mathbb Q},J)\) (pp. 258-259 of Section 5).

A better way to compute the local images of \(J({\mathbb Q}_p)/2J({\mathbb Q}_p)\) (Section 6).

Use of the parity of the dimension of the \(2\)-torsion of the Shafarevich-Tate group (Section 7).

Reviewer: Bjorn Poonen (Berkeley)