We suggest that the following plan will provide a powerful tool for trying to find the set of Q-rational points C(Q) on a curve C of genus>1:\ud
(1) Attempt to find J(Q)/2J(Q) via descent on J, the Jacobian of C.\ud
(2) Deduce generators for J(Q) via an explicit theory of heights.\ud
(3). Apply local techniques to try to deduce C(Q) via an embedding of C(Q) inside J(Q).\ud
We describe work just completed, which gives versions of (1),(2),(3) which are often workable in practice for genus 2, and outline the potential for a computationally viable generalisation. We note that (1),(2),(3) (quite aside from being part of this plan) have their own independent applications to other branches of the Mathematics of Computation, such as the search for large rank, the higher dimensional testing of well known conjectures, and algorithms for symbolic integration

Flynn, E. V.

English

Mathematical Institute Eprints Archive

Solving Diophantine Problems on Curves via Descent on the Jacobian
E. V. Flynn, Mathematical Institute, University of Oxford
§0. Introduction
The theory of Jacobians of curves has largely been developed in a vacuum, with
little computational counterpart to the abstract theory. A recent development has been
the explicit construction of Jacobians & formal groups, and workable methods of descent
[6],[7] to find the rank. We suggest that the following plan will provide a powerful tool for
finding the set of Q-rational points C(Q) on a curve C of genus > 1:
(1). Find J(Q)/2J(Q) via descent on J , the Jacobian of C.
(2). Deduce generators for J(Q) via an explicit theory of heights.
(3). Apply local techniques to try to deduce C(Q) via an embedding of C(Q) inside J(Q).
We describe work just completed, which gives versions of (1),(2),(3) which are often
workable in practice for genus 2, and outline the potential for a computationally viable
generalisation. We note that (1),(2),(3) (quite aside from being part of this plan) have
their own independent applications to other branches of the Mathematics of Computation,
such as the search for large rank, the higher dimensional testing of well known conjectures
[8], and algorithms for symbolic integration.
§1. Rank of the Mordell-Weil group using descent by isogeny.
It should first be mentioned that, in principle, there already exists a technique due to
Gordon and Grant [7] which computes the Mordell-Weil group J(Q) by complete 2-descent
for the highly special case when the curve of genus 2 has all of its Weierstrass points defined
over Q. Such curves can be written in the form: Y 2 = Π5i=1(X−αi), αi ∈ Z. The problem
here is that only a tiny handful of such curves have 3 or less primes of bad reduction, and
so there will typically be a large number of homogeneous spaces to be checked. We have
recently completed theory which fills this gap in the genus 2 situation; namely: a method
of descent by isogeny likely to yield rank tables for large numbers of curves. A helpful
feature recently publicised by Bost & Mestre in [1] (originally due to Richelot) is that
there are natural underlying pairs of curves whose Jacobians are isogenous; specifically,
there is an isogeny from the Jacobian of C to that of Ĉ, where:
C : Y 2 = g(X)h(X)i(X) = (g2X2 + g1X + g0)(h2X2 + h1X + h0)(i2X2 + i1X + i0) (∗)
Ĉ : ∆Y 2 = (h′(x)i(X)− h(X)i′(X))(i′(x)g(X)− i(X)g′(X))(g′(x)h(X)− g(X)h′(X))
1
where ∆ = det
 g0 g1 g2h0 h1 h2
i0 i1 i2
 .
Bost & Mestre investigate the geometry behind the isogeny, work over R rather than
Q, and do not make any use of the Jacobian as a variety. Our first task was to develop
the isogeny explicitly from J to Ĵ , the Jacobians of C and Ĉ, respectively, each embedded
into P15. We have found the quadratic map over the ground field φ : J 7→ Ĵ which defines
the isogeny, and have performed a similar computation for the dual isogeny φˆ : Ĵ 7→ J .
The computation of Ĵ(Q)/φ(J(Q)) is equivalent to determining, for a finite set of
pairs (d1, d2) ∈
(
Q∗/(Q∗)2
)2, whether the homogeneous spaces Cd1,d1 has a rational point.
The space Cd1,d2 is defined by a set of 72 equations over Q, which are twists of the
defining equations of the Jacobian. The symbolic algebra package Maple was essential for
constructing the defining equations of Cd1,d2 , as described in [6]. By applying the dual
isogeny, J(Q)/φ(Ĵ(Q)) may also be found, giving J(Q)/2J(Q), and hence the rank of
J(Q). We can show (for example) that the Jacobian of the curve:
C(k) : Y 2 = k(X2 + 1)(X2 + 2)(X2 +X + 1)
has rank 1 when k = 1, 2, 6. Note that there are at least 60 curves of the form (∗) with
small integer coefficients and only two primes (including p = 2) of bad reduction (which
would not have been possible in the completely split situation), and a much larger number
of curves available with three such primes. Even from these curves alone, We expect
quickly to have about 60 ranks calculated and considerably more to follow. Quite aside
from the applicability to solving Diophantine problems on curves (as will be described), it
is to be hoped that such rank calculations will also lay the eventual foundation for testing
and motivating higher dimensional analogues of conjectures [8] currently only tested for
elliptic curves, such as the conjectures of Birch and Swinnerton-Dyer.
§2. Explicit theory of heights.
For the theory of heights, We consider a = (a0, . . . a15) ∈ J(Q), where J is the
embedding of the Jacobian in P15 mentioned in §1. Any such a can be written so that
each ai ∈ Z and gcd(a0, . . . , a15) = 1. By analogy with elliptic curves, let:
H(a) = max(|a0|, . . . , |a15|).
which defines a height function. We have expressed the relevant constants in terms of the
coefficients of the original curve. That is to say, for C : Y 2 = f6X6 + . . .+ f0 and for any
2
fixed b ∈ J(Q), We have found k1 = k1(f0, . . . , f6,b) and k2 = k2(f0, . . . , f6), expressed
in terms of absolute values of expressions in Z[f0, . . . , f6], such that for any a ∈ J(Q):
H(a+ b) 6 k1H(a)2, H(2a) > k2H(a)4.
This form of k1 follows from the biquadratic forms defining the group law in [5]. The
constant k2 required considerably more symbolic computation (again, using Maple). It
required the application of many resultant calculations to the duplication law on the Jaco-
bian and Kummer surface (given by quartic forms). We have used this theory to show that,
for the curves in [6], [7], the given generators of J(Q)/2J(Q) are also generators of J(Q).
This represents the first non-trivial calculation of this type in dimension > 1. My intention
is to do the same for those curves whose ranks are determined by my method of descent
by isogeny. This should be facilitated by enhancements in progress using the restriction of
the height function to the Kummer surface. There are also immediate applications of our
explicit theory of heights to algorithms for finding the torsion subgroup of J(Q); these, in
turn, are relevant to the theory of computer integration of algebraic functions [4].
§3. Local techniques to find C(Q).
We have also investigated a theorem of Chabauty ([2],[3]) which states that a curve
has only finitely many rational points when its Jacobian has rank less than the genus of
the curve. This theorem was established (in 1941) long before Falting’s Theorem, and it is
an eccentric feature of the literature that there has been virtually no opportunity during
the intervening 50 years to apply it non-trivially to Diophantine problems, due to the lack
of rank calculations for genus > 1 (for elliptic curves, the result merely asserts in rank
0 that there are finitely many rational torsion points). This result can be interpreted in
genus 2 (with rank 1 Jacobian) in a form amenable to practical computation. One first
finds the first few terms of the formal group law, a calculation greatly facilitated by the
form of the global group law in [5]. One can then, for any prime p of good reduction, use
the generators of J(Q) (determined by the descent techniques already described) to obtain
a power series over Qp which has at least as many solutions as Q-rational points on the
original curve. Bounds on the p-adic values of the coefficients together with Strassman’s
theorem then gives an upper bound on the p-adic solutions to the power series, and hence
on the number of Q-rational point on the curve. Should this bound be attained by the
number of known Q-rational points then C(Q) is determined completely. Otherwise, one
can repeat the process for different values of p, and hope that the bound is attained for
some p. It is not claimed that this is an effective procedure, merely a workable tool which
appears almost always to resolve C(Q). We have applied it to a number of curves of genus
2 of rank 1, and found that the bound is attained in each case.
3
§4. Potential future progress.
The ideas of §1 are highly amenable to generalisation; there are already signs that
descent procedures to find J(Q)/2J(Q), and hence the rank of J(Q), are being made
viable at least for hyperelliptic curves of genus 3 [9], and possibly higher. The material of
§2 and §3 should in principle be amenable to generalisation, although the search area in
the heights computation may increase quickly with respect to genus.
REFERENCES
[1] Bost, J. B. and Mestre, J.-F.Moyenne arithme´tico-ge´ometrique et pe´riodes des courbes
de genre 1 et 2. Gaz. Math. Soc. France, 38 (1988), 36-64.
[2] Chabauty, C. Sur les points rationels des courbes alge´briques de genre supe´rieur a´
l’unite´. Comptes rendus, Paris 212 (1941), 882-885.
[3] Coleman, R. F. Effective Chabauty. Duke Math. J. 52 (1985), 765-770.
[4] Davenport, J. H. On the Integration of Algebraic Functions, Springer Lecture Notes
in Computer Science 102, Springer-Verlag (1981).
[5] Flynn, E.V. The group law on the Jacobian of a curve of genus 2. J. Reine Angew.
Math. 439 (1993), 45-69.
[6] Flynn, E.V. Descent via isogeny in dimension 2. To appear in Acta Arith.
[7] Gordon, D.M. and Grant, D. Computing the Mordell-Weil rank of Jacobians of curves
of genus 2. Transactions of the American Mathematical Society. To appear.
[8] Hulsbergen, W.W.J. Conjectures in arithmetic algebraic geometry: a survey, Aspects
of Mathematics 18, Vieweg (1992).
[9] Schaefer, E.F. 2-descent on the Jacobians of hyperelliptic curves. Draft manuscript,
Dec. 1992.
4


2-descent on the Jacobians of hyperelliptic curves.

Computing the Mordell-Weil rank of Jacobians of curves of genus 2.

Conjectures in arithmetic algebraic geometry: a survey,

Descent via isogeny in dimension 2.

Effective Chabauty.

Moyenne arithme´tico-ge´ometrique et pe´riodes des courbes de genre 1 et 2.

On the Integration of Algebraic Functions,

Sur les points rationels des courbes alge´briques de genre supe´rieur a´ l’unite´. Comptes rendus,

The group law on the Jacobian of a curve of genus 2.

Solving Diophantine problems on curves via descent on the jacobian

We suggest that the following plan will provide a powerful tool for trying to find the set of Q-rational points C(Q) on a curve C of genus>1: (1) Attempt to find J(Q)/2J(Q) via descent on J, the Jacobian of C. (2) Deduce generators for J(Q) via an explicit theory of heights. (3). Apply local techniques to try to deduce C(Q) via an embedding of C(Q) inside J(Q). We describe work just completed, which gives versions of (1),(2),(3) which are often workable in practice for genus 2, and outline the potential for a computationally viable generalisation. We note that (1),(2),(3) (quite aside from being part of this plan) have their own independent applications to other branches of the Mathematics of Computation, such as the search for large rank, the higher dimensional testing of well known conjectures, and algorithms for symbolic integration

Oxford University Research Archive

Solving Diophantine Problems on Curves via Descent on the JacobianE. V. Flynn, Mathematical Institute, University of Oxford§0. IntroductionThe theory of Jacobians of curves has largely been developed in a vacuum, withlittle computational counterpart to the abstract theory. A recent development has beenthe explicit construction of Jacobians & formal groups, and workable methods of descent[6],[7] to find the rank. We suggest that the following plan will provide a powerful tool forfinding the set of Q-rational points C(Q) on a curve C of genus > 1:(1). Find J(Q)/2J(Q) via descent on J , the Jacobian of C.(2). Deduce generators for J(Q) via an explicit theory of heights.(3). Apply local techniques to try to deduce C(Q) via an embedding of C(Q) inside J(Q).We describe work just completed, which gives versions of (1),(2),(3) which are oftenworkable in practice for genus 2, and outline the potential for a computationally viablegeneralisation. We note that (1),(2),(3) (quite aside from being part of this plan) havetheir own independent applications to other branches of the Mathematics of Computation,such as the search for large rank, the higher dimensional testing of well known conjectures[8], and algorithms for symbolic integration.§1. Rank of the Mordell-Weil group using descent by isogeny.It should first be mentioned that, in principle, there already exists a technique due toGordon and Grant [7] which computes the Mordell-Weil group J(Q) by complete 2-descentfor the highly special case when the curve of genus 2 has all of its Weierstrass points definedover Q. Such curves can be written in the form: Y 2 = Π5i=1(X−αi), αi ∈ Z. The problemhere is that only a tiny handful of such curves have 3 or less primes of bad reduction, andso there will typically be a large number of homogeneous spaces to be checked. We haverecently completed theory which fills this gap in the genus 2 situation; namely: a methodof descent by isogeny likely to yield rank tables for large numbers of curves. A helpfulfeature recently publicised by Bost & Mestre in [1] (originally due to Richelot) is thatthere are natural underlying pairs of curves whose Jacobians are isogenous; specifically,there is an isogeny from the Jacobian of C to that of Ĉ, where:C : Y 2 = g(X)h(X)i(X) = (g2X2 + g1X + g0)(h2X2 + h1X + h0)(i2X2 + i1X + i0) (∗)Ĉ : ∆Y 2 =(h′(x)i(X)− h(X)i′(X))(i′(x)g(X)− i(X)g′(X))(g′(x)h(X)− g(X)h′(X))1where ∆ = det g0 g1 g2h0 h1 h2i0 i1 i2 .Bost & Mestre investigate the geometry behind the isogeny, work over R rather thanQ, and do not make any use of the Jacobian as a variety. Our first task was to developthe isogeny explicitly from J to Ĵ , the Jacobians of C and Ĉ, respectively, each embeddedinto P15. We have found the quadratic map over the ground field φ : J 7→ Ĵ which definesthe isogeny, and have performed a similar computation for the dual isogeny φ̂ : Ĵ 7→ J .The computation of Ĵ(Q)/φ(J(Q)) is equivalent to determining, for a finite set ofpairs (d1, d2) ∈(Q∗/(Q∗)2)2, whether the homogeneous spaces Cd1,d1 has a rational point.The space Cd1,d2 is defined by a set of 72 equations over Q, which are twists of thedefining equations of the Jacobian. The symbolic algebra package Maple was essential forconstructing the defining equations of Cd1,d2 , as described in [6]. By applying the dualisogeny, J(Q)/φ(Ĵ(Q)) may also be found, giving J(Q)/2J(Q), and hence the rank ofJ(Q). We can show (for example) that the Jacobian of the curve:C(k) : Y 2 = k(X2 + 1)(X2 + 2)(X2 + X + 1)has rank 1 when k = 1, 2, 6. Note that there are at least 60 curves of the form (∗) withsmall integer coefficients and only two primes (including p = 2) of bad reduction (whichwould not have been possible in the completely split situation), and a much larger numberof curves available with three such primes. Even from these curves alone, We expectquickly to have about 60 ranks calculated and considerably more to follow. Quite asidefrom the applicability to solving Diophantine problems on curves (as will be described), itis to be hoped that such rank calculations will also lay the eventual foundation for testingand motivating higher dimensional analogues of conjectures [8] currently only tested forelliptic curves, such as the conjectures of Birch and Swinnerton-Dyer.§2. Explicit theory of heights.For the theory of heights, We consider a = (a0, . . . a15) ∈ J(Q), where J is theembedding of the Jacobian in P15 mentioned in §1. Any such a can be written so thateach ai ∈ Z and gcd(a0, . . . , a15) = 1. By analogy with elliptic curves, let:H(a) = max(|a0|, . . . , |a15|).which defines a height function. We have expressed the relevant constants in terms of thecoefficients of the original curve. That is to say, for C : Y 2 = f6X6 + . . . + f0 and for any2fixed b ∈ J(Q), We have found k1 = k1(f0, . . . , f6,b) and k2 = k2(f0, . . . , f6), expressedin terms of absolute values of expressions in Z[f0, . . . , f6], such that for any a ∈ J(Q):H(a + b) 6 k1H(a)2, H(2a) > k2H(a)4.This form of k1 follows from the biquadratic forms defining the group law in [5]. Theconstant k2 required considerably more symbolic computation (again, using Maple). Itrequired the application of many resultant calculations to the duplication law on the Jaco-bian and Kummer surface (given by quartic forms). We have used this theory to show that,for the curves in [6], [7], the given generators of J(Q)/2J(Q) are also generators of J(Q).This represents the first non-trivial calculation of this type in dimension > 1. My intentionis to do the same for those curves whose ranks are determined by my method of descentby isogeny. This should be facilitated by enhancements in progress using the restriction ofthe height function to the Kummer surface. There are also immediate applications of ourexplicit theory of heights to algorithms for finding the torsion subgroup of J(Q); these, inturn, are relevant to the theory of computer integration of algebraic functions [4].§3. Local techniques to find C(Q).We have also investigated a theorem of Chabauty ([2],[3]) which states that a curvehas only finitely many rational points when its Jacobian has rank less than the genus ofthe curve. This theorem was established (in 1941) long before Falting’s Theorem, and it isan eccentric feature of the literature that there has been virtually no opportunity duringthe intervening 50 years to apply it non-trivially to Diophantine problems, due to the lackof rank calculations for genus > 1 (for elliptic curves, the result merely asserts in rank0 that there are finitely many rational torsion points). This result can be interpreted ingenus 2 (with rank 1 Jacobian) in a form amenable to practical computation. One firstfinds the first few terms of the formal group law, a calculation greatly facilitated by theform of the global group law in [5]. One can then, for any prime p of good reduction, usethe generators of J(Q) (determined by the descent techniques already described) to obtaina power series over Qp which has at least as many solutions as Q-rational points on theoriginal curve. Bounds on the p-adic values of the coefficients together with Strassman’stheorem then gives an upper bound on the p-adic solutions to the power series, and henceon the number of Q-rational point on the curve. Should this bound be attained by thenumber of known Q-rational points then C(Q) is determined completely. Otherwise, onecan repeat the process for different values of p, and hope that the bound is attained forsome p. It is not claimed that this is an effective procedure, merely a workable tool whichappears almost always to resolve C(Q). We have applied it to a number of curves of genus2 of rank 1, and found that the bound is attained in each case.3§4. Potential future progress.The ideas of §1 are highly amenable to generalisation; there are already signs thatdescent procedures to find J(Q)/2J(Q), and hence the rank of J(Q), are being madeviable at least for hyperelliptic curves of genus 3 [9], and possibly higher. The material of§2 and §3 should in principle be amenable to generalisation, although the search area inthe heights computation may increase quickly with respect to genus.REFERENCES[1] Bost, J. B. and Mestre, J.-F. Moyenne arithmético-géometrique et périodes des courbesde genre 1 et 2. Gaz. Math. Soc. France, 38 (1988), 36-64.[2] Chabauty, C. Sur les points rationels des courbes algébriques de genre supérieur ál’unité. Comptes rendus, Paris 212 (1941), 882-885.[3] Coleman, R. F. Effective Chabauty. Duke Math. J. 52 (1985), 765-770.[4] Davenport, J. H. On the Integration of Algebraic Functions, Springer Lecture Notesin Computer Science 102, Springer-Verlag (1981).[5] Flynn, E. V. The group law on the Jacobian of a curve of genus 2. J. Reine Angew.Math. 439 (1993), 45-69.[6] Flynn, E.V. Descent via isogeny in dimension 2. To appear in Acta Arith.[7] Gordon, D.M. and Grant, D. Computing the Mordell-Weil rank of Jacobians of curvesof genus 2. Transactions of the American Mathematical Society. To appear.[8] Hulsbergen, W.W.J. Conjectures in arithmetic algebraic geometry: a survey, Aspectsof Mathematics 18, Vieweg (1992).[9] Schaefer, E.F. 2-descent on the Jacobians of hyperelliptic curves. Draft manuscript,Dec. 1992.4

We suggest that the following plan will provide a powerful tool for trying to find the set of Q-rational points C(Q) on a curve C of genus>1:
(1) Attempt to find J(Q)/2J(Q) via descent on J, the Jacobian of C.
(2) Deduce generators for J(Q) via an explicit theory of heights.
(3). Apply local techniques to try to deduce C(Q) via an embedding of C(Q) inside J(Q).
We describe work just completed, which gives versions of (1),(2),(3) which are often workable in practice for genus 2, and outline the potential for a computationally viable generalisation. We note that (1),(2),(3) (quite aside from being part of this plan) have their own independent applications to other branches of the Mathematics of Computation, such as the search for large rank, the higher dimensional testing of well known conjectures, and algorithms for symbolic integration

Solving Diophantine problems on curves via descent on the jacobian

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