Two-descent on some genus two curves
aa r X i v : . [ m a t h . N T ] F e b TWO-DESCENT ON SOME GENUS TWO CURVES
TIM EVINK, GERT-JAN VAN DER HEIDEN, AND JAAP TOP Introduction
In this paper we study for an arbitrary prime number p the curve C p of genus 2defined by the equation(1) y = x ( x − p )( x − p ) . Specifically, we start by bounding the rank of its Jacobian J p over Q in termsof the 2-Selmer group S ( J p / Q ). Next we show for three infinite sets of primenumbers p how to improve the upper bound on rank J p ( Q ) by using a 2-Selmergroup computation over Q ( √± p ) of the Jacobian of the curve C = C defined by y = x ( x − x − X ( J p / Q ) isnontrivial. As an example: for primes p ≡
23 mod 48 it turns out that J p ( Q ) isfinite and X ( J p / Q )[2] ∼ = ( Z / Z ) .We also discuss the Q -rational points of the curve C p . This is easy in case thegroup J p ( Q ) is finite (as occurs, for example, for all primes p ≡ p = 241; the group J ( Q ) turns out to have rank 2. Usingso-called ‘Two-Selmer sets’, it is shown that C ( Q ) consists of only the obviousWeierstrass points (the one at infinity and the ones with y = 0).Studying genus 1 curves depending on a prime number p is a very classicalsubject; the survey paper [12] already lists various examples; more recent ones arefound, e.g., in [2], [11], [17]. The natural question of investigating analogous ideasin the case of genus 2 curves so far seems to have obtained less attention. The 1998master’s thesis [8] by one of us provides a first step (not yet involving Shafarevich-Tate groups). As shown in loc. sit. Prop. 4.3.3 and Thm. 4.3.4, this already sufficesto conclude for the curves discussed in the present paper that C p ( Q ) consists ofthe 6 Weierstrass points only, whenever p ≡ Q allow the authors to identifycongruence conditions on the prime p such that the corresponding Mordell-Weilgroup is finite. As a consequence, for those primes the only rational points on thecurve are the rational Weierstrass points.Many results in the present paper originate from two master’s projects [8], [6](1998 resp. 2019) by the second and the first author, supervised by the third one.2. Notation and results
The first step in order to obtain information on the rank of Jacobian J p of thehyperelliptic curve C p defined by the equation y = x ( x − p )( x − p ) for a prime p , is the relatively basic computation of the 2-Selmer group of J p / Q . Itfits in the well know short exact sequence(2) 0 → J p ( Q ) / J p ( Q ) → S ( J p / Q ) → X ( J p / Q )[2] → . This Selmer group was computed in [8] (with minor corrections in [6, Appendix B]).The computation is based on the method described in [14] and uses (see [7, Sec-tion 7]) J p ( K v ) / J p ( K v ) = | | − v · J p ( K v )[2] = | | − v · K v ⊃ Q ℓ is a finite extension with valuation ring O v , and | | v = vol(2 O v ) / vol( O v ) = (cid:26) ℓ = 2 , − [ K v : Q ] if ℓ = 2 . The result is as follows. (A calculation illustrating this type of result is the proofof Lemma 5.1 below.)
Proposition 2.1.
For a prime number p > , the F -vectorspace S ( J p / Q ) of theJacobian J p of the curve defined by y = x ( x − p )( x − p ) has dimension asgiven in the next table. p mod 24 dim F S ( J p / Q )1 85 , , ,
19 57 417 ,
23 6Since all Weierstrass points on C p are Q -rational, one has J p ( Q )[2] ∼ = ( Z / Z ) .Either by observing that in the present situation J p ( Q )[2] ֒ → S ( J p / Q ), or us-ing that the torsion subgroup J p ( Q ) tor ⊂ J p ( Q ) yields a 4-dimensional subspace J p ( Q ) tor / J p ( Q ) tor of J p ( Q ) / J p ( Q ), the short exact sequence (2) implies(3) rank J p ( Q ) + dim F X ( J p / Q )[2] = dim F S ( J p / Q ) − . We state an immediate consequence of this:
Corollary 2.2.
For any prime number p ≡ one has rank J p ( Q ) = 0 and C p ( Q ) consists of only the Weierstrass points of C p .Proof. The proof of the statement about the rank is indicated above. Note thatfor p = 5 one has J p ( F ) = 16 independent of p . Moreover J ( F ) = 48 and J ( F ) = 128. Since for primes ℓ ≥ ℓ map is an injectivegroup homomorphism on rational torsion points, it follows that J p ( Q ) has torsionsubgroup ( Z / Z ) for every prime p . Embedding C p ⊂ J p via P [ P − ∞ ] with ∞ ∈ C p the Weierstrass point at infinity, one concludes that C p ∩ J p ( Q ) consistsof the divisor classes [ W − ∞ ] for W any Weierstrass point on C p , implying theresult. (cid:3) For the primes p ≡ , , ,
19 mod 24 the structure of the group J p ( Q ) is infact also predicted by Proposition 2.1: Corollary 2.3.
For any prime p > , assume that X ( J p / Q ) is finite. Thenrank J p ( Q ) ≡ (cid:26) if p ≡ , , ,
19 mod 24;0 mod 2 otherwise.
In particular, if for a prime p ≡ , , ,
19 mod 24 the group X ( J p / Q ) is finitethen for this prime J p ( Q ) ∼ = Z × ( Z / Z ) . WO-DESCENT ON SOME GENUS TWO CURVES 3
Proof.
By a result of Poonen and Stoll [13, § §
8] finiteness of X and the factthat C p contains a rational point, implies that dim F X ( J p / Q )[2] is even. HenceEquation (3) and Proposition 2.1 imply the first assertion as well as rank J p ( Q ) = 1whenever p ≡ , , ,
19 mod 24. The result follows since the proof of Corollary 2.2in particular determines the torsion subgroup of J p ( Q ). (cid:3) The remainder of this paper deals with improvements of Proposition 2.1 andvariations on Corollary 2.2. Specifically, this is possible in all remaining congruenceclasses (so, p ≡ , ,
23 mod 24). We show the following.
Theorem 2.4.
Let p ≡
23 mod 48 be a prime number. The Jacobian J p of the curvecorresponding to y = x ( x − p )( x − p ) satisfies J p ( Q ) = J p ( Q )[2] ∼ = ( Z / Z ) and X ( J p / Q )[2] ∼ = ( Z / Z ) . Theorem 2.5.
Let p ≡
17 mod 24 be a prime number that does not split completelyin Q ( √ . The Jacobian J p of the curve corresponding to y = x ( x − p )( x − p ) satisfies J p ( Q ) = J p ( Q )[2] ∼ = ( Z / Z ) and X ( J p / Q )[2] ∼ = ( Z / Z ) . Theorem 2.6.
Let p ≡ be a prime number satisfying one of the conditions ( a ) p splits completely in Q ( √ and not in Q ( p √ ; ( b ) p ≡ and p splits completely in Q ( p √ and not in Q ( √ ; ( c ) p ≡
25 mod 48 and p does not split completely in either of Q ( √ and Q ( p √ .The Jacobian J p of the curve corresponding to y = x ( x − p )( x − p ) satisfies J p ( Q ) = J ( Q )[2] ∼ = ( Z / Z ) and X ( J p / Q )[2] ∼ = ( Z / Z ) . Using Chebotar¨ev’s density theorem (see, e.g., [15]), one observes that the setof prime numbers satisfying the condition given in Theorem 2.4 has a positiveDirichlet density. The same holds for the set of primes satisfying the conditionin Theorem 2.5 and for each of the three sets corresponding to Theorem 2.6 (a),2.6 (b), and 2.6 (c). 3.
R´edei Symbols
In this section we recall the definition and various properties of the R´edei symbol.It is a tri-linear symbol taking values in µ and it satisfies a reciprocity law basedon the product formula for quadratic Hilbert symbols. This reciprocity allows us tolink the splitting behaviour of certain primes in dihedral extensions over Q of degree8 in a non-trivial way, which functions as a useful supplement to various 2-Selmergroup computations. The reciprocity of the R´edei-symbol is a recent result due toP. Stevenhagen in [16]; his text is the basis for the exposition in this section.Let a, b be square-free integers representing non-trivial elements in Q ∗ / Q ∗ , andsuppose their local quadratic Hilbert symbols are all trivial:(4) ( a, b ) p = 1 , for all primes p. By the local-global principle of Hasse and Minkowski, condition (4) is equivalentto the existence of a non-zero rational solution ( x, y, z ) to the equation(5) x − ay − bz = 0 . T. EVINK, G.J. VAN DER HEIDEN, AND J. TOP
Take such a solution and put(6) α = 2( x + z √ b ) , β = x + y √ a. Then F := E ( √ α ) = E ( √ β ) defines a quadratic extension of E = Q ( √ a, √ b )that is normal over Q , cyclic of degree 4 over Q ( √ ab ), and dihedral of degree 8over Q when Q ( √ ab ) = Q , see [16, Lemma 5.1, Corollary 5.2]. The extension F can be twisted to F t for t ∈ Q ∗ by scaling the solution ( x, y, z ) to ( tx, ty, tz ). By[16, Propositions 7.2, 7.3] choosing t appropriately ensures that F t /E is unramifiedat all finite primes of odd residue characteristic, but in some cases ramificationover 2 cannot be avoided. With ∆( d ) = ∆( O Q ( √ d ) ) for d ∈ Q ∗ / Q ∗ denoting thediscriminant, one makes the following definition. Definition 3.1.
Let K = Q ( √ ab ) for non-trivial a, b ∈ Q ∗ / Q ∗ , and let F be thequadratic extension of E = Q ( √ a, √ b ) corresponding with a solution of (5) . Theextension F/K is minimally ramified if the following conditions hold: ( a ) The extension
F/K is unramified over all odd primes p gcd(∆( a ) , ∆( b )) . ( b ) The extension
F/K is unramified over if ∆( a )∆( b ) is odd, or if one of ∆( a ) , ∆( b ) is . ( c ) If { ∆( a ) , ∆( b ) } ≡ { , } mod 8 , take s ∈ { a, b } such that ∆( s ) ≡ .The local biquadratic extension Q ( √ s ) ⊂ F ⊗ Q must have conductor . By [16, Lemma 7.7] it is possible to twist a given F to a suitable F t which isminimally ramified over Q ( √ ab ). For convenience, a degree 8 dihedral extension of Q is called minimally ramified if it is so over its subfield defined by the order 4 cyclicsubgroup of the Galois group. Observe that the definition imposes no restrictionsover the prime 2 in case 2 is totally ramified in Q ( √ a, √ b ). Definition 3.2.
For non-trivial a, b, c ∈ Q ∗ / Q ∗ with local quadratic Hilbert sym-bols (7) ( a, b ) p = ( a, c ) p = ( b, c ) p = 1 for all primes p and moreover (8) gcd(∆( a ) , ∆( b ) , ∆( c )) = 1 , set K = Q ( √ ab ) and E = Q ( √ a, √ b ) , and take a corresponding F/K which isminimally ramified. Define [ a, b, c ] ∈ Gal(
F/E ) = µ by [ a, b, c ] = ( Art( c , F/K ) if c > c ∞ , F/K ) if c < where Art( · , · ) is the Artin symbol and c ∈ I ( O K ) has norm | c | with c the square-free integer representing c , and ∞ denotes an infinite prime of K .If at least one of a, b and c is trivial then one sets [ a, b, c ] = 1 . Proposition 3.3.
For a, b, c ∈ Q ∗ / Q ∗ satisfying (7) and (8) , the R´edei sym-bol [ a, b, c ] ∈ µ is well-defined. Moreover, the symbol is tri-linear, and perfectlysymmetrical in all three arguments.Proof. (Sketch.) If p | c , then ( c, b ) p = ( c, a ) p = 1 implies that p is either split or ram-ified in both Q ( √ a ) and Q ( √ b ). Condition (8) implies that p cannot ramify in both,hence a prime p K | p in K has norm p and splits in E . The prime p K is unramified in F by the minimal ramification of F , where the parity of p determines whether this WO-DESCENT ON SOME GENUS TWO CURVES 5 is due to condition ( a ) or ( b ). It follows that indeed Art( p K , F/K ) ∈ Gal(
F/E ),and as Gal(
F/E ) is in the center of Gal( F/ Q ) this Artin symbol is independent of p K . When c < K is real, the Artin symbol in F of any infinite prime of K measures whether F is real or complex and hence is independent of the choice ofinfinite prime of K . As [ a, b, c ] is the product of such Artin symbols we see that[ a, b, c ] does not depend on the choice of c or ∞ . For the independence of F werefer to [16, Corollary 8.2].The set of triples ( a, b, c ) in Q ∗ / Q ∗ for which (7) and (8) hold is ‘tri-linearlyclosed’, and the R´edei symbol [ a, b, c ] is clearly linear in c , hence tri-linearity followsfrom the symmetry. The symmetry in the first two arguments is immediate, whilethe identity [ a, b, c ] = [ a, c, b ]is a non-trivial reciprocity depending on the product formula for quadratic Hilbertsymbols in Q ( √ a ). The proof of this reciprocity is the subject of [16, Section 8]. (cid:3) Example 3.4.
Consider the case when a = b = 2. Then the invariant fields F and F ′ of the subgroups generated by − Q ( ζ ) / Q ),respectively, are two minimally ramified extensions of Q which can be used tocompute a R´edei symbol of the form [2 , , c ] provided that the symbol is defined,i.e. when ∆( c ) is odd and (2 , c ) = 1, i.e. when c ≡ c = − p for a prime p ≡ − F is totally real and p splits completelyin F precisely when p ≡ ± , , − p ] = ( p ≡ − − p ≡ F ′ as p splitscompletely in F ′ precisely when p ≡ , Nonexample 3.5.
Continuing the setup of Example 3.4, we see that ‘[2 , , − ′ isnot defined as ∆( −
1) is even (although (2 , − = 1). We can nonetheless consideran Artin symbol ‘that should define [2 , , − F is real and F ′ complex,such a symbol is not independent of the minimally ramified extension. Example 3.6.
Let p ≡ π ∈ Z [ √
2] be an element ofnorm p with conjugate π ′ . Then [2 , p, p ] = 1 precisely when π is a square mod π ′ . Since [2 , p, p ] = [ p, p, E of Q ( ζ p ). As E corresponds with the subgroup of fourth powersin Gal( Q ( ζ p ) / Q ) = ( Z /p Z ) ∗ and 2 mod p = Frob p ∈ Gal( Q ( ζ p ) / Q ), we see that 2splits completely in E precisely when 2 mod p is a fourth power, i.e. when p splitscompletely in Q ( √ , − , p ] = 1. We thus have the identity[2 , p, p ] = [2 , − , p ] . With this we obtain a generalisation of [17, Prop. 4.1], where it is used to provethat ( Z / Z ) ⊂ X ( E/ Q )[2] for the elliptic curve E defined by y = ( x + p )( x + p )for a prime p ≡ √− ∈ F p is a square. Corollary 3.7.
Let p ≡ be a prime, let π ∈ Z [ √ have norm p withconjugate π ′ and let i ∈ F p be a primitive fourth root of unity. Consider thefollowing statements. T. EVINK, G.J. VAN DER HEIDEN, AND J. TOP (a) π is a square mod π ′ .(b) i is a square mod p .Then the statements are equivalent when p ≡ , while for p ≡ exactly one of the statements holds.Proof. Statement ( a ) holds when [2 , p, p ] = [2 , − , p ] = 1, while statement ( b ) holdswhen [2 , − , p ] = 1. The result follows because[2 , − , p ] · [2 , − , p ] = [2 , , p ] = ( p ≡ , − p ≡ . (cid:3) Computation of -Selmer groups We start by recalling the explicit form of 2-descent that will be used. Let K bea number field and C the hyperelliptic curve defined by y = f ( x ), for f ∈ K [ x ]square-free and of odd degree 2 g + 1. We have the short exact sequence0 → J ( K ) / J ( K ) → S ( J/K ) → X ( J/K )[2] → , where S ( J/K ) and X ( J/K ) are respectively the 2-Selmer group and the Shafarevich-Tate group defined in terms of Galois cohomology by S ( J/K ) := ker (cid:16) H ( G K , J ( K )[2]) → Y p H ( G K p , J ( K p )) (cid:17) , X ( J/K ) := ker (cid:16) H ( G K , J ( K )) → Y p H ( G K p , J ( K p )) (cid:17) . By [14, Theorems 2.1 & 2.2] one has H ( G K , J ( K )[2]) ∼ = ker( A ∗ /A ∗ N −→ K ∗ /K ∗ ),where A = K [ x ] / ( f ( x )) and N is induced by the norm map A → K . This identifies S ( J/K ) with the elements in ker( A ∗ /A ∗ N −→ K ∗ /K ∗ ) that are mapped, accordingto the commutative diagram J ( K ) / J ( K ) A ∗ /A ∗ J ( K p ) / J ( K p ) A ∗ p /A ∗ p , δδ p into im( δ p ) for all primes p of K .We consider the special case that f ∈ O K [ x ] is monic and completely splits, so f = Q g +1 i =1 ( x − α i ) for distinct α j ∈ O K . In this case A ∼ −→ L g +1 i =1 K determinedby x ( α , . . . , α g +1 ), and the norm map A → K corresponds to multiplication ⊕ g +1 i =1 K → K . Hence the kernel of the norm ⊕ g +1 i =1 K ∗ /K ∗ N −→ K ∗ /K ∗ consistsof the ‘hyperplane’ of those (2 g + 1)-tuples for which the product of all coordinatesis trivial.Let S consist of the real primes of K together with the finite primes dividing2∆( f ), and put K ( S ) := { x ∈ K ∗ /K ∗ : ord p ( x ) ≡ p / ∈ S } .One has (compare [14, pp. 226-227])(9) S ( J/K ) ⊂ ker (cid:18) g +1 M i =1 K ( S ) → K ( S ) (cid:19) , and S ( J/K ) consists of those elements in the kernel of (9) that map into im( δ p )for each p ∈ S in the following diagram. WO-DESCENT ON SOME GENUS TWO CURVES 7 J ( K ) / J ( K ) g +1 M i =1 K ∗ /K ∗ J ( K p ) / J ( K p ) g +1 M i =1 K ∗ p /K ∗ p . δδ p Here the injective homomorphism δ and similarly δ p is given by(10) r X i =1 [ P i ] − r [ ∞ ] r Y i =1 ( x ( P i ) − α , . . . , x ( P i ) − α g +1 ) , for P , . . . , P r ∈ C ( K ) forming a G K -orbit not containing a Weierstrass point.The j -th coordinate of the δ -image of [( α i , − [ ∞ ] for i = j is α i − α j . The i -thcoordinate is then determined by the hyperplane condition: it equals Q j = i ( α i − α j ).As already remarked in Section 2 the cardinality of J ( K p ) / J ( K p ) and hence thatof im( δ p ) is known. In practise this makes it fairly straightforward to describeexplicit representants of the elements in im( δ p ), for each p ∈ S .The group K ( S ) fits in the exact sequence0 R ∗ S /R ∗ S K ( S ) Cl( R S )[2] 0 β where R S = { } ∪ { x ∈ K ∗ : ord p ( x ) ≥ p / ∈ S } is the ring of S -integers in K . Here β sends xK ∗ to the class [ IR S ], where x O K = a I with a and I co-prime fractional ideals such that a is supported on prime ideals of S and thesupport of I does not contain any prime of S . This is well-known; for completenesssee [6, Prop. 2.4.4]. The case of interest to us is when K has odd class number. Proposition 4.1. If K has odd class number then the map R ∗ S /R ∗ S → K ( S ) is anisomorphism. Moreover, for each finite p ∈ S writing p k p = ( x p ) with k p the orderof p in the class group of K , the x p together with an F -basis for O ∗ K / O ∗ K form an F -basis for K ( S ) .Proof. A detailed proof of this standard fact is provided in [6, Cor. 2.4.7]. (cid:3)
For an odd prime p write p ∗ = ( − ( p − / p , so Q ( √ p ∗ ) is the quadratic subfieldof the cyclotomic field Q ( ζ p ). In what follows we will compute 2-Selmer groups overthese quadratic fields. One has Lemma 4.2.
For any odd prime p the field K = Q ( √ p ∗ ) has odd class number,and if K is real (i.e., p ≡ ) then a fundamental unit of K has norm − .Proof. For a proof using genus theory, see for example [16, Thm 2.1]. A slightlymore direct argument is given in [6, Appendix A.2]. (cid:3) Proofs of the rank and Shafarevich-Tate group results
Consider the genus two hyperelliptic curves C/ Q : y = f ( x ) := x ( x − x − , and, for p any prime number, C p / Q : y = x ( x − p )( x − p ) . T. EVINK, G.J. VAN DER HEIDEN, AND J. TOP
Then C p is a quadratic twist of C over both Q ( √ p ) and Q ( √− p ). Let J and J p denote the Jacobians of C and C p , respectively. Observe that(11) rank J p ( Q ) + rank J ( Q ) = rank J ( Q ( √± p )) , for both possibilities of the sign ± . A quick computation (Lemma 5.1) yieldsrank J ( Q ) = 0. Since for the Jacobians at hand the torsion subgroup yields asubgroup of the 2-Selmer group of dimension 4, it follows thatrank J p ( Q ) ≤ dim F S ( J/ Q ( √± p )) − . Using Q ( √ p ) in case p ≡ ,
17 mod 24 and Q ( √− p ) for p ≡
23 mod 24, it will beshown that for certain subsets of these primes the bound for rank J p ( Q ) obtainedin this way sharpens the one which follows by directly applying Proposition 2.1.Specifically, this results in proofs for Theorems 2.4 - 2.6.Label the roots of f as ( α , α , α , α , α ) = ( − , − , , , F ⊃ Q and a point ( ξ, η ) ∈ C ( F ) write D ξ ∈ J ( F ) for the point corresponding to thedivisor [( ξ, η )] − [ ∞ ] on C . Note that although D ξ depends on η , its image in the2-Selmer group S ( J/F ) does not. The image of J ( Q )[2] under δ is spanned by x + 2 x + 1 x x − x − D − − − − − D − − − − − D − − D − − x − α i denotes the map [ P ] − [ ∞ ] x ( P ) − α i as in (10), compare [14].The local fields for which we need the images im δ p are Q , Q , Q ( i ) and R .Much of this was already done in [8, pp. 43-45]. One has Q ∗ / Q ∗ = h− , , i , Q ∗ / Q ∗ = h− , i , for F = Q ( i ) moreover F ∗ /F ∗ = h , r i , where r = 1 + i , andof course R ∗ / R ∗ = h− i . The local images are then spanned as follows. Q x + 2 x + 1 x x − x − D − − − − − D − − − − − D − − D − − D − − D − − Q x + 2 x + 1 x x − x − D − − − − − D − − − D − − D − − Q ( i ) x + 2 x + 1 x x − x − D − D − D i r r r rD i r r WO-DESCENT ON SOME GENUS TWO CURVES 9 R x + 2 x + 1 x x − x − D − − − − − D − − Lemma 5.1.
We have rank J ( Q ) = 0 .Proof. It suffices to show dim F S ( J/ Q ) = 4. Note ∆( f ) = 2 · , so S = { , , ∞} and K ( S ) = h− , , i . Then S ( J/ Q ) injects into the 2-adic image, and S ( J/ Q ) = A ⊕ δ ( J ( Q )[2])where A consists of all x ∈ S ( J/ Q ) with 2-adic image in the span of( 2 , − , , − , , ( 1 , , − , , − . If x = ( e , . . . , e ) ∈ A , then the 3-adic image forces e ∈ h− i , hence x is in thespan of (1 , , − , , − x is trivial because (1 , , − , , − im( δ ).Thus A = 0 and S ( J/ Q ) has F -dimension 4. (cid:3) We now compute S ( J/ Q ( √− p )) for p ≡
23 mod 24 and S ( J/ Q ( √ p )) for p ≡ ,
17 mod 24. The computation follows [6, § loc. sit. Consider a prime p ≡
23 mod 24 and let K = Q ( √− p ). Then K is complex andboth 2 and 3 split in K , so as set S of places of K needed for embedding S ( J/K )in ⊕ i =1 K ( S ) we take the four primes dividing 6. The completion of K at a primein S equals Q or Q .Write p , q for the prime ideals in O K dividing 3 and let k be the order of [ p ]in Cl K . Then p k = ( x ) for some x ∈ O K . Since q ∤ ( x ) and K q = Q , this x maps to ± K ∗ q /K ∗ q . Multiplying x by − x is a square in K q . The conjugate y ∈ O K of x satisfies q k = ( y )and x y = 3 k .Let p , q be the prime ideals in O K over 2. In the p -adic completion, x and y yieldelements of h− , i ⊂ Q ∗ / Q ∗ . By Lemma 4.2 the order k of [ p ] ∈ Cl K is odd,so the product x y yields 3 ∈ Q ∗ / Q ∗ . Hence exactly one of x , y after p -adiccompletion has image 1 or − Q ∗ / Q ∗ . As im p ( y ) = im q ( x ), this implies that x maps into h− i ⊂ Q ∗ / Q ∗ for precisely one of p , q . Denote this ideal by p , then p is unramified in K ( √ x ).Let x ∈ p be a generator for p k , with k the order of [ p ]. As above, multiply-ing x by − x maps q -adically into h− i ⊂ Q ∗ / Q ∗ , where q is the conjugate of p . Let y be the conjugate of x , so q k = ( y ) and x y = 2 k .Proposition 4.1 implies K ( S ) = h− , x , y , x , y i . We collect the local imagesin K ∗ p /K ∗ p of these generators, for p ∈ S = { p , q , p , q } , as follows.(12) p q p q − − − − − x y x y For l ∈ { , } recall im p ( x l ) = im q ( y l ) for conjugate p and q in S and x l y l = l k l with k l odd. Hence the 2 × x l , y l and conjugate p , q is determined by any one entry in the block.The normal closure of K ( √ x ) / Q yields a minimally ramified extension over Q ( √− p ), as p is unramified in K ( √ y ). Hence y is p -adically a square if andonly if [ − p, ,
2] = 1. Thus the top left block in (12) is determined by the R´edeisymbol [2 , , − p ]. As suggested by the coloring, the two blocks away from thediagonal are both determined by the same R´edei symbol. To see this, note that thenormal closure of K ( √ x ) / Q yields a minimally ramified extension of Q ( √− p ).This extension has trivial inertia degree over 3, hence im p ( x ) = 1 if and onlyif [ − p, ,
6] = 1. Similarly, the normal closure of K ( √ x y ) / Q yields a minimallyramified extension of Q ( √− p ). Since im p ( y ) = 1, this implies im p ( x ) = 1 ifand only if [ − p, ,
3] = 1. Hence Table (12) is determined by the values of the twoR´edei symbols [2 , , − p ] and [3 , , − p ]. Below, the four possibilities for this pair ofsymbols will be considered. Remark 5.2.
Since (3 , = −
1, the similar statement ‘[ − p, ,
3] = [ − p, , a, b, c ] = [ a, b, c ] relies on the product formulafor quadratic Hilbert symbols in Q ( √ a ); there is nothing against using this productformula in Q ( √− p ). Here the identity Q p ( x , x ) p = 1 leads to im p ( x ) = 1 ⇔ im p ( x ) = 1, but one still needs the symbol [3 , , − p ] to link the two blocks tosplitting behaviour of primes in a fixed (i.e., not depending on p ) number field.For the Selmer group computations, observe that S ( J/K ) = A ⊕ im( J ( K )[2])for A = { ( e , . . . , e ) ∈ S ( J/K ) : e p e p } . First consider the case [2 , , − p ] = [3 , , − p ] = 1, which means the table is as follows. p q p q − − − − − x − y − x y x = ( e , . . . , e ) ∈ A . The p -adic and q -adic image implies e ∈ h x , − y i ,and therefore im p ( e ) ⊂ h− , i and im q ( e ) ⊂ h− i . This removes the fifth rowof the Q -table from consideration. As im p ( e ) = 1, one concludes im p ( x ) is inthe span of ( 6 , , − , , − , ( 6 , , − , , − . Together with im p ( e ) ⊂ h− i this gives e ∈ h y , y i . Next, im q ( e ) ⊂ h i andim q ( e ) ⊂ h− i implies im q ( x ) is in the span of( 2 , , , − , − , ( 3 , , , − , − , so e ∈ h x i . Since n = (1 , y , x , , x y ) ∈ A , a complement inside A of h n i isobtained by setting e = 1. For x in this complement im p ( x ) is trivial, hence WO-DESCENT ON SOME GENUS TWO CURVES 11 e i ∈ h y , x i for all i , implying im q ( x ) is in the span of (6 , , , , e , e ∈h y x i and e , e ∈ h y i . A nontrivial im q ( x ) can only occur for e , e , e , e all = 1, so this complement is at most one dimensional. Since ( y x , y , , y x , y ) ∈ A one concludes that A is two-dimensional, and dim F S ( J/K ) = 6.In the remaining three cases (i.e., [2 , , − p ] and [3 , , − p ] not both 1) the com-putation is analogous; for details see [6, § , , − p ] [3 , , − p ] dim F S ( J/K ) additional generators1 1 6 (1 , y , x , , x y ) , ( y x , y , , y x , y )1 − − y y , y , , − y y , y ) , (1 , − y , y , , − y y ) − − − Proof of Theorem 2.4.
Let p ≡
23 mod 48 be prime. Then p ≡ , , − p ] = −
1. The table above implies dim F S ( J/ Q ( √− p )) = 4and as a consequence rank J ( Q ( √− p )) = 0. Hence rank J p ( Q ) = 0 by equation (11).Since p ≡
17 mod 24, Proposition 2.1 yields dim F S ( J p / Q ) = 6 hence the exactsequence (3) shows X ( J p / Q )[2] ∼ = ( Z / Z ) . (cid:3) Remark 5.3.
Part of what is proven above is that dim F S ( J/ Q ( √− p )) for primes p ≡
23 mod 24 depends only on the values of [2 , , − p ] and [3 , , − p ]. Hence insteadof the provided calculations for an undetermined p ≡
23 mod 24 one may take afixed prime for each of the four possibilities for the pair of R´edei symbols, and usee.g. Magma [1] to compute the Selmer group for this prime. The smallest primescovering all cases are given in the table below. p [2 , , − p ] [3 , , − p ]191 1 147 1 − − − − Proposition 5.4.
For K = Q ( √ p ) with p ≡
17 mod 24 prime, dim F S ( J/K ) iscompletely determined by the R´edei symbols [2 , , p ] and [2 , − , p ] .Proof. Let σ , σ : K ֒ → R be the two real embeddings of K . Take a fundamentalunit ε ∈ O ∗ K with σ ( ε ) >
0. Lemma 4.2 implies εε = −
1, hence there is a uniqueprime ideal p ⊂ O K over 2 that is unramified in K ( √ ε ). Let q be the conjugateof p and write p k = ( x ) where k is the order of [ p ] in Cl K . Multiplying x by ± ε if necessary we can and will assume that x has positive norm and moreover q is unramified in K ( √ x ). Let y be the conjugate of x , so x y = 2 k . Put S = { p , q , (3) , σ , σ } , then K ( S ) = h− , ε, x , y , i . The table of images in K ∗ v /K ∗ v of the generators of K ( S ) is as follows (as before, r = 2 i ∈ Q ( i )). p q (3) σ σ − − − − − ε r − x ry r ε, x , y follow by observing that the inertia degree of 3 Z inthe normal closures of K ( √ x ) and K ( √ ε ) over Q equals 4. As p is unramifiedin K ( √ ε ) and in K ( √ y ), the normal closures over Q yield minimally ramifiedextensions. Hence im p ( ε ) = 1 ⇔ [ p, − ,
2] = 1 and im p ( y ) = 1 ⇔ [ p, ,
2] = 1and im σ ( x ) = 1 ⇔ [ p, , −
1] = 1. R´edei reciprocity completes the proof. (cid:3)
Aided by Magma for the rightmost column, one computes the following table. p [2 , , p ] [2 , − , p ] dim F S ( J/ Q ( √ p ))113 1 1 617 1 − − − − Proof of Theorem 2.5.
Take p ≡
17 mod 24 prime and put K = Q ( √ p ). Proposi-tion 5.4 and the table above show dim F S ( J/K ) = 4 ⇔ [2 , , p ][2 , − , p ] = − , − , p ] = −
1, which by Example 3.6 means p is not completely split in Q ( √ F S ( J/K ) = 4 ⇒ rank J ( K ) = 0 ⇔ rank J p ( Q ) = 0.Proposition 2.1 and the exact sequence (3) now finish the proof. (cid:3) Lastly we cover the case p ≡ Proposition 5.5.
For K = Q ( √ p ) with p ≡ prime, dim F S ( J/K ) iscompletely determined by the R´edei symbols [2 , , p ] , [2 , − , p ] , [3 , − , p ] and [3 , , p ] .Proof. Let p ≡ K = Q ( √ p ). As in the proof of Propo-sition 5.4 let σ , σ : K ֒ → R be the real embeddings, take a fundamental unit ε ∈ O K with σ ( ε ) >
0, let p be the prime over 2 that is unramified in K ( √ ε ),and denote the conjugate of p by q . Then p k = ( x ) with k = ord([ p ]), whereone chooses x ∈ O K of positive norm and such that q is unramified in K ( √ x ).Let p be the prime over 3 that splits in K ( √ x ), and let q be its conjugate. With k = ord([ p ]), write p k = ( x ) with x ∈ O K of positive norm, chosen so that p is unramified in K ( √ x ). For i ∈ { , } let y i be the conjugate of x i , so x i y i = i k i . WO-DESCENT ON SOME GENUS TWO CURVES 13
Put S = { p , q , p , q , σ , σ } , then K ( S ) = h− , ε, x , y , x , y i ⊂ K ∗ /K ∗ . In-formation on local images of K ( S ) is presented in the following table. p q p q σ σ − − − − − − − ε − x − y − x y × p, ,
3] = 1if and only if im p ( y ) = im p ( x y ) = 1, and similarly [ p, ,
6] = 1 precisely whenthe equivalence im q ( y ) = 1 ⇔ im p ( y ) = 1 holds. Since [ p, ,
3] = [ p, , q ( y ) = 1 and moreover im p ( y ) = 1 if and only if [ p, ,
3] = 1. Thechoice of x and the equality x y = 3 k for k = ord([ p ]) odd, implies the bottomleft block. The remaining assertions about the table (in particular: the regionscolored in the same shade of grey are determined by any one entry in that region)are straightforward and/or analogous to what we did in other mod 24 cases.As in the 17 mod 24 case, im p ( ε ) = 1 ⇔ [ p, − ,
2] = 1, and im σ ( x ) = 1 ⇔ [ p, , −
1] = 1, and im p ( y ) = 1 ⇔ [ p, ,
2] = 1.Finally, im p ( ε ) = im p ( εx ) = 1 precisely when [ p, − ,
3] = 1. Since im p ( x ) = 1,one has im σ ( x ) = 1 ⇔ [ p, , −
2] = 1. R´edei reciprocity finishes the proof. (cid:3)
Using Magma for the rightmost column results in the following table (in factimplying a stronger version of Proposition 5.5: dim F S ( J/ Q ( √ p )) for the primes p ≡ , , p ] , [2 , − , p ], and [3 , − , p ]). p [2 , , p ] [2 , − , p ] [3 , − , p ] [3 , , p ] dim F S ( J/ Q ( √ p ))2593 1 1 1 1 81153 1 1 1 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Proof of Theorem 2.6.
Let p ≡ K = Q ( √ p ). Proposi-tion 2.1 implies dim F S ( J p / Q ) = 8, hence as in the proofs of Theorems 2.4 and2.5 it suffices to show that dim F S ( J/K ) = 4 if p satisfies one of the conditions(a), (b), or (c) mentioned in the statement of Theorem 2.6.Note: p splits completely in Q ( √ ⇔ [2 , , p ][2 , − , p ] = [2 , − , p ] = 1. Also, p splits completely in Q ( p √ ⇔ [3 , − , p ] = 1, and [2 , , p ] = 1 ⇔ p ≡ p ∈ { , , , } in the tableabove. Condition (b) corresponds to p ∈ { , } in the table, and condition (c)to p ∈ { , } . In all these cases the table shows dim F S ( J/K ) = 4, hencethe result follows by using Proposition 5.5. (cid:3)
We finish this section by presenting an analogous result for elliptic curves; werestrict to p ≡ p mod 24. Proposition 5.6.
Let E/ Q be an elliptic curve with good reduction away from , and with E ( Q )[2] = E ( Q )[2] . For a prime p ≡ , the size of the -Selmergroup S ( E/ Q ( √ p )) is determined by E/ Q together with the R´edei symbols [2 , , p ] , [2 , − , p ] , [3 , − , p ] , [3 , , p ] . Proof.
We use the notation introduced in the proof of Proposition 5.5. Descentyields an embedding δ : E ( K ) / E ( K ) ֒ → ker M i =1 K ( S ) → K ( S ) ! and S ( E/K ) consists of the elements in L i =1 K ( S ) that locally are in the imageof the corresponding maps δ v , for all v ∈ S = { σ , σ , p , q , p , q } . For these v ,the image in K ∗ v /K ∗ v of a basis for K ( S ) is described in the table presented inthe proof of Proposition 5.5. As this table is determined by the four given R´edeisymbols and S ( E/K ) consists of triples of elements in K ( S ) that for v ∈ S locallyare in δ v ( E ( K v )), the result follows. (cid:3) Remark 5.7.
The finite list of elliptic curves satisfying the conditions from Propo-sition 5.6 was already presented in the PhD thesis of F.B. Coghlan [5]. In fact helisted all elliptic curves over Q having good reduction away from 2 and 3. Precisely28 of these have full rational 2-torsion. In the LMFDB tables [10] contain themunder the conductors { , , , , , , , , , } .6. The Q -rational points Here briefly Q -rational points of the curves C p are discussed. The proof ofCorollary 2.2 shows that for primes p such that rank J p ( Q ) = 0, the set C p ( Q )consists of the Weierstrass points only. Below a less immediate case is discussed,namely a situation with rank J p ( Q ) = 2. We remark that in this case rank J p ( Q ) isnot strictly smaller than the genus of C p so the standard Chabauty method doesnot apply.Take the prime p = 241. Using rank J p ( Q ) ≤ dim F S ( J/ Q ( √ p ))) −
4, the row p = 241 in the table preceding the proof of Theorem 2.6 yields rank J ( Q ) ≤ P = (cid:0) x − x + , x − (cid:1) ,Q = (cid:0) x − x + , x + (cid:1) turn out to define points in J ( Q ). The homomorphism δ : J p ( Q ) → S ( J p / Q )yields δ ( P ) = (2 , p, , p,
2) and δ ( Q ) = (1 , p, p, p, p ). These images are independent WO-DESCENT ON SOME GENUS TWO CURVES 15 of δ ( J p ( Q ) tor ) which is generated by (6 , − p, − p, − p, − p ), ( p, − , − p, − p, − p ),(2 p, p, , − p, − p ) and (3 p, p, p, − , − p ). Hence rank J ( Q ) = 2. Moreover byProposition 2.1 and equality (3) one concludes X ( J / Q )[2] ∼ = ( Z / Z ) .To determine C ( Q ) the methods developed in [3] will now be used. Althoughthis works in much greater generality, here it is only briefly recalled in the specialcase of the curves C p . Consider the composition C p ( Q ) −→ J p ( Q ) δ −→ S ( J p / Q )mapping ( a, b ) ∈ C p ( Q ) with b = 0 to ( a + 2 p, a + p, a, a − p, a − p ) ∈ S ( J p / Q ). Is s = ( e , . . . , e ) ∈ S ( J p / Q ), then being in the image of C p ( Q ) implies that one hasa rational point on the smooth, complete curve X s / Q corresponding to the affineequations x + 2 p = e y , x + p = e y , x = e y , x − p = e y , x − p = e y . Here by abuse of notation e j represents the class e j ∈ Q ∗ / Q ∗ ; the result is in-dependent of this representing element. The curve X s is what in [3] is called atwo-cover of C p over Q . The “Two-Selmer set” of C p / Q is (cid:8) s ∈ S ( J p / Q ) : X s has rational points everywhere locally (cid:9) . As an example, for p = 241 let s := δ ( P ) = (2 , p, , p, X s one has x + 2 p = 2 y and x − p = py , defining the conic Q : 2 y − py = 3 p .One obtains a finite morphism X s → Q defined over Q . Since Q ( Q ) (as well as Q ( Q )) is empty, this shows δ ( P ) is not in the Two-Selmer set of C p / Q . In otherwords: although δ ( P ) is everywhere locally (even globally!) in δ v ( J p ( Q v )), it is notin the image of C p ( Q ) ⊂ J p ( Q ).The Magma command TwoCoverDescent(); computes the curves X s corre-sponding to the Two-Selmer set. In our case it turns out that of the 2 elements in S ( J / Q ), only the six δ ([ W ] − [ ∞ ]) for W ∈ C p ( Q ) a Weierstrass point, are inthe Two-Selmer set. We now show that for each of these six elements s one findsthat { R ∈ C p ( Q ) : δ ([ R ] − [ ∞ ]) = s } consists of only a Weierstrass point. As aconsequence, C ( Q ) = {∞ , (0 , , ( ± , , ( ± , } . We use the notation D ξ (here for certain elements in J p ) as introduced on page 8. • s := δ (0) = (1 , , , , a, b ) ∈ C ( Q ) with b = 0 would result in δ -image s , then in particular the elliptic curve E : y = x ( x + p )( x + 2 p )admits a point in E ( Q ) with x = a and y = 0. Since E ( Q ) ∼ = Z / Z × Z / Z ,no such point exists. • s := δ ( D − p ) = (6 , − p, − p, − p, − p ). In this case, considering the 1st, 3rd,and 4th entry results in the elliptic curve E : y = x ( x +2 p )( x − p ) satisfying E ( Q ) ∼ = Z / Z × Z / Z . Hence only the Weierstrass point ( − , ∈ C p ( Q )yields δ -image s . • s := δ ( D − p ) = ( p, − , − p, − p, − p ). Here the 2nd, 4th, and 5th entryresults in the elliptic curve E : − y = ( x + p )( x − p )( x − p ) whose onlyrational points are the points of order at most 2. Reasoning as before, thisimplies that only the Weierstrass point ( − , ∈ C p ( Q ) yields δ -image s . • s := δ ( D ) = (2 p, p, , − p, − p ). Using entries 1, 2, and 3 results in theelliptic curve E : 2 y = x ( x + p )( x + 2 p ), whose only rational points are thepoints of order at most 2. As above, this implies that only the Weierstrasspoint (0 , ∈ C p ( Q ) yields δ -image s . • s := δ ( D p ) = (3 p, p, p, − , − p ). Here we use entries 1, 2, and 4, leading to E : − y = ( x + 2 p )( x + p )( x − p ). Also here the only rational points are thepoints of order dividing 2. So only the Weierstrass point (241 , ∈ C p ( Q )yields δ -image s . • s := δ ( D p ) = ( p, p, p, p, E : 2 y = ( x + 2 p )( x + p )( x − p ). Here as well, the only rational pointsare the points of order dividing 2. So (482 , ∈ C p ( Q ) is the only rationalpoint with δ -image s .This completes the determination of the rational points on C .Note that for p = 5 there are two additional points: one has C ( Q ) = 8, where thetwo non-Weierstrass points are (20 , ± p > C p ( Q ) > X := X (1 , , , , of C p / Q correspondsto the affine model x + 2 p = y , x + p = y , x = y , x − p = y , x − p = y . The maps y j
7→ − y j define a group ( Z / Z ) in Aut Q ( X ). Using appropriatesubgroups one obtains up to isogeny the decomposition of Jac( X ) over Q givenas follows. Let E : y = ( x − x −
4) and E : y = x − x and finally E a : y = x ( x + 1)( x −
2) be elliptic curves over Q . For any such E/ Q andany d ∈ Q / Q ∗ we write E ( d ) for the quadratic twist of E defined by d . ThenJac( X ) is isogenous over Q to the product J p × ( E ) × E ( − × E ( p )24 × E ( − p )24 × (cid:16) E ( p )32 (cid:17) × E (2 p )32 × (cid:16) E ( − a (cid:17) × (cid:16) E ( p )96 a (cid:17) × (cid:16) E ( − p )96 a (cid:17) . In particular the rank of Jac( X ) is determined by that of J p and of the giventwists of the three elliptic curves E , E , and E a . Using analogs of Proposi-tion 5.6 for various classes of primes p provides a natural approach towards boundingrank Jac( X )( Q ). Acknowledgement
Several people provided valuable suggestions during this work. We mentionPeter Stevenhagen who in the context of [17] already in the 90’s showed one of usrelations with 4-ranks of class groups, and much more recently referred us to hiswork on R´edei symbols. We also mention Nils Bruin who explained one of us theMagma implementation of the elliptic Chabauty method, originally while workingon [4, Section 3] but also very relevant for the much simpler situation given inSection 6 of the present paper. Michael Stoll directed us to the theory of Two-Selmer sets and envisioned the result given in Section 6. Finally, Steffen M¨uller,Jeroen Sijsling, and Marius van der Put all showed their interest in this work andin that way encouraged us to complete it.
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Institute of Algebra and Number Theory, Ulm University, Helmholtzstr. 18, 89081Ulm, Germany.
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