CCONSTRUCTING CURVES OF HIGH RANK VIA COMPOSITEPOLYNOMIALS
ARVIND SURESH
Abstract.
We improve on a construction of Mestre–Shioda to produce some families ofcurves X/ Q of record rank relative to the genus g of X . Our first main result is that for anyinteger g (cid:62) g ≡ g hyperelliptic curvesover Q with at least 8 g + 32 Q -points and Mordell–Weil rank (cid:62) g + 15 over Q . Our secondmain theorem is that if g + 1 is an odd prime and K contains the g + 1-th roots of unity,then there exist infinitely many genus g hyperelliptic curves over K with Mordell–Weil rankat least 6 g over K . Introduction
Let K be a number field, let X/K be a nice (i.e. smooth, projective, geometrically integral)curve of genus g , and let J X /K be the Jacobian of X . The Mordell-Weil Theorem says that J X ( K ) is a finitely generated abelian group. The rank of X/K is the integerrank
X/K := dim Q J X ( K ) ⊗ Z Q . Faltings’ Theorem [Fal83, Satz 7], formerly known as the Mordell Conjecture, says that if g (cid:62)
2, then the set of rational points X ( K ) of X is finite. Given these finiteness theorems, itis natural to wonder, for a fixed genus g and number field K , how large the integers X ( K )and rank X/K can be. We define two integers, R ( K, g ) and N ( K, g ), if they exist, as follows: R ( K, g ) := sup { R | rank X/K (cid:62) R for infinitely many genus g curves X/K } N ( K, g ) := sup { N | X ( K ) (cid:62) N for infinitely many genus g curves X/K } . (0.1)Above, and for the rest of the article, by “infinitely many curves,” we mean “ infinitely manynice curves that are pairwise non-isomorphic over an algebraic closure K .”For any number field K and positive g , whether the constant R ( K, g ) exists is an openquestion. On the other hand, it is a theorem of Caporaso, Harris, and Mazur [CHM97,Theorem 1.2] that if the Weak Lang Conjecture [CHM97, Conjecture A] is true (which is notcurrently known), then the integer N ( K, g ) exists for any number field K and g (cid:62) X/K with a rational point is an elliptic curve, and X ∼ = J X . The existenceof R ( Q ,
1) has been the subject of multiple folklore conjectures; the papers [PPVW19, Wat15]give heuristics that suggest R ( Q , (cid:54)
21. For g (cid:62)
2, to the best of our knowledge, thereexist no such heuristics, nor a folklore conjecture in either direction.Producing an infinite set of genus g curves X/K satisfying rank
X/K (cid:62) R and X ( K ) (cid:62) N would yield lower bounds R ( K, g ) (cid:62) R and N ( K, g ) (cid:62) N . The standard way to do this is bythe method of specialization ; one constructs over the function field K ( B ) of a K -variety B acurve X/K ( B ) that has the desired properties (genus g , many rational points, large rank),and then one takes specializations of X that preserve these properties. Date : February 4, 2021. a r X i v : . [ m a t h . N T ] F e b The use of the specialization method to construct curves of high rank originated in thework of Néron [Nér52, Nér56], who proved a theorem [Nér52, Théorème 6] on the injectivity ofthe specialization map (cf. 2.2) for Jacobians of curves
X/K ( B ), with B rational over K , andapplied it to construct infinitely many elliptic curves of rank at least 10 (i.e. R ( Q , (cid:62) g (cid:62)
2, infinitely many genus g curves of rank at least 3 g + 6 (see also [SU99] for acorrection of Néron’s original claim of 3 g + 7).Mestre [Mes91a, Mes91b] gave a way to construct genus g curves X/K ( t , . . . , t n ) with atleast 8 g + 12 rational points; here, n is an integer that depends on g . His construction yields N ( Q , g ) (cid:62) g + 12, and he used it in the case g = 1 to prove R ( Q , (cid:62)
12. Shioda [Shi98]generalized Mestre’s construction and proved R ( Q , g ) (cid:62) g + 7 and N ( Q , g ) (cid:62) g + 16 for all g (cid:62)
2; we refer to this as the
Mestre–Shioda construction (see Section 3).In this paper, largely inspired by the ideas in [Shi98], we produce a refinement of theMestre–Shioda construction and use it to improve on the known lower bounds for the constantsdefined in (0.1). Our first theorem is the following.
Theorem 0.2 (see 13.11) . Let g be a positive integer not divisible by . Then R ( Q , g ) (cid:62) g + 8 if g (cid:62) g ≡ g + 11 if g (cid:62) g ≡ , g + 15 if g (cid:62) g ≡ , ,N ( Q , g ) (cid:62) g + 17 if g ≡ g + 24 if g ≡ , g + 32 if g ≡ , . The following table summarizes the current records for lower boundson R ( Q , g ). The records below for g = 2 and 5 have not explicitly appeared in print; the formerfollows by applying [Kul01, Théorème 2.2.1] to the elliptic curve E/ Q with rank E/ Q (cid:62) y = f ( x ), with deg f = 6. The record for g = 5 then follows by considering the set of genus5 curves given by y = f ( x + bx ), with b ∈ Q ; we leave the details to the reader. g , , (cid:62) R ( Q , g ) (cid:62)
19 [Elk06] 32 26 [Kul01, Thm. 3.2.1] 4 g + 7 [Shi98, Cor. 2]Table 0.4: Rank recordsThe curves of Shioda’s family that witness the bound R ( Q , g ) (cid:62) g + 7 also yield the currentrecord N ( Q , g ) (cid:62) g + 16 for all g (cid:62) g = 4, for which the recordis N ( Q , (cid:62)
126 [Elk96, Page 9]; the record for g = 2 is N ( Q , (cid:62)
150 [Elk09]. Thus,Theorem 0.2 gives a new record for R ( Q , g ) for all g (cid:62) N ( Q , g ) for all g (cid:62) g = 4.Our next theorem produces a “better-than-constant” improvement in the bound for R ( K, g )for certain K and g . Theorem 0.5 (see 14.5) . Let g (cid:62) be an integer such that g + 1 is prime, and let K be anumber field that contains the ( g + 1) -th roots of unity. Then R ( K, g ) (cid:62) g . Let T be a finite abelian group and consider, if it exists, the constant: R ( K, g ; T ) := sup { R | J X ( K ) ⊃ Z R ⊕ T for infinitely many genus g curves X/K } . (0.6)Fix T = Z / Z and K = Q . The current record for g = 1 (elliptic curves) is R ( Q , Z / Z ) (cid:62)
11, due to Elkies [Elk07], and for g (cid:62)
2, the record is R ( Q , g ; Z / Z ) (cid:62) g + 2, due toShioda [Shi98, Theorem 7] (see also Remark 10.16). The next theorem improves the recordfor certain g ’s. Theorem 0.7 (see 11.4) . Let g be a positive integer divisible by . Then R ( Q , g ; Z / Z ) (cid:62) g + 6 . Remark 0.8.
A few remarks are in order.(1) The definition in (0.1) implies that R ( K, g ) (cid:62) R ( Q , g ) for any number field K . It isperhaps natural to expect that if we fix g , then R ( K, g ) goes to infinity as [ K : Q ]goes to infinity. It appears, however, to be a non-trivial task to exhibit a K and g forwhich R ( K, g ) enjoys a bound better than one of the form R ( K, g ) (cid:62) g + C , andwe know of no results in this direction in the literature. In fact, we know of onlyone other construction that yields a bound of the form 4 g + C , namely, the authorsof [HKL +
20] prove R ( K, g ) (cid:62) g + 2, conditional on a conjecture of Nagao, using amethod distinct from Mestre–Shioda.(2) There exist number fields K and small genera g for which N ( K, g ) is known tohave a significantly better lower bound than what is mentioned above (see [Elk96]).In particular, for large g and K containing the primitive ( g + 1)-th roots of unity,the current record is N ( K, g ) (cid:62) g + 1), due to Brumer [Cap95, Section 5] (seeRemark 14.7).(3) Fix a number field K and consider, if it exists, the integer ρ K := limsup g →∞ ( R ( K, g ) /g ) . Néron’s construction in the 1950’s gives ρ K (cid:62)
3. In the 1990’s, this was improvedto ρ K (cid:62) ρ K for anynumber field K . It would be very interesting if one could produce a construction thatimproves this bound for some K .(4) The analogues of Faltings’ Theorem and the Mordell-Weil Theorem also hold forcurves X/K , with K the function field of a curve in positive characteristic. It isproved in [CUV12] (resp. [Ulm07]) that N ( F p ( t ) , g ) (resp. R ( F p ( t ) , g )) does not exist. In Section 1, we recall some facts about curves and Jacobians. InSection 2, we review the specialization method described in the introduction. In Section 3, wereview the Mestre–Shioda construction of curves with many rational points. In Section 4, weoutline the strategy behind our refinement of the Mestre–Shioda construction. In Section 5,we discuss how our strategy is related to a classical problem of number theory, the Prouhet-Tarry-Escott problem. In Sections 6 – 11, we prove Theorem 0.7 and establish the bounds inTheorem 0.2 for the case g ≡ , , g ≡ , This work is part of the author’s doctoral dissertation at theUniversity of Georgia. The author is grateful to UGA and its math department for providinga stimulating environment for research. The author is greatly indebted to his advisor, DinoLorenzini, for carefully reading the manuscript and providing thoughtful feedback.
Contents
Introduction 11. Preliminaries 42. The specialization method 53. The Mestre–Shioda construction 64. Strategy behind improvement 75. Composite tuples 96. Parametrizing (6 , n )-composite tuples 107. Normality and rationality of B n K ( B n ) 149. Tools for bounding ranks of curves 1810. Bounding the ranks of curves parametrized by B d N ( K, g ) and R ( K, g ) for g ≡ , Z n /K N ( K, g ) and R ( K, g ) for odd g Preliminaries
A scheme
X/K is a K -variety if X is geometricallyintegral, separated, and of finite type over K ; the function field of X is denoted by K ( X ).A nice curve is a smooth, projective, one-dimensional K -variety. We will sometimes regard K -points of X as closed points with residue field K . We use bold font as short-hand todenote tuples of indeterminates; for instance, if we say “ t denotes the tuple of indeterminates t , . . . , t n ,” then K [ t ] denotes K [ t , . . . , t n ], K ( t ) the function field K ( t , . . . , t n ), and A n ( t )the affine space Spec K [ t ]. If we make an assertion about the Galois group Gal( L/F ) of anextension of fields
L/F , then it is implied that
L/F is Galois.
Let K be a field, and X/K a nice curve of genus g . Wedenote by Div( X ) the group of divisors on X , and by Pic ( X ) the group of degree 0 divisorsmodulo the subgroup of principal divisors. If L/K is a field extension, then we will often usethe natural injection Div( X ) , −→ Div( X L ) to regard divisors on X as divisors on X L . For D ∈ Div( X ), we denote its class in Pic ( X ) by [ D ]. The Jacobian of X , denoted J X /K , isan abelian variety of dimension g such that J X ( L ) ∼ = Pic ( X L ) for any field extension L/K (not necessarily finite) that verifies the condition X ( L ) = ∅ .For a K -morphism f : X −→ Y of curves, we denote by f ∗ (resp. f ∗ ) the induced pull-back(resp. pushforward) homomorphism on Div and Pic. Assume char K = 2. Let f ( x ) ∈ K [ x ] be a non-constantpolynomial. The affine curve X /K defined by the equation y = f ( x ) is smooth if and onlyif f ( x ) is separable. The degree of the equation y = f ( x ) (or of X ) is the integer d := deg f .Assume now that X is smooth. If g denotes the genus of X , then we have d = 2 g + 1 (resp.2 g + 2) if d is odd (resp. even). Let h ( x, z ) ∈ K [ x, z ] denote the binary form z g +2 f ( x/z ), andlet K [ x, y, z ] denote the graded K -algebra in which deg x = deg z = 1 and deg y = g +1. Then, X := Proj K [ x, y, z ] / ( y − h ( x, z )) is a nice curve of genus g , and X is naturally isomorphic to the open subset { z = 0 } ⊂ X . The natural projection π X : X −→ P K := Proj K [ x, z ] is finiteof degree 2, with branch divisor given by { h ( x, z ) = 0 } ; this is the quotient by the involution ι : X −→ X defined by ( x, y ) ( x, − y ). If d = 2 g + 1 is odd, then X has a single ramified K -point lying over ∞ := [1 : 0] ∈ P K , and if d = 2 g + 2 is even, then X has two (geometric)points lying over ∞ ; in either case, we let D X, ∞ ∈ Div( X ) denote the degree 2 divisor π ∗ ( ∞ ),and for any degree k divisor E ∈ Div( X ), we let (cid:15) ( E ) := [2 E − kD X, ∞ ] ∈ Pic ( X ).Since char K = 2, a nice curve X/K is birational to an affine curve y = f ( x ) if and onlyif X admits a degree 2 morphism to P K ; for g (cid:62)
2, such curves are called hyperelliptic curves . Lemma 1.4.
Let
X/K (resp.
Y /K ) be the nice curve associated (as in 1.3 above) to thesmooth, affine curve given by y = g ( x ) (resp. y = g ( x ) ). Suppose f : X −→ Y is afinite K -morphism of degree k , and h : P K −→ P K is finite of degree k and totally ramifiedat ∞ , and suppose that f ◦ π Y = h ◦ π X . Then, f ∗ D X, ∞ = D Y, ∞ and f ∗ D Y, ∞ = kD X, ∞ . Consequently, for points P ∈ X ( K ) and Q ∈ Y ( K ) , we have f ∗ ( (cid:15) ( P )) = (cid:15) ( f ( P )) ∈ Pic( Y ) , and f ∗ ( (cid:15) ( Q )) = (cid:15) ( f ∗ Q ) ∈ Pic( X ) . Proof.
The diagram immediately gives f ∗ D Y, ∞ = f ∗ π ∗ Y ∞ = π ∗ X h ∗ ∞ = π ∗ X ( k ∞ ) = kD X, ∞ . On the other hand, since ι Y ◦ f = f ◦ ι X , we have( ι Y ) ∗ f ∗ D X, ∞ = f ∗ ( ι X ) ∗ D X, ∞ = f ∗ D X, ∞ . So, f ∗ D X, ∞ ∈ Div( Y ) is a degree 2 divisor lying over ∞ ∈ P K , and which is fixed by ι Y ; thisimplies that f ∗ D X, ∞ = D Y, ∞ . The reader can now easily verify that the last statement in thelemma also holds. (cid:3) The specialization method
Now, and for the rest of this article, K will denote a number field . Let K /K be an extension of fields such that trdeg K K = n > K is algebraically closed in K . Given a nice curve X/ K of genus g (cid:62)
2, we can finda K -variety V with K ( V ) ∼ = K , and a smooth, projective family of curves X −→ V withgeometrically integral fibers, and generic fiber X/ K (cf. [Poo17, Theorem 3.2.1 (i)]). Thisfamily induces a K -morphism V −→ M g,K , where M g,K denotes the coarse moduli schemeof genus g curves over K (cf. [MFK94, Page 103, Proposition 5.4]); we say that X/ K is non-constant if this morphism is non-constant (this definition is independent of the choiceof V and family X /V ). An elliptic curve X/ K is called non-constant if its j -invariant is in K \ K . With X /V and X/K ( V ) as above, for a point s ∈ V , thefiber X s /κ ( s ) is called a specialization of X . The relative Jacobian of X /V is an abelian V -scheme J /V [BLR90, Proposition 9.4.4] such that, for any point s ∈ V , the fiber J s /κ ( s ) isisomorphic to the Jacobian of X s /κ ( s ); in particular, the generic fiber of J /V is isomorphicto J X /K ( V ).If V is regular, then we have an isomorphism of groups J X ( K ( V )) ∼ = J ( V ); indeed, [GLL15,Proposition 6.2] implies that the natural injective map J ( V ) , −→ J X ( K ( V )) is also surjective.For any P ∈ V ( K ), we also have a natural map J ( V ) −→ J P ( K ); pre-composing thiswith the isomorphism J X ( K ( V )) ∼ = J ( V ), we get the specialization map σ P : J X ( K ( V )) −→ J P ( K ) , which is a group homomorphism. The Lang-Néron Theorem (see [LN59, Theorems 1, 2]), a generalization of the Mordell-WeilTheorem, implies that J X ( K ( V )) is a finitely generated abelian group; we define as usual theinteger rank X/K ( V ) := dim Q J X ( K ( V )) ⊗ Z Q . Let V be a K -variety. We recall from [Ser97, Page 121, Section 9.5]or [CT20, Section 2] the notion of a thin set T ⊂ V ( K ). Thin sets are useful for thefollowing reason: if V ( K ) is not thin, then { V ( K ) \ T | T ⊂ V ( K ) is thin } is a collection ofZariski-dense subsets of V ( K ) that is closed under finite intersections. It is well-known (see[Ser97, Section 9.6]) that if X/K is a K -rational variety (i.e. X is K -birational to an affinespace A nK ), then X ( K ) is not thin. Theorem 2.4 ([CT20, Theorem 2.4], [Ser97, Section 11.1]) . Let V be a smooth K -variety,and let A /V be an abelian scheme with generic fiber A/K ( V ) . The set { P ∈ V ( K ) | σ P : A ( K ( V )) −→ A P ( K ) is not injective } is thin. Recall the constants defined in (0.1) and (0.6).
Corollary 2.5.
Let R and N be non-negative integers, let T be a finite abelian group, andlet B /K be a K -rational variety. Suppose there exists a curve X/K ( B ) of positive genus suchthat(a) X/K ( B ) is non-constant,(b) X ( K ( B )) (cid:62) N ,(c) J X ( K ( B )) contains a subgroup isomorphism to Z R ⊕ T .Then N ( K, g ) (cid:62) N and R ( K, g ; T ) (cid:62) R (and therefore also R ( K, g ) = R ( K, g ; { } ) (cid:62) R ).Proof. Suppose V ⊂ B is an open subscheme that is smooth, and which admits a smooth,projective family of curves X /V with generic fiber X/K ( V ) (as in 2.1). Since V is K -rational,for any thin set S ⊂ V ( K ), the complement V ( K ) \ S is Zariski-dense in V . Combiningthis with assumption (a), we see that: if S ⊂ V − . emK − . em is thin, then the set { X P /K } P ∈ V ( K ) \ S contains infinitely many nice genus g curves over K . By assumption (b), there exist distinct points P , . . . , P N ∈ X ( K ( V )), which extend to N sections V −→ X of X /V (cf. [GLL15, Proposition 6.2]); by replacing V with a non-emptyopen subset (if necessary), we may assume that these sections specialize to N distinct K -pointson any K -fiber of V . Then, the curves { X P /K } P ∈ V ( K ) witness the claim N ( K, g ) (cid:62) N .Similarly, applying Theorem 2.4 to the relative Jacobian J /V of X /V , we find that thereis a thin set S ⊂ V ( K ) such that J P ( K ) ⊃ Z R ⊕ T for any P ∈ V ( K ) \ S , so the curves { X P /K } P ∈ V ( K ) \ S witness the claim R ( K, g ; T ) (cid:62) R . (cid:3) The Mestre–Shioda construction
In this section, we outline the Mestre–Shioda construction.
Lemma 3.1.
Let d (cid:62) be an integer. Let m , h , and ‘ denote the tuples of indeterminates ( m , . . . , m d − ) , ( h , . . . , h d − ) , and ( l , . . . , l d − ) respectively, and define: m ( x ) := x d + m d − x d − + · · · + m ∈ K [ m ][ x ] h ( x ) := x d + h d − x d − + · · · + h ∈ K [ h ][ x ] ‘ ( x ) := l d − x d − + · · · + l ∈ K [ ‘ ][ x ] . (3.2) The K -morphism φ d : A d ( h ) × A d ( ‘ ) −→ A d ( m ) , defined on coordinate rings by m i i th coefficient of ( h ( x ) − ‘ ( x )) , i = 0 , . . . , d − , is an isomorphism. Thus, given a monic M ( x ) ∈ K [ x ] of degree d , there exists a unique monic H ( x ) ∈ K [ x ] of degree d and unique L ( x ) ∈ K [ x ] of degree at most d − , satisfyingthe relation M ( x ) = H ( x ) − L ( x ) . (3.3) Proof.
To show that φ d is an isomorphism, we need to express each h i , l i as a polynomialin the variables m , . . . , m d − ; for this, we can compare coefficients (in descending order)in the identity m ( x ) = h ( x ) − ‘ ( x ) and sequentially “solve” for the variables h i , l i in termsof m i . Indeed, we have h d − = m d − /
2. For j = 2 , . . . , d , the coefficient of x d − j in h ( x ) is of the form 2 h d − j + g j for some polynomial g j ∈ K [ h d − , h d − , . . . , h d − j +1 ], which yields h d − j = ( m d − j − g j ) / , for j = 2 , . . . , d . This expresses each coefficient of h ( x ) as a polynomialin the m i . For 0 (cid:54) i (cid:54) d −
1, the coefficient of x i in h ( x ) − ‘ ( x ) is of the form q i − l i ,where q i ∈ K [ h , . . . , h d − ], which yields l i = q i − m i , for i = 0 , . . . , d −
1. This expresseseach coefficient of ‘ ( x ) as a polynomial in the m i , and concludes the proof that φ d is anisomorphism.For the second part, consider the inverse φ − d : A d ( m ) −→ A d ( h ) × A d ( ‘ ), and suppose M ( x ) ∈ K [ x ] is monic of degree 2 d . There is a unique point P ∈ A d ( m )( K ) such that M ( x )is the specialization of m ( x ) at P . By construction, the specializations H ( x ) and L ( x ) of h ( x )and ‘ ( x ) respectively at φ − d ( P ) satisfy the identity (3.3), L ( x ) is of degree at most d − H ( x ) is monic of degree d . The uniqueness is obvious. (cid:3) Definition 3.4.
In the setting of Lemma 3.1, we refer to H ( x ) (resp. L ( x )) as the square-rootapproximation (resp. square-root remainder ) of M ( x ). Let u denote the tuple of indeterminates u , . . . , u d , let s j ( u ) denote the j -th elementary symmetric polynomial in the u i ’s, and considerthe morphism F : A d ( u ) −→ A d ( m ), defined on coordinate rings by m i ( − d − j s j ( u ).The classical Fundamental theorem of symmetric functions says that F is finite and surjective,and the extension K ( u ) /K ( m ) is Galois, with group isomorphic to the symmetric group S d .Using the isomorphism K [ ‘ , h ] ∼ = K [ m ] provided by Lemma 3.1, along with the injection K [ m ] , −→ K [ u ] corresponding to F , we can regard m ( x ) , ‘ ( x ) , h ( x ) as polynomials in K [ u ][ x ],which satisfy the identity( x − u ) · · · ( x − u d ) = m ( x ) = h ( x ) − ‘ ( x ) ∈ K [ u ][ x ] . (3.6)Now, consider the smooth affine curve X /K ( u ) given by the degree d − y = ‘ ( x ).The identity (3.6) implies that X has the K ( u )-rational points P i := ( u i , ± h ( u i )), for i = 1 , . . . , d . Shioda showed in [Shi98, Theorems 5, 6] that the divisor classes [2 P i − D X, ∞ ](cf. 1.3) generate a subgroup of rank 2 d − J X ( K ( u )), where X/K ( u ) is a smooth, propermodel of X . 4. Strategy behind improvement
We outline in this section our strategy to produce curves with more rational points relativeto their genus that what is obtained from the Mestre–Shioda construction.Continuing with the notation of 3.5, for a point P ∈ A d ( u )( K ), let ‘ P ( x ) , h P ( x ) , m P ( x ) ∈ K [ x ] denote the specializations of the polynomials ‘ ( x ) , h ( x ) , m ( x ) ∈ K [ u ][ x ] respectively. Our basic strategy is to find K -points of the closed subset { l d − = 0 } ⊂ A d ( u ), i.e. points P ∈ A d ( u )( K ) for which deg ‘ P (cid:54) d −
2. If deg ‘ P is small enough, then the genus of thespecialized curve C P /K defined by y = ‘ P ( x ) will be smaller than that of X /K ( u ); wewant to find such K -points P for which C P ( K ) (cid:62) d and rank C P /K (cid:62) d − Example 4.1.
Let d = 6, so that A dK = A K . In [Mes91a, Page 142, Section 2], Mestredescribed a way to systematically find Q -rational curves inside { l = 0 } ∩ { l = 0 } ⊂ A Q ;he used it to produce a non-isotrivial elliptic curve over Q ( t ) of rank at least 12 over Q ( t ).Subsequently, Nagao [Nag94] used the same method to find one of rank at least 13 over Q ( t ),and Kihara [Kih01] used it to find one of rank at least 14 over Q ( t ). We refer the readerto [Elk07] for an exposition of these and other constructions of high rank elliptic curves over Q ( t ) and Q .The starting point for the results of this paper is Observation 4.3 below, which gives a“natural” sufficient condition for a point P ∈ A d ( u )( K ) to satisfy deg ‘ P ( x ) (cid:54) d −
2. First,we need the following definition.
Definition 4.2.
A point ( a , . . . , a en ) ∈ A en ( K ) is ( e, n ) -composite if there exist polynomials G ( x ) , M ( x ) ∈ K [ x ], of degree e and n respectively, such that ( x − a ) · · · ( x − a en ) = M ( G ( x )). Observation 4.3.
Recall the space A d ( u ) = Spec K [ u , . . . , u d ] from the construction in 3.5.Suppose d factors as d = ed , with e (cid:62) d (cid:62)
2. Then, if P = ( a , . . . , a d ) ∈ A d ( u )( K )is ( e, d )-composite, then deg ‘ P ( x ) = d − ej , for some j (cid:62)
1. In particular, deg ‘ P ( x ) (cid:54) d − e . Proof.
By assumption, there exist polynomials G ( x ) , M ( x ) ∈ K [ x ], of degree e and 2 d respectively, such that m P ( x ) := ( x − a ) · · · ( x − a d ) = M ( G ( x )). If b denotes the leadingcoefficient of G ( x ), we can replace G ( x ) with G ( x ) /b and M ( x ) with M ( bx ), and henceassume without loss of generality that M ( x ) and G ( x ) are both monic. Let H ( x ) , L ( x ) denotethe square-root approximation and remainder of M ( x ), so that H ( x ) ∈ K [ x ] is monic ofdegree d , L ( x ) ∈ K [ x ] is of degree d − j for some j (cid:62)
1, and we have M ( x ) = H ( x ) − L ( x ) asin (3.3). Replacing x with G ( x ), we get m P ( x ) = M ( G ( x )) = H ( G ( x )) − L ( G ( x )) . Observenow that H ( G ( x )) is monic of degree d , anddeg L ( G ( x )) = e ( d − j ) = d − ej (cid:54) d − e. By the uniqueness statement in Lemma 3.1, we see that h P ( x ) = H ( G ( x )) and ‘ P ( x ) = L ( G ( x )), so we conclude deg ‘ P = d − ej for some j (cid:62)
1, as desired. (cid:3)
Let us now summarize how we will use Observation 4.3 going forward.For certain integers e (cid:62)
2, and for any integer d (cid:62)
2, we will define a positive dimensionalaffine K -rational variety V := Spec S with the following property: There exist elements t , . . . , t de ∈ S and monic polynomials g ( x ) and m ( x ) in S [ x ], of degree e and 2 d respectively,such that m ( g ( x )) = ( x − t ) · · · ( x − t de ) ∈ S [ x ] . Thus, V parametrizes ( e, d ) -composite tuples in the sense that to any point P ∈ V ( K ), corresponding to a homomorphism P : S −→ K ,we can associate the tuple ( P ( t ) , . . . , P ( t de )) ∈ K de , which is ( e, d )-composite.Then, we will apply Lemma 3.1 to obtain polynomials h ( x ) and ‘ ( x ) in S [ x ], of degree d and d − m ( x ) = h ( x ) − ‘ ( x ), giving the identity ( x − t ) · · · ( x − t de ) = h ( g ( x )) − ‘ ( g ( x )) ∈ S [ x ] . We will then consider (variants of) the curve f X/K ( V ) defined by y = ‘ ( g ( x )), which has the K ( V )-points P i := ( t , ± h ( g ( t i ))), for i = 1 , . . . , de . We willdetermine the rank of the subgroup of J X ( K ( V )) generated by the divisor classes [2 P i − D e X, ∞ ](cf. 1.3), and finally, we will apply Corollary 2.5 to get lower bounds on N ( K, g ) and R ( K, g ). Remark 4.5.
Mestre (cf. [Elk96, Page 4]) and Shioda (see 5.5 (i)) produced improvementsof the construction in 3.5 by applying Observation 4.3 in the special case when the “innerpolynomial” G ( x ) is x e . In fact, we came upon Observation 4.3 by noting that there isnothing special about x e , and that any choice of G ( x ) would yield a similar improvement.5. Composite tuples
The purpose of this section is to provide context to the reader for how we find K -rationalvarieties V that parametrize ( e, n )-composite tuples (as in 4.4); the results in this section arenot needed or used in the subsequent sections. Let e (cid:62) n (cid:62) i = 1 , . . . , n , let t i denote the tuple of indeterminates t i , . . . , t ie . For each j = 1 , . . . , e , let s j ( t i ) denote the j th elementary symmetric polynomial inthe tuple t i . The Q -variety of ( e, n ) -composite tuples , denoted G ( e, n ), is the closed subschemeof A en Q = Spec Q [ t , . . . , t n ] defined by the equations: s j ( t ) = · · · = s j ( t n ) , j = 1 , . . . , e − . (5.2)Note that if we set g ( x ) := x e + P e − j =1 ( − j s j ( t ) x e − j and m ( x ) := Q ni =1 ( x + ( − e s e ( t i )),then we have Q i,j ( x − t ij ) = m ( g ( x )) ∈ S [ x ], where S denotes the coordinate ring of G ( e, n ).It is not hard to show that G ( e, n ) / Q is geometrically integral, i.e. it is indeed a Q -variety.It is an easy consequence of [Ric11, Proposition 1] that, for a field K ⊃ Q , a K -point P of the ambient affine space A en Q is an ( e, n )-composite tuple if and only if there is apermutation σ of the coordinates of P such that σ ( P ) ∈ G ( e, n )( K ). Thus, to carry outthe method of 4.4 for some e (cid:62) d (cid:62)
2, we need to: find a K -rational sub-variety V ⊂ G − . eme, d − . em K . Remark 5.3.
For a positive integer j , let p j ( t i ) := t ji + · · · + t jie . The classical Prouhet–Tarry–Escott (PTE) problem of size e and length n asks for integer solutions to the system ofequations: p j ( t ) = · · · = p j ( t n ) , j = 1 , . . . , e − . (5.4)We note here that the equations in (5.4) define the closed subscheme G − . eme, n − . em ⊂ Spec Q pt t , . . . , t n − . em ; in other words, a tuple ( a i ) ∈ K en is ( e, n )-composite if andonly if the coordinates give a ( K -rational) solution to the PTE problem of size e and length n . The equivalence between the systems of equations (5.2) and (5.4) is an easy application ofthe Newton Identities (cf. [Mea92]); we leave the details to the reader.
Most investigations of G ( e, n )( Q ) havefocused on the case n = 2, and even in this case, non-trivial points in G ( e, n )( Q ) are knownonly for e (cid:54)
12; here, by “ non-trivial ” we mean that the coordinates of the point are distinct.For our purposes (cf. 4.4), we need n = 2 d (cid:62)
4, so we do not dwell on what is known for n (cid:54)
3, instead referring the reader to the survey article [RN19]. We summarize below thecases, with n (cid:62)
4, in which G ( e, n ) is known to have a non-trivial Q -point. First, we define aclosed subscheme G ( e, n ) := { s ( t ) = 0 } ⊂ G ( e, n ) that will appear below.(i) The case ( e, n ) = (2 , n ) . The reader can easily verify that G (2 , n ) ∼ = A n +1 Q , and G (2 , n ) ∼ = A n Q . In [Shi98], Shioda carried out the method of 4.4 by taking V = G (2 , n ),and thereby proved N ( Q , g ) (cid:62) g + 16 and R ( Q , g ) (cid:62) g + 7 for g (cid:62) The case ( e, n ) = (3 , n ) . Dickson showed in [Dic29, Theorem 47, Page 52] that G (3 ,
2) is Q -rational; Gloden showed in [Glo44, Pp. 74-81] that for arbitrary n , the subscheme B n := G (3 , n ) (see 6.1) contains infinitely many non-trivial Q -points. Weshow in Theorem 7.6 that B n is a Q -rational variety; in fact, one can show that G (3 , n )is Q -birational to B n × A Q , so G (3 , n ) is itself a Q -rational variety.(iii) The case ( e, n ) = (4 , n ) . Chernick showed in [Che37] that G (4 ,
2) is Q -rational;Gloden showed in [Glo44, Pp. 81-83] that for arbitrary n , G (4 , n ) contains infinitelymany non-trivial Q -points. In Proposition 12.3, we show that these points live on a Q -rational sub-variety Z n ⊂ G (4 , n ), and in Section 13, we carry out the method of4.4 with V = Z d ; this culminates in Theorem 13.10.(iv) The case ( e, n ) = (6 , n ) . Gloden noted in [Glo44, Page 87] that if P is a Q -point of B n = G (3 , n ), and “ − P ” denotes the tuple whose coordinates are the negatives ofthe coordinates of P , then the tuple ( P, − P ) ∈ Q n is (6 , n )-composite. In Sections 6– 11, we carry out the method of 4.4 with V = B d ; this culminates in Theorems 11.2,11.3 and 0.7.(v) The “Kummer” case.
There is a closed subscheme K ( e, n ) ⊂ G ( e, n ), whosegeometric irreducible components are affine spaces defined over the e -th cyclotomicfield Q ( ζ e ), such that for any field K ⊃ Q ( ζ e ), the K -points of K ( e, n ) are tuples( a i ) ∈ K en such that Q i ( x − a i ) = M ( x e ), for some M ( x ) ∈ K [ x ] of degree n (seeRemark 14.6). In Section 14, we carry out the method of 4.4 by taking V to be oneof these affine spaces; this culminates in Theorem 0.5. Remark 5.6.
Choudhry [Cho19] has constructed elliptic curves of large rank (at least 12)using solutions to the PTE problem, but his method is distinct from the method of thispaper. We note (see Tables 10.18 and 10.25) that for genus g = 1, our method cannot do anybetter than the “baseline” method of 3.5, which produces elliptic curves of rank at least 8.6. Parametrizing (6 , n ) -composite tuples Sections 6 – 11 form one block, culminating in Theorems 11.2, 11.3, and the proof ofTheorem 0.7 in 11.4. In this section, we fix an integer n (cid:62)
2, an arbitrary number field K ,and define a K -variety B n (see 5.5 (ii) and (iv)), which parametrizes (6 , n )-composite tuples(as in 4.4). B n . Let B n denote the closed subscheme of A nK := Spec K [ T i , T i , T i | i = 1 , . . . , n ] defined by the equations: T + T + T = · · · = T n + T n + T n = 0 ,T T + T T + T T = · · · = T n T n + T n T n + T n T n . (6.2)More formally, if we let P i (resp. G i ) denote the polynomial T i + T i + T i (resp. T i T i + T i T i + T i T i ), then we define: S := K [ T i , T i , T i | i = 1 , . . . , n ]( P , . . . , P n , G − G , . . . , G n − − G n ) , B n := Spec S. The following notation will recur (until Section 11). (i) We define the following elements of S : b := T T + T T + T T = G ,t i := T i T i T i i = 1 , . . . , n,u i := t i , i = 1 , . . . , n,m j := ( − n − j s n − j ( u , . . . , u n ) , j = 0 , . . . , n − ,U ij := T ij , i = 1 , . . . , n, j = 1 , , . (6.4)We let t , u , and m denote the tuples of elements ( t , . . . , t n ) , ( u , . . . , u n ) , and ( m , . . . , m n − )respectively. We let R denote the K -subalgebra of S generated by the U ij ’s, and weset H n := Spec R .(ii) We define the following polynomials in S [ x ]: g ( x ) := x ( x + b ) , b g ( x ) := x ( x + b ) ,m ( x ) := x n + m n − x n − + · · · + m . (6.5)The fourth line of (6.8) below implies that B n parametrizes (6 , n )-composite tuples (as in4.4). Proposition 6.6.
Continuing with the notation of 6.3, we have the following identities in S [ x ] :(i) For i = 1 , . . . , n , we have g ( x ) + t i = ( x + T i )( x + T i )( x + T i ) g ( x ) − t i = ( x − T i )( x − T i )( x − T i ) b g ( x ) − u i = ( x − U i )( x − U i )( x − U i ) g ( x ) − u i = ( x − T i )( x − T i )( x − T i )( x + T i )( x + T i )( x + T i )= b g ( x ) − u i . (6.7) (ii) We have m ( x ) = ( x − u ) · · · ( x − u n ) m ( x ) = n Y i =1 ( x + t i )( x − t i ) m ( b g ( x )) = n Y i =1 ( x − U i )( x − U i )( x − U i ) m ( g ( x ) ) = n Y i =1 ( x − T i )( x − T i )( x − T i )( x + T i )( x + T i )( x + T i )= m ( b g ( x )) . (6.8) Proof.
The reader can check that these identities follow from the definitions in 6.3. (cid:3) Normality and rationality of B n As in the previous section, n (cid:62) B n is anormal, K -rational variety; the normality of B n is used in the next section, and the rationalityin Section 11. Recall that an affine K -scheme Y is a complete intersection over K if it admits a presentation Y ∼ = Spec K [ x , . . . , x m ] / ( f , . . . , f r ), with dim Y = m − r . Lemma 7.1.
Let f m denote the tuple of indeterminates f m , . . . , f m n − .(a) The K -morphism f : B n −→ A n +1 K := Spec K [ e b, f m ] , defined on coordinate rings by e b b, f m i m i , is finite and surjective.(b) Each of the tuples ( b, m ) , ( b, u ) and ( b, t ) generates a K -subalgebra of S that isisomorphic to an n + 1 -dimensional polynomial ring over K .(c) The K -scheme B n is an n + 1 -dimensional complete intersection over K .Proof. We first note that (c) follows from (a); indeed, (a) implies that dim B n = n + 1, and itis then clear from the definition in 6.1 that B n is a complete intersection over K .We now prove (a). To see that f is surjective, suppose Q = ( b , k , . . . , k n − ) is a K -pointof the target. Let b G ( x ) := x ( x + b ) ∈ K [ x ], let M ( x ) := x n + k n − x n − + · · · + k ∈ K [ x ],and let c , . . . , c n ∈ K denote the roots of M ( x ) (counted with multiplicity). For each i = 1 , . . . , n , let a i , a i , a i , − a i , − a i , − a i ∈ K denote the roots of b G ( x ) − c i (counted withmultiplicity). Then, P = ( a ij ) is a K -point of the fiber f − ( Q ). Since f is dominant and thetarget is reduced, the homomorphism K [ e b, f m ] −→ S is injective and identifies K [ e b, f m ] withthe K -subalgebra of S generated by b, m (which we denote by K [ b, m ]). The finiteness of f then follows from the last line of (6.8), which implies that each T ij is the root of the monicpolynomial m ( g ( x ) ) ∈ K [ b, m ][ x ], and hence, that S is a finite K [ b, m ]-module.For (b), we need to show that each n + 1-tuple in the statement consists of elements of S that are algebraically independent over K ; we already saw above that this is true of thetuple ( b, m ). For the remaining tuples, is suffices to show they each generate a field F oftranscendence degree n + 1 over K . For this, we note that K ( b, m ) ⊂ F ⊂ K , and since[ K : K ( b, m )] is finite (by (a)), we have trdeg K F = trdeg K K = n + 1, so we are done. (cid:3) B n . For each i = 1 , . . . , n , we have T i = − ( T i + T i ) ∈ S , and the reader can check that we have an isomorphism S ∼ = K [ T i , T i | (cid:54) i (cid:54) n ] / ( F , . . . , F n − ), where F i := ( T i + T i T i + T i ) − ( T i +1 , + T i +1 , T i +1 , + T i +1 , ) for i = 1 , . . . , n −
1. Note that the element b ∈ S from (6.4) is identified with − ( T + T T + T )under this isomorphism.For each i = 1 , . . . , n , if we let N : A K −→ Spec K [ z i ] be the morphism defined oncoordinate rings by z i ( T i + T i T i + T i ), then B n fits into a cartesian diagram : B n ( A K ) n A K ( A K ) n , (cid:2) ( N ) n ∆ (7.3)in which the bottom horizontal morphism is the diagonal closed immersion.We recall before the next lemma that a K -group scheme Y is an algebraic K -torus if thereis an isomorphism of K -group schemes Y K ∼ = G rm,K , for some r . Lemma 7.4.
The open subscheme V n := { b = 0 } ⊂ B n is an n + 1 -dimensional algebraic K -torus that is rational over K .Proof. Let L denote the K -algebra K [ x ] / ( x + x + 1), and let ζ ∈ L denote the class of x . The set { , − ζ } is a K -basis for L , and the bijection A K ( R ) = R L/K A L ( R ) obtained by identifying ( a i , a i ) ∈ A K ( R ) with ( a i − a i ζ ) ∈ R ⊗ K L is natural in R , so we can identifySpec K [ T i , T i ] with the Weil Restriction R L/K A L (cf. [BLR90, Chapter. 7.6]). An element a i − ζa i ∈ R ⊗ K L is invertible if and only if its norm N R ⊗ L/R ( a i − ζa i ) := a i + a i a i + a i is invertible in R . Thus, the open subscheme { T i + T i T i + T i = 0 } ⊂ R L/K A L is naturallyidentified with R L/K G m,L , and moreover, the map N : R L/K A L −→ A K from 7.2 restricts toa group scheme homomorphism N : R L/K G m,L = Spec K [ T i , T i , ( T i + T i T i + T i ) − ] −→ G m,K := Spec K [ z i , z − i ]. By restricting everything in (7.3) to ( G m,K ) n ⊂ ( A K ) n , we see that V n fits into a cartesian diagram : V n ( R L/K G m,L ) n G m,K ( G m,K ) n . (cid:2) ( N ) n ∆ (7.5)Since the right vertical map and the bottom closed map are both group-scheme homomor-phisms, V n is a subgroup scheme of ( R L/K G m,L ) n . Note that L/K is étale, so R L/K G m,L isan algebraic K -torus. We claim that there is an isomorphism of algebraic K -tori: φ : V n ∼−−→ H := R L/K G m,L × ( R L/K G m,L ) n − , where R L/K G m,L denotes the norm-one torus , i.e. the kernel of the norm map N : R L/K G m,L −→ G m,K . Since R L/K G m,L is K -rational and one-dimensional K -tori are K -rational, this iso-morphism implies that H (and therefore, also V n ) is K -rational.Now, we construct the isomorphism φ . For any commutative group scheme G , let m : G −→ G denote the multiplication, inv : G −→ G the inversion, p i : G n −→ G the i th -projection, and let p ij := ( p i , p j ) : G n −→ G . We define the homomorphism φ : G n −→ G n tobe the composition G n ( inv ◦ p , p , ... , p n ) −−−−−−−−−−→ G n ( Id , m ◦ p , ... , m ◦ p n ) −−−−−−−−−−−−−→ G n . The reader can easily check that φ ◦ φ = Id, so any subgroup scheme of G n is mappedisomorphically onto its image. In particular, if G = G m,K , then φ : G nm,K −→ G nm,K is given by( z , . . . , z n ) ( z − , z z − , . . . , z n z − ) , and visibly, φ gives an isomorphism from the diagonal∆( G m,K ) = { z = · · · = z n } ⊂ G nm,K onto the subgroup scheme G m,K × (1 , . . . , ⊂ G nm,K (i.e: the subgroup scheme on which the i th -coordinate is 1 for i = 2 , . . . , n ). We have acommutative diagram: ( R L/K G m,L ) n ( R L/K G m,L ) n G nm,K G nm,K . ( N ) n φ ( N ) n φ By (7.5), the pre-image under ( N ) n of the diagonal ∆( G m,K ) ⊂ G nm,K is V n , and this ismapped isomorphically onto the pre-image of G m,K × (1 , . . . , H = R L/K G m,L × ( R L/K G m,L ) n , so we are done. (cid:3) Theorem 7.6.
The K -scheme B n is a normal, K -rational variety of dimension n + 1 .Proof. Recall that S denotes the coordinate ring of B n . First, we note that the closed subset { b = 0 } ⊂ B n is simply the n -fold product of the affine curve given by x + xy + y = 0, soit is of dimension n . Since B n is a complete intersection (cf. Lemma 7.1), its irreducible components are equi-dimensional, of dimension n + 1, so { b = 0 } cannot contain an irreduciblecomponent of B n . Thus, b ∈ S is not a zero-divisor, the localization map S −→ S [ b − ] isinjective, and Lemma 7.4 implies that B n is a K -rational variety of dimension n + 1.Next, we prove that B n is normal; in light of [Sta18, Lemma 038O], we may assumethat K is algebraically closed, and in particular, that K contains a primitive third root ofunity ζ . Lemma 7.1 (c) implies (using [Sta18, Lemma 00SB] and [Sta18, Lemma 045N])that B n is Cohen-Macaulay, so it satisfies Serre’s condition (S2) (by [Sta18, Lemma 0342]).Using [Sta18, Lemma 031S], we are therefore reduced to showing that: B n is regular incodimension
1. To see that this is true, we first note that B n is K -isomorphic to the closedsubscheme of Spec K [ z i , z i | i = 1 , . . . , n ] given by the equations z z = · · · = z n z n , withthe isomorphism defined on coordinate rings by z ij T i − ζ j T i , for i = 1 , . . . , n and j = 1 , i = 1 , . . . , n , let D i denote the closed subset { z i = z i = 0 } ⊂ B n ; note that D i is isomorphic to the ( n − xy = 0, so it is ofcodimension ( n + 1) − ( n −
1) = 2 in B n . We claim that the singular locus of B n is containedin the codimension 2 subset D := D ∪ · · · ∪ D n . Indeed, any point in the complement of D is contained in an open subset of the form U ( (cid:15) (1) , . . . , (cid:15) ( n )) := { z ,(cid:15) (1) · · · z n,(cid:15) ( n ) = 0 } ⊂ B n , where each (cid:15) ( i ) ∈ { , } . This open subset is isomorphic to Spec K [ w, v i , v − i | i = 1 , . . . , n ] viathe morphism defined on coordinate rings by w z , − (cid:15) (1) and v i z i,(cid:15) ( i ) for i = 1 , . . . , n ;the inverse is defined by z i,(cid:15) ( i ) v i and z i, − (cid:15) ( i ) wv v − i , for each i = 1 , . . . , n . Thus U ( (cid:15) (1) , . . . , (cid:15) ( n )) is smooth, and we are done. (cid:3) Some properties of the function field K ( B n )As in the previous section, n (cid:62) K := K ( B n ). If n = 2 d for some integer d (cid:62)
2, we define a certainsubfield K ( b, ‘ ) ⊂ K and show that it is algebraically closed in K . Propositions 8.3 and 8.9together allow us to apply the results of the next section to determine (in Section 10) theranks of various curves parametrized by B d . K ( B n ) . Let K denote the function field K ( B n ), and recall the variouselements of S defined in (6.4). By Lemma 7.1, we have subrings K [ b, m ] , K [ b, u ] and K [ b, t ]of S . Note moreover that K [ b, u ] ⊂ R ; indeed, the identity ( x − U i )( x − U i )( x − U i ) = x + 2 bx + b x − u i (cf. (6.7)) implies that b = − ( U i + U i + U i ) / u i = U i U i U i .Putting it all together, we get the following diagrams of rings and their function fields: S K K [ b, t ] R K ( b, t ) K ( H n ) K [ b, u ] K ( b, u ) K [ b, m ] K ( b, m ) . (8.2)Let S n denote the symmetric group on n letters. Recall that for a finite group G , the wreathproduct of G by S n , denoted G o S n , is the semi-direct product G n (cid:111) q S n , where q : S n −→ Aut( G n ) is the natural permutation action. We refer the reader to [Odo85, Section 4] for adiscussion of wreath products arising as Galois groups of composite polynomials (note thatOdoni’s notation G [ H ] corresponds to our notation H o G ). Proposition 8.3. (i) The following isomorphisms hold: α : ( S ) n ∼−−→ Gal( K /K ( b, t )) α : ( S ) n ∼−−→ Gal( K ( H n ) /K ( b, u ))( σ , . . . , σ n ) ( T ij T i,σ i ( j ) ) ( σ , . . . , σ n ) ( U ij U i,σ i ( j ) ) β : ( S ) n ∼−−→ Gal( K /K ( H n )) β : ( S ) n ∼−−→ Gal( K ( b, t ) /K ( b, u ))( e , . . . , e n ) ( T ij e i T ij ) ( e , . . . , e n ) ( t i e i t i ) , (8.4) where each e i ∈ {± } = S .(ii) The maps α and β induce an isomorphism ( α , β ) : ( S × S ) n ∼−−→ Gal( K /K ( b, u )) . (iii) We have an injection γ : S n −→ Gal( K /K ( b, m )) defined by τ ( T ij T τ ( i ) ,j ) ,which induces homomorphisms: γ : S n , −→ Aut K ( b, m ) ( K ( b, t )) γ : S n ∼−−→ Gal( K ( b, u ) /K ( b, m )) τ ( t i t τ ( i ) ) τ ( u i u τ ( i ) ) (8.5) (iv) The maps ( α , β ) and γ induce an isomorphism δ : ( S × S ) o S n ∼−−→ Gal( K /K ( b, m ))( σ, e ) · τ (cid:16) T ij e τ ( i ) T τ ( i ) ,σ τ ( i ) ( j ) (cid:17) , (8.6) where ( σ, e ) denotes a tuple (( σ , e ) , · · · , ( σ n , e n )) ∈ ( S × S ) n .Proof. Recall the presentation of B n given in 6.1. Since K /K ( b, m ) is the splitting field ofthe polynomial m ( g ( x ) ) ∈ K ( b, m )[ x ] (cf. fourth line of (6.8)), any σ ∈ Gal( K /K ( b, m )) isdetermined by a permutation of {± T ij } ; any such permutation that preserves the identitiesin (6.2) induces an automorphism of B n , and hence, of K . The reader can use this to easilyverify that the formulas above define honest automorphisms of K . Using the various identitiesin 6.3, the reader can also verify that the maps above, regarded a priori as maps to Aut( K ),have image contained in the asserted subgroups.First, we prove (i). Since K /K ( b, t ) is the splitting field of the polynomials g ( x ) − t i = x − bx − t i (cf. (6.7)), any element of Gal( K /K ( b, t )) is given by a simultaneous permutationof the triplets T i , T i , T i , for i = 1 , . . . , n . So, we have α (( S ) n ) = Gal( K /K ( b, t )). To seethat α is injective, suppose for a contradiction that there exists an element ( σ , . . . , σ n ) =1 ∈ ker φ . Then for some k , we have σ k = 1 ∈ S , so there exist 1 (cid:54) p = q (cid:54) T kp = σ k ( T kp ) = T kq ; but this is absurd, since disc( x + bx − t k ) = − b − t k = 0 ∈ K ( b, t ).We conclude that φ is an isomorphism.Composing α with the natural restriction Aut( K ) −→ Aut( K ( H n )), we get α . Note that K ( H n ) /K ( b, u ) is the splitting field of the polynomials b g ( x ) − u i = x + 2 bx + b x − u i ∈ K ( b, u )[ x ], and disc( b g ( x ) − u i ) = − b u i − u i = 0 ∈ K ( b, u ). Thus, we can repeat theabove argument by replacing T ij with U ij , g ( x ) − t i with b g ( x ) − u i , and K /K ( b, t ) with K ( H n ) /K ( b, u ), to find that α is an isomorphism.Next, we note that β is obtained by composing β with the restriction Aut( K ) −→ Aut( K ( b, t )), and β is clearly an isomorphism (cf. (6.4)). This implies that β is injective, and since (i) implies that [ K /K ( H n )] = [ K : K ( b, t )][ K ( b, t ) : K ( b, u )] / [ K ( H n ) /K ( b, u )] = 2 n ,we find that β is also an isomorphism. This concludes the proof of (i).Part (i) implies that K ( b, u ) and K ( H n ) are linearly disjoint extensions of K ( b, u ), withcompositum K . So, Galois theory gives Gal( K /K ( b, u )) ∼ = ( S ) n × ( S ) n , and the latter isnaturally isomorphic to ( S × S ) n , which establishes part (ii).For part (iii), note that if we compose γ with the restriction Aut( K ) −→ Aut( K ( b, t ), thenwe get γ , and by composing further with the restriction Aut( K ( b, t )) −→ Aut( K ( b, u )), weget γ . Thus, the injectivity of γ and γ would follow if γ is an isomorphism; the latterfact is nothing but the Fundamental theorem of symmetric functions (cf. [Hun80, Page 254,Theorem 2.18]).For part (iv), we first note that δ is induced by ( α , β ) and γ in the sense that δ ( σ, e ·
1) =( α , β )( σ, e ) = ( α ( σ ) , β ( e )) and δ (1 · τ ) = γ ( τ ). To verify that δ is a homomorphism, wecheck that the map respects the semi-direct product structure; for any σ, e ∈ ( S × S ) n , any τ ∈ S n , and any i = 1 , . . . , n, j = 1 , ,
3, we compute: (cid:16) δ ( τ − ) δ ( σ, e ) δ ( τ ) (cid:17) ( T ij ) = (cid:16) δ ( τ − ) δ ( σ, e ) (cid:17) ( T τ ( i ) ,j )= (cid:16) δ ( τ − ) (cid:17) ( e τ ( i ) T τ ( i ) ,σ τ ( i ) ( j ) = e τ ( i ) T i,σ τ ( i ) ( j ) = (cid:16) q ( τ )( σ, e ) (cid:17) ( T ij ) , so δ is a homomorphism. That it is an isomorphism follows from parts (ii) and (iii), and thefollowing commutative diagram of exact sequences:1 ( S × S ) n ( S × S ) o S n S n
11 Gal( K /K ( b, u )) Gal( K /K ( b, m )) Gal( K ( b, u ) /K ( b, m )) 1 . ( α ,β ) δ γ (cid:3) For the rest of this section, we set n = 2 d for an integer d (cid:62)
2, so that B d is a K -rationalvariety of dimension 2 d + 1. The coordinate ring S contains a 2 d + 1-dimensional polynomialalgebra K [ b, m ] (cf. 8.1); using the isomorphism K [ b, ‘ , h ] ∼ = K [ b, m ] from Lemma 3.1, where ‘ and h denote the tuples of indeterminates l , . . . , l d − and h , . . . , h d − respectively, we getinclusions K [ b, ‘ ] ⊂ K [ b, ‘ , h ] ∼ = K [ b, m ] ⊂ S. Recall the finite, surjective K -morphism f : B d −→ A d +1 ( b, m ) (cf. Lemma 7.1), andlet ∆ ⊂ A d +1 ( b, m ) denote the branch locus (i.e. set of branch points) of f . Note that ∆ isnon-empty; for instance, (0 , . . . , ∈ A d +1 ( b, m ) is a branch point of f , since the fiber consistsof the single point of B d with all coordinates 0. Since B d is normal (cf. Theorem 7.6), the Theorem of purity of branch locus (cf. [Zar58, Proposition 2] or [Sta18, Lemma 0BMB]) thenimplies that ∆ is a hypersurface in A d +1 ( b, m ), so it is defined by an ideal ( F ) ⊂ K [ b, m ],with F square-free and well-defined up to multiplication by a unit in K [ b, m ]. Lemma 8.8.
There are no irreducible factors of F ∈ K [ b, m ] contained in the subring K [ b, ‘ ] (cid:40) K [ b, m ] .Proof. We first note from the proof of Proposition 8.3 that the Galois group G = Gal( K /K ( b, m ))naturally acts on B d via permutations of the set {± T ij | i = 1 , . . . , d, j = 1 , , } , and moreover, G = Aut( B d / A d ( b, m )) (we are abusing notation here). Let P := ( a ij ) ∈ B d ( K )be the point corresponding to the ring homomorphism S −→ K defined by T ij a ij . Then, P is not a ramification point for f if and only if its orbit under G is of size G , if andonly if the elements of the multiset {± a ij } are distinct, if and only if the specialization m P ( g P ( x ) ) is separable. Thus, the irreducible factors of F are precisely the irreduciblefactors of disc m ( g ( x ) ), which we now compute.(i) Suppose a ij = − a ij for some 1 (cid:54) i (cid:54) d . Then P lies in { T ij = 0 } , so T ij divides F in S ; taking the product over all i and j , we get that m (which is irreducible in K [ b, m ]) divides F in K [ b, m ].(ii) Suppose that a ij = ± a ik , or equivalently, that a ij = a ik , for some 1 (cid:54) i (cid:54) d andsome 1 (cid:54) j = k (cid:54)
3. Then P lies in the closed set { T ij = T ik } = { U ij = U ik } ,so U ij − U ik divides F in S . This implies that disc( b g ( x ) − u i ) divides F in R (thecoordinate ring of H d ), and so ω := Q i disc( b g ( x ) − u i ) divides F in K [ b, m ]. The groupGal( K ( b, u ) /K ( b, m )) ∼ = S d acts transitively on { disc( b g ( x ) − u ) . . . disc( b g ( x ) − u d ) } ,which implies that ω is irreducible in K [ b, m ].(iii) Suppose that a ij = ± a pj , or equivalently, that a ij = a pj , for some 1 (cid:54) i = p (cid:54) d .Then, u i − u p = ( x − U i )( x − U i )( x − U i ) − ( x − U p )( x − U p )( x − U p )vanishes at P , which implies that Q i = j ( u i − u j ) = disc m ( x ) divides F in K [ b, m ].Note that disc m ( x ) is irreducible in K [ b, m ] (cf [Odo85, Lemma 8.1]).Thus, we have showed that F factors (upto units) into irreducibles as F = ω · m · (disc m ) ∈ K [ b, m ]; that these irreducible factors are co-prime follows by noting how they split in K [ b, u ].Letting h ( x ) , ‘ ( x ) ∈ K [ b, h , ‘ ] ∼ = K [ b, m ][ x ] denote the square-root approximation andremainder of the degree 2 d polynomial m ( x ) (as in (3.2)), we re-write this factorization as: F = ω · ( h − ‘ ) · disc( h ( x ) − ‘ ( x )) ∈ K [ b, h , ‘ ] . Clearly, h − ‘ / ∈ K [ b, ‘ ]; we now show the same for the remaining two irreducible factors.To show that ω / ∈ K [ b, ‘ ], it suffices to show that upon setting b = 0, ω does not specialize toan element of K [ ‘ ]. Indeed, if we set b = 0, then each b g ( x ) − u i specializes to x − u i ∈ K [ u ][ x ],whose discriminant has the single irreducible factor u i ∈ K [ u ]. The product ω thereforespecializes to a power of u · · · u d = m = h − l ∈ K [ ‘ , h ], which is clearly not containedin K [ ‘ ].Similarly, to show disc( h ( x ) − ‘ ( x )) / ∈ K [ b, ‘ ], we specialize the tuple ( b, ‘ d − , . . . , ‘ , ‘ ) tothe tuple (0 , , . . . , , − ∈ K d +1 , and show that disc m specializes to a non-constant elementof K [ h ]. Indeed, m ( x ) specializes to h ( x ) − ∈ K [ h ][ x ], which admits the specialization x d − ∈ K [ x ], which is separable, and also the specialization ( x d − − ∈ K [ x ], which isnot separable. This establishes that disc( h ( x ) − ∈ K [ h ] is non-constant, and concludesthe proof of the lemma. (cid:3) Proposition 8.9.
The field K ( b, ‘ ) = K ( b, l , . . . , l d − ) is algebraically closed in K .Proof. Assume for a contradiction that there is some finite extension L (cid:41) K ( b, ‘ ) contained in K , and let Y be the normalization of A d ( b, ‘ ) in L (i.e. Y = Spec A , where A is the integralclosure of K [ b, ‘ ] in L ). By standard facts, Y −→ A d ( b, ‘ ) is finite and surjective. We alsohave a morphism B d −→ A d +1 ( b, m ) −→ A d +1 ( b, ‘ ). Since B d is normal (cf. Theorem 7.6),we have a containment A ⊂ S , which corresponds to a morphism B d −→ Y , so we get a morphism B d −→ Y := Y × A d ( b, ‘ ) A d ( b, m ), giving a commutative diagram: B d Y A d ( b, m ) Y A d ( b, ‘ ) . (cid:2) Since the composite morphism B d −→ Y −→ A d ( b, m ) has points that ramify completelyin B d (for example, the point P = (0 , . . . , ∈ A d ( b, m )( K )), we get that Y −→ A d ( b, m ) is ramified. So, Y −→ A d ( b, ‘ ) is also ramified, and its branch locus δ isnon-empty and defined by a proper ideal ( f ) ⊂ K [ b, ‘ ]. The pre-image (call it δ ) of δ under A d ( b, m ) −→ A d ( b, ‘ ) is the branch locus of Y −→ A d ( b, m ), and δ is defined by( f ) ⊂ K [ b, m ]. If ∆ (cf. 8.7) is defined by an ideal ( F ), then the containment δ ⊂ ∆ impliesthat f divides F in K [ b, m ]. This contradicts Lemma 8.8 above, so we are done. (cid:3) Tools for bounding ranks of curves
In this section, we develop some tools that will be used to bound the ranks of the curvesappearing in Section 10, 13, and 14. Corollary 9.2 below is used in the proof of Lemma 10.5by taking ( K , K , K ) = ( K ( b, ‘ ) , K ( b, m ) , K ) . Lemma 9.3 below is used in the proofs ofTheorems 13.6 and 14.3 (we have included it in this section because it fits into the theme ofthis section).We recall first that if
L/K is an extension of fields, a point P : Spec L −→ X of a K -scheme X is algebraic over K if the image of P is a closed point of X . Lemma 9.1.
Let K ⊂ K ⊂ K be a chain of field extensions such that K is algebraicallyclosed in K and K /K is finite Galois. Let A/K be an abelian variety and P ∈ A ( K ) atorsion point. Then, P is fixed by the action of Gal( K /K ) on A ( K ) .Proof. The key point is that torsion points of A are algebraic over K ; indeed, for any n (cid:62) n morphism [ n ] : A −→ A is a K -isogeny, so the kernel is finite over K . Thus, the image of P : Spec K −→ A is a closed point x with residue field κ ( x ) a finiteextension of K . We have an injection κ ( x ) , −→ K given by P , and the hypotheses implythat κ ( x ) = K , so P ∈ A ( K ) (cid:40) A ( K ), or more precisely, P is the base change to K of apoint in A ( K ). The desired conclusion then follows because Gal( K /K ) ⊂ Aut K K . (cid:3) Corollary 9.2.
With K ⊂ K ⊂ K as above, let X/K be a nice curve of positive genus,which has distinct K -points P , . . . , P n , Q , . . . , Q m , with n (cid:62) . Suppose there exists asubgroup H (cid:54) Gal( K /K ) isomorphic to S n , which acts by permutation on { P , . . . , P n } andfixes each Q i . Let D ∈ Div( X ) be a divisor of positive degree k , so that (cid:15) ( P i ) := [ kP i − D ] and (cid:15) ( Q j ) := [ kQ j − D ] are points in J X ( K ) for each i = 1 , . . . , n and j = 1 , . . . , m (uponregarding D as a divisor on X K ). Then, any linear dependence relation n X i =1 a i (cid:15) ( P i ) + m X j =1 b i (cid:15) ( Q i ) = 0 ∈ J X ( K ) satisfies a = · · · = a n .Proof. Assume for a contradiction that we have a relation as above with a r = a s for some1 (cid:54) r = s (cid:54) n . Let σ ∈ H be a transposition that exchanges P r and P s and fixes all the other P i ’s and the Q i ’s. Applying σ to the relation above and subtracting the result from theoriginal, we get:( a r − a s ) · ( (cid:15) ( P r ) − (cid:15) ( P s )) = k ( a r − a s ) · [ P r − P s ] = 0 ∈ J X ( K ) , so [ P r − P s ] ∈ J X ( K ) is a torsion point . Let p ∈ { , . . . , n } \ { r, s } (this is where we usethat n (cid:62) σ ∈ H (cid:54) G that interchanges P r and P p and fixes P s . Applying Lemma 9.1, we have σ ([ P r − P s ]) = [ P r − P s ], and therefore[ P r − P s ] = σ ([ P r − P s ]) = [ σ ( P r ) − σ ( P s )] = [ P p − P s ] ∈ J X ( K ) . But this implies that [ P r − P p ] = 0 ∈ Pic ( X K ), which is absurd because X is of positivegenus. (cid:3) Lemma 9.3.
Let
M/L be an extension of fields, and let
X/L be a nice curve of positivegenus having a point D ∈ X ( L ) . If P ∈ X ( M ) is a point such that [ P − D ] ∈ J X ( M ) istorsion, then P is algebraic over L .Proof. We can use D as a base-point to embed X in its Jacobian, i.e. we have a closedimmersion r : X , −→ J X , defined over L , such that for any field F ⊃ L and point Q ∈ X ( F ),we have r ( Q ) = [ Q − D ] ∈ J X ( F ). In particular, we have r ( P ) = [ P − D ] ∈ J X ( M ), andsince this is a torsion point, the image of the compositionSpec M P −−→ X r , −→ J X is a closed point x ∈ J X . Since r is a closed immersion, the pre-image r − ( x ) (which is theimage of P ) is a closed point of X , so we are done. (cid:3) Bounding the ranks of curves parametrized by B d Fix an integer d (cid:62)
2. Recall the notation in 6.3, and in particular the various functions in S (the coordinate ring of B d ) defined in (6.4). Recall from the setup in 8.7 that we have anequality of polynomials m ( x ) = h ( x ) − ‘ ( x ) ∈ S [ x ], where deg ‘ ( x ) = d −
1. Combining thiswith the identities in (6.8), we get the following identities in S [ x ]: h ( x ) − ‘ ( x ) = d Y i =1 ( x − u i ) ,h ( x ) − ‘ ( x ) = d Y i =1 ( x + t i )( x − t i ) ,h (cid:16)b g ( x ) (cid:17) − ‘ (cid:16)b g ( x ) (cid:17) = d Y i =1 ( x − U i )( x − U i )( x − U i ) ,h (cid:16) g ( x ) (cid:17) − ‘ (cid:16) g ( x ) (cid:17) = h (cid:16)b g ( x ) (cid:17) − ‘ (cid:16)b g ( x ) (cid:17) = d Y i =1 ( x + T i )( x − T i )( x + T i )( x − T i )( x + T i )( x − T i ) . (10.1) The table below contains a list of curves with explicit points; thefirst three curves (resp. last three curves) are defined over K ( ‘ ) (resp. K ( b, ‘ )), and thepoints are K ( t )-rational (resp. K -rational). In the first column we denote the smooth,proper model of the corresponding affine curve y = f ( x ) in the second column (as in f ( x ) that appears is separable; indeed, upon setting( b, l , l , . . . , l d − , l d − ) = (0 , , , . . . , , f ( x ) specializes to a separable polynomial in K [ x ]. The reader can verify using (6.4) and (10.1) that these curves come with the explicit K -rational points listed in the table, where we recall from 8.1 that K = K ( B d ). Curve Equation K -rational pointsΓ /K ( ‘ ) y = ‘ ( x ) P i := ( u i , h ( u i )) i = 1 , . . . , d Γ /K ( ‘ ) y = x‘ ( x ) Q i := ( u i , t i h ( u i )) i = 1 , . . . , d e Γ /K ( ‘ ) y = ‘ ( x ) e P ± i := ( ± t i , h ( u i )) i = 1 , . . . , d Θ /K ( b, ‘ ) y = ‘ ( b g ( x )) P ij := ( U ij , h ( b g ( U ij ))) i = 1 , . . . , d, j = 1 , , /K ( b, ‘ ) y = x‘ ( b g ( x )) Q ij := ( U ij , T ij h ( b g ( U ij )) i = 1 , . . . , d, j = 1 , , e Θ /K ( b, ‘ ) y = ‘ ( b g ( x )) e P ± ij := ( ± T ij , h ( b g ( U ij ))) i = 1 , . . . , d, j = 1 , , Table 10.3: Curves with K -points By an abuse of notation, for each curve X appearing in Table 10.3, we let D ∞ denote thedegree 2 divisor π ∗ X ∞ (cf. 1.3). Regarding D ∞ as a divisor on X K , for a divisor E ∈ Div( X K )of degree k , we let (cid:15) ( E ) := [2 E − kD ∞ ] ∈ J X ( K ) . If X = Γ or Θ , we let D denote the closed point (0 , P ∈ X ( K ), we let r ( P ) := [ P − D ] ∈ J X ( K ) . Since [2 D − D ∞ ] = 0 ∈ J X ( K ), for any P ∈ X ( K ), we have (cid:15) ( P ) = 2 r ( P ). Lemma 10.5.
Consider the curves Γ , Γ , Θ , and Θ from Table 10.3 above.(i) If d X i =1 a i (cid:15) ( P i ) = 0 ∈ J Γ ( K ( t )) (10.6) is a linear dependence relation, then a = · · · = a d . Moreover, the same is true if wereplace each (cid:15) ( P i ) ∈ J Γ ( K ( t )) with r ( Q i ) ∈ J Γ ( K ( t )) .(ii) If d X i =1 3 X j =1 a ij (cid:15) ( P ij ) = 0 ∈ J Θ ( K ) (10.7) is a linear dependence relation, then for each i = 1 , . . . , d , we have a i = a i = a i .Moreover, the same is true if we replace each (cid:15) ( P ij ) ∈ J Θ ( K ) with r ( Q ij ) ∈ J Θ ( K ) .Proof. The curves in question are defined over K ( b, ‘ ), which is algebraically closed in K (cf. Proposition 8.9); Proposition 8.3 tells us that Gal( K /K ( b, m )) has plenty of subgroupsisomorphic to symmetric groups. So, we are well-poised to apply Corollary 9.2.Recall from Proposition 8.3 (iii) that there is a subgroup S n (cid:54) Gal( K /K ( b, m )) that actsby permutation on the elements u , . . . , u d (resp. t , . . . , t d ) of K . Visibly, this induces apermutation action of γ ( S n ) on P , . . . , P d ∈ Γ ( K ( b, t )) (resp. Q , . . . , Q d ∈ Γ ( K ( b, t ))).Part (i) now follows from Corollary 9.2. For each i = 1 , . . . , d , let G i (cid:54) ( S ) d denote the subgroup consisting of tuples with everycoordinate being 1 except for the i -th coordinate; note that G i ∼ = S . Then, S ∼ = α ( G k ) (cid:54) Gal( K /K ( b, m )) is a subgroup that acts by permutation on the elements T k , T k , T k (resp. U k , U k , U k ) and fixes all the other T ij ’s (resp. U ij ’s) (cf. Proposition 8.3 (i)); visibly, thissubgroup acts by permutation on P k , P k , P k ∈ Θ ( K ) (resp. Q k , Q k , Q k ∈ Θ ( K )) andfixes all the other P ij ’s (resp. Q ij ’s). Part (ii) now follows from Corollary 9.2. (cid:3) Thus far, for X = Γ , Γ , Θ , or Θ , we have produced many points in J X ( K ) and obtainedrestrictions on the possible dependence relations between them. In order to determine theranks of the subgroups they generate, we need the fact that these curves fit together into adiagram: e ΘΘ e Γ Θ Γ Γ ψ ψ e ψϑ ϕ ϕ (10.8)in which the K ( b, ‘ )-morphisms are defined by ϕ : ( x, y ) ( x , y ) ϕ : ( x, y ) ( x , xy ) ψ : ( x, y ) ( x , y ) ψ : ( x, y ) ( x , xy ) ϑ : ( x, y ) ( b g ( x ) , y ) e ψ : ( x, y ) ( g ( x ) , y ) . (10.9)Above, we have abused notation and written Γ to denote the base-change (Γ ) K ( b, ‘ ) , andsimilarly for Γ and e Γ. Below, we show the effect of these maps on the explicit K -points:( T ij , h ( b g ( U ij )))( U ij , h ( b g ( U ij ))) ( t i , h ( u i )) ( U ij , T ij h ( b g ( U ij )))( u i , h ( u i )) ( u i , t i h ( u i )) . ψ ψ e ψϑ ϕ ϕ To bound the ranks of e Γ and e Θ, we use the following (echoing Shioda’s strategy in [Shi98]).
Proposition 10.10 (see [Shi98, Proposition 3]) . Let K be a field of characteristic , and let f ( x ) ∈ K [ x ] be a polynomial of degree (cid:62) such that f ( x ) , xf ( x ) and f ( x ) are all separable.Let X /K, X /K and f X/K be smooth, proper models of the affine curves y = f ( x ) , y = xf ( x ) , and y = f ( x ) respectively. Define φ : f X −→ X φ : f X −→ X ( x, y ) ( x , y ) ( x, y ) ( x , xy ) . Then, the induced morphism (( φ ) ∗ , ( φ ) ∗ ) : J e X −→ J X × J X is a K -isogeny. In particular,for any field L ⊃ K for which J e X ( L ) is finitely generated, we have rank f X/L = rank X /L +rank X /L . With this, we are finally ready to bound the ranks of the curves in Table 10.3, beginningwith the trio of curves Γ , Γ , and e Γ. The reader can check that each morphism appearing in(10.8) satisfies the condition of Lemma 1.4; below, we will use this lemma without mention.
Theorem 10.11.
Recall from Table 10.3 the nice curve Γ /K ( ‘ ) .(i) The genus of Γ is if and only if d = 2 or .(ii) If d (cid:62) and d is odd (resp. even), then ( t ) (cid:62) d (resp. d + 1 ).(iii) If d (cid:62) , then the points (cid:15) ( P ) , . . . , (cid:15) ( P d ) satisfy the single relation (cid:15) ( P ) + · · · + (cid:15) ( P d ) = 0 and generate a subgroup of J Γ ( K ( t )) of rank d − .Proof. For part (i), note that if d = 2 or 3, then deg ‘ ( x ) = d − is of genus 0.Assume now that d (cid:62)
4. If deg ‘ ( x ) = d − has the 4 d K ( t )-points P i , ι ( P i ), and if d − has in addition to these a K ( t )-point at infinity,giving a total of 4 d + 1 K ( t )-points. This establishes (ii). Continuing with the assumption d (cid:62)
4, for (iii), we note that the identity Q i ( x − u i ) = h ( x ) − ‘ ( x ) (cf. (10.1)) implies thatdiv( y − h ( x )) = P i P i − dD ∞ , so d X i =1 (cid:15) ( P i ) = d X i =1 [2 P i − D ∞ ] = 2[div( y − h ( x ))] = 0 ∈ J Γ ( K ( t )) . The desired conclusion then follows from Lemma 10.5 (i). (cid:3)
Theorem 10.12.
Recall from Table 10.3 the nice curve Γ /K ( ‘ ) .(i) The genus of Γ is if and only if d = 2 .(ii) If d (cid:62) and d is even (resp. odd) then ( K ( t )) (cid:62) d + 1 (resp. d + 2 ).(iii) If d (cid:62) , then the points r ( Q ) , . . . , r ( Q d ) are linearly independent in J Γ ( K ( t )) , so rank J Γ ( K ( t ) (cid:62) d .Proof. Since deg x‘ ( x ) = d , the case d = 2 is the only case in which Γ is of genus 0. Assume d (cid:62)
3. For part (ii), note that if deg ‘ ( x ) = d is even, then Γ has the 4 d K ( t )-points Q i , ι ( Q i ),and also the point D = (0 , d + 1 K ( t )-points. If d is odd, then it has inaddition to these a K ( t )-rational point at infinity, giving a total of 4 d + 2 K ( t )-points.For (iii), assume d (cid:62)
3. By Lemma 10.5, any relation among the r ( Q i )’s is of the form a · d X i =1 r ( Q i ) = 0 ∈ J Γ ( K ( t )) . (10.13)Let e ∈ ( S ) d ∼ = Gal( K ( b, t ) /K ( b, u )) (cf. Proposition 8.3 (i)) be an involution that sends t to − t , and fixes all the other t i ’s. Then, e sends r ( Q ) to r ( ι ( Q )) = − r ( Q ) and fixes allthe other r ( Q i )’s. Applying e to the relation (10.13), we get − a r ( Q ) + a · d X j =2 r ( Q i ) = 0 ∈ J Γ ( K ( t )) . Lemma 10.5 then forces a = − a , so we conclude that the points r ( Q ) , . . . , r ( Q d ) are linearlyindependent in J Γ ( K ( t )). (cid:3) Suppose H is an abelian group generated by elements R , . . . , R n , andconsider the homomorphism f : Z n −→ H which sends each “coordinate vector” e i to theelement R i , so that rank Z H = n − rank Z ker f. An element P i a i e i of the kernel corresponds to an identity P i a i R i = 0 in H ; such an identity will be called a relation . Two relations P i a i R i = 0 and P i b i R i = 0 will be called linearly independent if the associated elements P i a i e i , P i b i e i ∈ ker f are Z -linearly independent. We will use without mention the followingsimple observation: for any disjoint subsets S and S of { , . . . , n } , the relations P i ∈ S R i = 0and P j ∈ S R j = 0 are linearly independent. Theorem 10.15.
Recall from Table 10.3 the nice curve e Γ /K ( ‘ ) . For i = 1 , . . . , d , let p i := (cid:15) ( e P + i ) + (cid:15) ( e P − i ) ∈ J e Γ ( K ( t )) .(i) The genus of e Γ is if and only if d = 2 .(ii) If d (cid:62) , then e Γ( K ( t )) (cid:62) d .(iii) If d = 3 , then e Γ is of genus , and the points (cid:15) ( e P ± ) , . . . , (cid:15) ( e P ± ) satisfy the linearlyindependent relations p = · · · = p = 0 and generate a subgroup of J e Γ ( K ( t )) of rank .(iv) If d (cid:62) , then the points (cid:15) ( e P ± ) , . . . , (cid:15) ( e P ± d ) satisfy the single relation p + · · · + p d = 0 and generate a subgroup of J e Γ ( K ( t )) of rank d − .Proof. For (i), we have deg ‘ ( x ) = 2( d −
1) = 2 if d = 2, and clearly this is the only case whenthe genus of e Γ is 0. For (ii), we note that e Γ has the 8 d K ( t )-points e P + i , e P − i , ι ( e P + i ) , ι ( e P − i ), for i = 1 , . . . , d .For parts (iii) and (iv), we first let q i := (cid:15) ( e P + i ) − (cid:15) ( e P − i ) ∈ J e Γ ( K ( t )), we let H (resp. H )denote the subgroup of J e Γ ( K ( t )) generated by p i , q i (resp. e P + i , e P − i ) for i = 1 , . . . , d . Clearly, H (cid:54) H ; on the other hand, we have 2 (cid:15) ( e P + i ) = p i + q i , and 2 (cid:15) ( e P − i ) = q i − p i , so H (cid:54) · H ,which implies that rank Z H = rank Z H . Note that p i = ϕ ∗ ( (cid:15) ( P i )) and q i = ϕ ∗ ( (cid:15) ( Q i )), sounder the isogeny ( ϕ ∗ + ϕ ∗ ) : J Γ × J Γ −→ J e Γ (that is the dual isogeny of the one providedby Proposition 10.10), the subgroup H of J Γ ( K ( t )) × J Γ ( K ( t )) generated by the (cid:15) ( P i )’sand (cid:15) ( Q i )’s is mapped surjectively onto H , giving rank Z H = rank Z H = rank Z H .Now, if d (cid:62)
3, then the points (cid:15) ( Q i ) = 2 r ( Q i ) (cf. 10.4) are linearly independent in J Γ ( K ( t )) (cf. Theorem 10.12 (iii)), so the points q , . . . , q d are linearly independent in J e Γ ( K ( t )). If d = 3, then Γ is of genus 0 (cf. Theorem 10.11 (i)), so we have the six linearlyindependent relations p = · · · = p = 0 ∈ J e Γ ( K ( t )) (cf. 10.14), so we get rank Z H =rank Z H = 6, which establishes (iii). If d (cid:62)
4, then the (cid:15) ( P i )’s satisfy only the single relation (cid:15) ( P i ) + · · · + (cid:15) ( P d ) = 0 ∈ J Γ ( K ( t )) (cf. Theorem 10.11 (iii)), so the p i ’s satisfy only thesingle relation p = · · · = p d = 0, and we have rank Z H = rank Z H = (2 d −
1) + 2 d = 4 d − (cid:3) Remark 10.16.
The curves Γ , Γ , and e Γ were considered (for d (cid:62)
4) by Shioda in [Shi98]and he obtained the same bounds as we do. Our proofs of the rank bounds in Theorems 10.11,10.12, and 10.15 are more elementary compared to Shioda’s proofs in that we do not use thetheory of the canonical height. Also, we have included some extra details in Theorem 10.15(not found in [Shi98]) that will be used in the proof of Theorem 13.5.We also note that the present record R ( Q , g ; Z / Z ) (cid:62) g + 2 (cf. (0.6)) is obtained bytaking specializations of the curve Γ in the cases corresponding to the last row of Table 10.18. In the table below, we summarize the results in Theorems 10.11, 10.12,and 10.15. For X = Γ , Γ , or e Γ, we denote the genus, number of explicit K ( t )-points, andlower bound on rank X/K ( t ) by g ( X ) , N ( X ), and R ( X ) respectively. We express the genera g (Γ ) and g ( e Γ) in terms of the genus g of Γ . Γ /K ( ‘ ) : y = ‘ ( x ) Γ /K ( ‘ ) : y = x‘ ( x ) e Γ /K ( ‘ ) : y = ‘ ( x )deg ‘ ( x ) = d − x‘ ( x ) = d deg ‘ ( x ) = 2( d − d g (Γ ) N (Γ ) R (Γ ) g (Γ ) N (Γ ) R (Γ ) g ( e Γ) N ( e Γ) R ( e Γ)2 0 0 03 0 1 14 6 1 24 62 g + 2 ( g (cid:62) g g + 9 4 g + 3 g g (Γ ) + 9 4 g (Γ ) + 4 2 g g ( e Γ) + 16 4 g ( e Γ) + 72 g + 3 ( g (cid:62) g g + 12 4 g + 5 g + 1 8 g (Γ ) + 6 4 g (Γ ) + 2 2 g + 1 8 g ( e Γ) + 16 4 g ( e Γ) + 7
Table 10.18: Lower bound on rank and rational points for Γ , Γ , e ΓNext, we establish lower bounds on the ranks of Θ , Θ , and e Θ. Theorem 10.19.
Recall from Table 10.3 the nice curve Θ /K ( b, ‘ ) .(i) If d is odd (resp. even), then ( K ) (cid:62) d (resp. d + 1 ).(ii) If d = 2 , , then the points (cid:15) ( P ij ) , i = 1 , . . . , d, j = 1 , , , satisfy the linearlyindependent relations (cid:15) ( P i ) + (cid:15) ( P i ) + (cid:15) ( P i ) = 0 , for i = 1 , . . . , d , and generate asubgroup of J Θ ( K ) of rank d .(iii) If d (cid:62) , then the points (cid:15) ( P ij ) , i = 1 , . . . , d, j = 1 , , , satisfy the single relation P i,j (cid:15) ( P ij ) = 0 and generate a subgroup of J Θ ( K ) of rank d − .Proof. For (i), note that Θ has the 12 d K -points P ij , ι ( P ij ). If d is odd then deg ‘ ( b g ( x )) =3( d −
1) is even. If d is even, then Θ has an K -rational point at infinity, giving a total of12 d + 1 K -points.For parts (ii) and (iii), we first note that ϑ : Θ −→ Γ satisfies the conditions ofLemma 1.4; here, we abused notation to write Γ instead of (Γ ) K ( b, ‘ ) . Since ϑ ∗ P i = P i + P i + P i ∈ Div(Θ ), we have ϑ ∗ ( (cid:15) ( P i )) = (cid:15) ( P i ) + (cid:15) ( P i ) + (cid:15) ( P i ), so we can rephraseLemma 10.5 (ii) as saying that any relation among the (cid:15) − . emP ij − . em ’s is of the form P i a i ϑ ∗ − . em(cid:15) − . emP i − . em − . em = 0 ∈ J Θ − . em K − . em . If d = 2 or 3, then Γ is of genus 0, which implies that ϑ ∗ ( (cid:15) ( P i )) = 0 ∈ J Θ ( K ) for each i = 1 , . . . , d ; these relations are visibly linearly independent (cf. 10.14), so (ii) follows.If d (cid:62)
4, then the relation P i (cid:15) ( P i ) = 0 ∈ J Γ ( K ) pulls back to the relation P i ϑ ∗ ( (cid:15) ( P i )) = P i,j (cid:15) ( P ij ) = 0 ∈ J Θ ( K ). On the other hand, any relation P i a i ϑ ∗ ( (cid:15) ( P i )) can be pushedforward to give a relation d X i =1 a i · ( ϑ ∗ ϑ ∗ ( (cid:15) ( P i ))) = d X i =1 a i · (cid:15) ( P i ) = 0 ∈ J Γ ( K ) . Theorem 10.11 (iii) then implies a = · · · = a d , so (iii) follows. (cid:3) Theorem 10.20.
Recall from Table 10.3 the nice curve Θ /K ( b, ‘ ) .(i) If d is even (resp. odd), then ( K ) (cid:62) d + 1 (resp. d + 2 ).(ii) If d = 2 , then the points r ( Q ij ) , i = 1 , . . . , , j = 1 , , , satisfy the linearly independentrelations (cid:16) r ( Q i ) + r ( Q i ) + r ( Q i ) (cid:17) = 0 , for i = 1 , . . . , , and generate a subgroup of J Θ ( K ) of rank d = 8 .(iii) If d (cid:62) , then the points r ( Q ij ) , i = 1 , . . . , d, j = 1 , , , generate a subgroup of J Θ ( K ) of rank d . Proof.
For part (i), we first note that Θ has the 12 d K -points Q ij , ι ( Q ij ), and also thepoint R = (0 , d is odd, then Θ has an additional K -point at infinity, so we get ( K ) (cid:62) d + 2.Now, Lemma 10.5 says that any relation among the r ( Q ij )’s is of the form P i a i ( r ( Q i ) + r ( Q i ) + r ( Q i )) = 0 ∈ J Θ ( K ). Parts (ii) and (iii) rely on the observation that for each i = 1 , . . . , d , the point r ( Q i )+ r ( Q i )+ r ( Q i ) ∈ J Θ ( K ) is “linked” to the point r ( Q i ) ∈ J Γ ( K )in the following way. From the diagram (10.8), we get group homomorphisms( ϕ ) ∗ ◦ e ψ ∗ ◦ ψ ∗ : Pic (Θ ) K −→ Pic (Γ ) K , ( ψ ) ∗ ◦ e ψ ∗ ◦ ( ϕ ) ∗ : Pic (Γ ) K −→ Pic (Θ ) K . We compute (cid:16) r ( Q i ) + r ( Q i ) + r ( Q i ) (cid:17) ψ ∗ (cid:20) X j =1 (cid:16) e P + ij + ι ( e P − ij ) − D ∞ (cid:17)(cid:21) = (cid:20) X j =1 e P + ij + X j =1 ι (cid:16) e P − ij (cid:17) − D ∞ (cid:21)e ψ ∗ h e P + i + 3 · ι (cid:16) e P − i (cid:17) − D ∞ i ( ϕ ) ∗ [6 Q i − D ∞ ]= 6 · r ( Q i ) , (10.21)and similarly, r ( Q i ) ( ϕ ) ∗ (cid:16) h e P + i + ι (cid:16) e P − i (cid:17) − D ∞ i (cid:17)e ϕ ∗ (cid:20) X j =1 e P + ij + X j =1 ι (cid:16) e P − ij (cid:17) − D ∞ (cid:21) = (cid:20) X j =1 (cid:16) e P + ij + ι ( e P − ij ) − D ∞ (cid:17)(cid:21) ( ψ ) ∗ h Q i + 2 Q i + 2 Q i − D ∞ i = 2 (cid:16) r ( Q i ) + r ( Q i ) + r ( Q i ) (cid:17) (10.22)If d = 2, then Γ is of genus 0 (cf. Theorem 10.12 (i)), so each r ( Q i ) = 0 ∈ J Γ ( K ), and (10.22)implies 2( r ( Q i ) + r ( Q i ) + r ( Q i )) = 0. This establishes that the relations in the statementof part (ii) are satisfied; moreover, these relations are visibly linearly independent (cf. 10.14),so we conclude that the r ( Q ij )’s generate a subgroup of J Θ ( K ) of rank 6 d − d = 4 d .For part (iii), if we have a relation P i a i ( r ( Q i ) + r ( Q i ) + r ( Q i )), then (10.21) impliesthat P i a i · r ( Q i ) = 0. Part (ii) of Theorem 10.12 then forces a i = 0 for all i . We concludethat if d (cid:62)
3, then the points r ( Q ij ) are linearly independent in J Θ ( K ), so they generate asubgroup of rank 6 d . (cid:3) Corollary 10.23.
Recall from Table 10.3 the nice curve e Θ /K ( b, ‘ ) .(i) We have e Θ( K ) (cid:62) d .(ii) If d = 2 , then rank e Θ / K (cid:62) d = 16 . (iii) If d = 3 , then rank e Θ / K (cid:62) d = 30 .(iv) If d (cid:62) , then rank e Θ / K (cid:62) d − .Proof. Part (i) follows from the observation that e Θ has the K -points e P + ij , e P − ij , ι ( e P + ij ) , ι ( e P − ij )for i = 1 , . . . , d and j = 1 , ,
3. The remaining parts follow immediately by combiningTheorem 10.19, Theorem 10.20, and Proposition 10.10. (cid:3)
In the table below, we summarize the results in Theorems 10.19, 10.20,and Corollary 10.23. For X = Θ , Θ , and e Θ, we denote the genus by g , g , and e g respectively,the number of explicit K -points by N , N , and f N respectively, and the lower bound onrank X/ K by R , R , and e R respectively. We express g , g , and e g in terms of the genus g ofΓ (cf. Table 10.18). Θ /K ( b, ‘ ) : y = ‘ ( b g ( x )) Θ /K ( b, ‘ ) : y = x‘ ( b g ( x )) e Θ /K ( b, ‘ ) : y = ‘ ( b g ( x ))deg ‘ ( b g ( x )) = 3( d −
1) deg x‘ ( b g ( x )) = 3 d − ‘ ( b g ( x )) = 6( d − d g N R g N R e g f N e R g + 2 ( g (cid:62)
1) 3 g + 1 8 g + 17 4 g + 7 3 g + 1 8 g + 17 4 g + 8 6 g + 2 8 e g + 32 4 e g + 152 g + 3 ( g (cid:62)
1) 3 g + 2 8 g + 20 4 g + 9 3 g + 3 8 g + 14 4 g + 6 6 g + 5 8 e g + 32 4 e g + 15 Table 10.25: Lower bound on rank and rational points for Θ , Θ , e Θ11.
Lower bounds on N ( K, g ) and R ( K, g ) for g ≡ , R ( K, g ) (cid:62) g + 7 and N ( K, g ) (cid:62) g + 16 for all g (cid:62)
2. In this section, we use the data in Table 10.25 to getimproved bounds for g ≡ , Proposition 11.1.
Let
X/K ( b, ‘ ) be one of the curves Θ , Θ , or e Θ . Then X/K ( b, ‘ ) isnon-constant. We recall before the proof that to any hyperelliptic curve of genus g (cid:62)
2, we can associate abinary form in K [ x, z ] of degree 2 g + 2, which is the branch divisor of a degree 2 morphism to P (cf. 1.3). If f ( x, z ) and f ( x, z ) are binary forms obtained in this way from hyperellipticcurves X and X , then X and X are K -isomorphic if and only if f and f are in the sameorbit of GL ,K (see [KSV05, Section 2]), which acts on the space of degree 2 g + 2 binaryforms by a bc d · g ( x, z ) = g ( ax + bz, cx + dz ) . We identify the space of degree n binary forms over K with the K -points of an affine space BF n := Spec K [ b , . . . , b n ]. A point P ∈ ( a , . . . , a n ) ∈ BF n ( K ) is identified with the form P i a i x i z n − i ∈ K [ x, z ]. Proof.
Assume for a contradiction that
X/K ( b, ‘ ) is constant. To prove non-constancy, wemay assume K = Q . Note that if the explicit affine equation for X/K ( b, ‘ ) from Table 10.25is y = f ( x ), then f ( x ) is a polynomial in K [ b, ‘ ][ x ], with deg f = 2 g + 1 or 2 g + 2, where g denotes the genus of X . We let h ( x, z ) := z g +2 f ( x/z ) be the binary form associated to thisequation (cf. 1.3). We now break the proof down into cases depending on d .First, assume d (cid:62)
4, and let v be an indeterminate. The ring homomorphism K [ b , . . . , b g +2 ] −→ K [ v, b, ‘ ] = K [ v ] ⊗ K K [ b, ‘ ] b i coefficient of x i z g +2 − i in h ( x + vz, z )defines a K -morphism φ : A K × Spec K [ b, ‘ ] −→ BF g +2 , which sends a K -point ( a, P ) tothe form h P ( x + az, z ) ∈ K [ x, z ], where h P ( x, z ) denotes the specialization of h ( x, z ) at P .It is easy to see that the morphism φ restricts to a closed immersion on the closed subset { v = b = 0 } ; the image Z is defined by the vanishing of some of the coefficients b i , and wehave dim Z = d (cid:62)
4. Note that Z is contained in the closed subset { b g +1 = 0 } ⊂ BF g +2 .For any K -point ( a, P ) of the locally closed set { v = 0 } ∩ { b = 0 } , the coefficient of x g +1 z in the form h P ( x + az, z ) is non-zero, i.e. φ ( a, P ) ∈ ( BF g +2 \ Z )( K ) so we see thatdim φ ( { b = 0 } ) = 1 + dim Z (cid:62) . The assumption that
X/K ( b, ‘ ) is non-constant implies that φ ( { b = 0 } ) is contained in anorbit of GL ,K , but this is a contradiction, since dim GL ,K = 4 < d = 3. If we set ( b, l , l , l ) = (0 , , a, a ∈ K , then Θ (resp. e Θ) specializes to (a smooth, proper model of) the curve y = x + ax + 1 (resp. thecurve y = x + ax + 1). We can then apply [LL18, Lemma 6.3.1] to conclude that aswe vary a ∈ K , we get infinitely many specializations of Θ (resp. e Θ) that are pair-wisenon-isomorphic.Next, with d = 3, consider Θ /K ( b, ‘ ). The binary form obtained from the explicit modelfor Θ (as in 1.3) is: h ( x, z ) = zx ( l ( x + 2 bx z + b xz ) + l ( x + 2 bx z + b xz ) z + l z )= l x z + 4 bl x z + 6 b l x z + ( l + 4 b l ) x z + (2 l + l b ) x z + l b x z + l xz Let φ : Spec K [ v, b, l , l , l ] −→ BF be as before; we will show that the image of φ is ofdimension 5, and obtain a contradiction as before. For this, we note that the image of { v = b = 0 } is of dimension 3 (exactly as before), and this image lands in the closed subset { b = 0 } . If ( a, P ) is a K -point of { v = 0 } ∩ { b = 0 } , then the coefficient of x z in h P ( x, z )is non-zero, i.e. φ ( a, P ) is a K -point of { b = 0 } ), which implies that dim φ ( { v = 0 } ) = 4.Note moreover that φ ( { v = 0 } ) is contained in the closed subset { b = 8 b b } ⊂ BF ,so to conclude, it suffices to produce a point ( a, P ) ∈ (Spec K [ v, b, ‘ ])( K ) such that thecoefficients of h P ( x + az, z ) satisfy the (non-)identity 3 b = 8 b b . For this, note that if we set( v, b, l , l , l ) = ( a, , , , ∈ K for some a ∈ K , then h ( x, z ) specializes to the binary form x z + (7 a + 4) x z + (21 a + 28 a + 6) x z + (l.o.t), and the identity 3 b = 8 b b is equivalentto the identity 21 a + 56 a = 0. The desired result follows by choosing an a ∈ K that doesnot satisfy the latter equation.Finally, assume d = 2. In this case, Θ and Θ are elliptic curves over K ( b, ‘ ), andthe reader can check that their j -invariants are non-constant polynomials in K [ b, ‘ ], sowe are left with the final case of the genus 2 and degree 6 curve e Θ /K ( b, l , l ) : y = l ( x + 2 bx + b x ) + l . To show that it is non-constant, it suffices to show that the curve X/K ( l ) = y = x + 2 x + x + l is non-constant, for which we make use of the Igusainvariants of the polynomial p ( x ) := x + 2 x + x + l (cf. [KSV05]). We use the function igusa_clebsch_invariants() in Sage [S +
20] to compute the
Igusa invariants ([KSV05])for the specializations at l = 1 and l = 2 , which we denote by p ( x ) and p ( x ) respectively.We have ( I ( p ) , I ( p ) , I ( p ) , I ( p )) = ( − , , − , − I ( p ) , I ( p ) , I ( p ) , I ( p )) = ( − , , − , − , and a quick computation shows that there is no r = 0 ∈ K for which I i ( p ) = r i I i ( p )for i = 1 , , ,
5. Thus, the binary forms associated to the two specializations are not GL ,K -conjugates, and the corresponding specializations of X are also not isomorphic, whichconcludes the proof. (cid:3) Theorem 11.2.
Let e g be a positive integer congruent to modulo . Then(i) N ( K, e g ) (cid:62) e g + 32 .(ii) If e g = 2 , then R ( K, e g ) (cid:62) .(iii) If e g = 5 , then R ( K, e g ) (cid:62) .(iv) If e g (cid:62) , then R ( K, e g ) (cid:62) e g + 15 .Proof. Let d (cid:62) e Θ / K from Table 10.25. This is anon-constant curve (cf. Proposition 11.1) over the function field of the K -rational variety B d (cf. Theorem 7.6), which satisfies e Θ( K )) (cid:62) f N and rank e Θ / K (cid:62) e R , so we can applyCorollary 2.5 to conclude that N ( K, e g ) (cid:62) f N and R ( K, e g ) (cid:62) e R , where e g is the genus of e Θ.We wrap up the proof by observing that as d ranges over the integers { , , . . . } , thegenus e g ranges over the positive integers congruent to 2 modulo 3, so we immediately get (i).Moreover, if d = 2 (resp. 3), then e g = 2 (resp. 5), so we get parts (ii) and (iii), and if d (cid:62) e g (cid:62)
8, so we get (iv). (cid:3)
Theorem 11.3.
Let g (cid:62) be an integer congruent to modulo . Then(i) N ( K, g ) (cid:62) g + 17 (ii) R ( K, g ) (cid:62) g + 8 .Proof. Consider the curve Θ from Table 10.25. Exactly as in the proof above, we have athand all the ingredients necessary to apply Corollary 2.5 to Θ / K .Note from the third line of Table 10.25 that as d ranges over the even integers { , , , . . . } ,the genus g ranges over { , , , . . . , } , that is, over the set of integers greater than 3 thatare congruent to 1 modulo 3, which immediately establishes both parts of the theorem. (cid:3) Proof of Theorem 0.7.
Observe that in Table 10.25, when d = 2 g + 3, we have g = 3( g + 1), so as g ranges over the positive integers, g ranges over the positive integersdivisible by 3. In each of these cases, the genus g curve Θ has two K -rational Weierstrasspoints, namely, the point D (cf. 10.4) and a point at infinity. The difference of these pointsgives a two torsion point in J ( K ), so J ( K ) ⊃ Z g +6 ⊕ ( Z / Z ). Theorem 2.4 then yields thebound R ( K, g ; Z / Z ) (cid:62) g + 6, as desired. Remark 11.5.
We conclude this section with a few words about the structure of theJacobians of the curves of Table 10.25. None of them is simple, as evidenced by the systemof morphisms (10.9), but J ( e Θ) decomposes in an interesting way. Indeed, there exist abelian varieties A /K ( B d ) and A /K ( B d ) such that J (Θ ) ∼ A × J (Γ ), and J (Θ ) ∼ A × J (Γ ),and J ( e Θ) ∼ J (Θ ) × J (Θ ) ∼ A × J ( e Γ) × A .The current record for R ( Q , g ) with simple Jacobians is due to Shioda and Terasoma[ST99], who showed that the curve Γ admits infinitely many specializations to curves over Q with the same lower bound 4 g + 5 on rank, and having (absolutely) simple Jacobians.12. The rational variety Z n /K This section and the next comprise one block. In this section, we define and establish the K -rationality of a variety Z n /K (see 5.5 (iii)), which parametrizes (4 , n )-composite tuples(as in 4.4). Z n . Fix an integer n (cid:62)
1. Let G i denote the polynomial z i + z i , and define: S := K [ z , z , . . . , z n , z n , G − ]( G − G , . . . , G n − − G n ) , Z n := Spec S. Let L be the étale K -algebra K [ x ] / ( x + 1), and ζ ∈ L the class of x . Following themethod described in 7.2, for each i = 1 , . . . , n , we identify Spec K [ z i , z i ] with the Weilrestriction R L/K A L by identifying, for any K -algebra A , the point ( a i , a i ) ∈ A ( A ) withthe element a i + a i ζ ∈ A ⊗ K L . The open subscheme Spec K [ z i , z i , ( z i + z i ) − ] isthen naturally identified with R L/K G m,L , and we have a group scheme homomorphism N : R L/K G m,L −→ G m,K . Observe that Z n fits into a cartesian diagram: Z n ( R L/K G m,L ) n G m,K ( G m,K ) n . (cid:2) ( N ) n ∆ (12.2) Proposition 12.3.
The scheme Z n is an n + 1 -dimensional K -rational variety.Proof. Let R L/K G m,L denote the norm-one torus, i.e. the kernel of the homomorphism N : R L/K G m,L −→ G m,K . We claim that there exists a K -isomorphism φ : Z n −→ R L/K G m,L × ( R L/K G m,L ) n − . We can produce such an isomorphism exactly as we did in Lemma 7.4; we omit the details,since the proof is identical to the proof of the lemma. This isomorphism implies that Z n is K -rational, since R L/K G m,L is K -rational and one-dimensional K -tori are K -rational. (cid:3) We now setup some notation for this section and the next.(i) Let K denote the function field K ( Z n ).(ii) We define the following elements in S : t ij := z ij , i = 1 , . . . , n, j = 1 , ,u i := − t i t i = − ( z i z i ) i = 1 , . . . , n,m j := ( − n − j s n − j ( u , . . . , u n ) , j = 0 , . . . , n − ,b := G = t + t = z + z . (12.5) (iii) Let R denote the K -subalgebra of S generated by the elements t ij , and let F denotethe fraction field of R . Let u and m denote the tuples of elements u , . . . , u n and m , . . . , m n − respectively, and abuse notation to let K ( b, u ) and K ( b, m ) denote thesubfield of K generated by these tuples respectively, so that K ( b, m ) ⊂ K ( b, u ) ⊂ F ⊂ K .(iv) We define the following polynomials in S [ x ]: g ( x ) := x − bxm ( x ) := x n + m n − x n − + · · · + m x + m . (12.6)The last line of (12.8) below implies that Z n parametrizes (4 , n )-composite tuples (as in4.4). Proposition 12.7.
Continuing with the notation of 12.4, we have the following identities in S [ x ] : g ( x ) − u i = ( x − z i )( x + z i )( x − z i )( x + z i ) , i = 1 , . . . , n,g ( x ) − u i = ( x − t i )( x − t i ) , i = 1 , . . . , n,m ( x ) = ( x − u ) · · · ( x − u n ) ,m ( g ( x )) = n Y i =1 ( x − t i )( x − t i ) ,m ( g ( x )) = n Y i =1 ( x − z i )( x + z i )( x − z i )( x + z i ) . (12.8) Proof.
The reader can check using the definitions in 12.4 that these identities hold. (cid:3)
Proposition 12.9 below is used (in the next section) in bounding the rank of the curveΛ /K ( Z d ) (defined in Table 13.3). Proposition 12.9. (i) The fields
F, K ( b, u ) and K ( b, m ) are n + 1 -dimensional rationalfunction fields.(ii) We have an isomorphism of groups: α : ( V ) n ∼−−→ Gal( K /F )(( e , e ) , . . . , ( e n , e n )) ( z ij e ij z ij ) , where V = ( S ) denotes the Klein-four group (so each e ij ∈ S = {± } ).Proof. For (i), we first note that the identities in (12.8) imply that K is the splitting field ofthe polynomial m ( g ( x )) ∈ K ( b, m )[ x ], so K /K ( b, m ) is a finite extension, and all the fieldsin (i) are of transcendence degree n + 1 over K . Since t i = b − t i for each i = 1 , . . . , n , thefield F is generated over K by b, t , . . . , t n . Similarly, the elements b, u (resp. b, m ) generate K ( b, u ) (resp. K ( b, m )), so we conclude all three of these are rational function fields.For (ii), first note that if we regard α a priori as a homomorphism from ( V ) n toAut( K [ z i , z i | i = 1 , . . . , n ]), then the image consists of automorphisms that fix the elements G i = z i + z i , and hence, also the ideal defining S . Thus, α is indeed a homomorphism from( V ) n to Aut( K ). It is clearly injective, and we leave it to the reader to verify that the imageis Gal( K /F ). (cid:3) Lower bounds on N ( K, g ) and R ( K, g ) for odd g We continue with the notation of the previous section. In this section, we construct curveswith many rational points over the function field K ( Z n ) and establish lower bounds on theirranks; this results in lower bounds on R ( K, g ) and N ( K, g ) for odd, positive integers g (seeTheorem 13.10).For the rest of this section, we set n = 2 d , where d (cid:62) d +1 elements b, m of S generate a polynomial algebra K [ b, m ] ⊂ S , and since b is invertible in S , we have an inclusion K [ b, b − , m ] ⊂ S . Let h ( x ) = x d + h d − x d − + · · · + h and ‘ ( x ) = l d − x d − + · · · + l denote the square-root approximation and remainder of m ( x )respectively (cf. Definition 3.4), so that we have inclusions of K -algebras: K [ b, b − , ‘ ] ⊂ K [ b, b − , ‘ , h ] ∼ = K [ b, b − , m ] , −→ R , −→ S, where, as usual, h and ‘ denote the tuples of indeterminates ( h , . . . , h d − ) and ( l , . . . , l d − )respectively, and the isomorphism K [ b, b − , ‘ , h ] ∼ = K [ b, b − , m ] is given by Lemma 3.1.Combining the identities in (12.8) with the identity m ( x ) = h ( x ) − ‘ ( x ) ∈ S [ x ], we get thefollowing identities in S [ x ]: h ( x ) − ‘ ( x ) = ( x − u ) · · · ( x − u d ) ,h ( g ( x )) − ‘ ( g ( x )) = d Y i =1 ( x − t i )( x − t i ) ,h ( g ( x )) − ‘ ( g ( x )) = d Y i =1 ( x − z i )( x + z i )( x − z i )( x + z i ) . (13.1) The second column of the table below contains three affine curvesover the function field K ( b, ‘ ), given by equations of the form y = f ( x ), with f ( x ) ∈ K ( b, ‘ ) a separable polynomial; to check separability, it suffices to note that upon setting( b, l , l , . . . , l d − , l d − ) = (0 , , , . . . , , f ( x ) specializes to a separable polynomial in K [ x ].In the first column, we denote the smooth, proper model of the corresponding affine curve(see 1.3). The reader can verify using (13.1) that the curves come with the explicit K -rationalpoints listed in the table, where we recall that K = K ( Z d ). Curve Equation K -rational pointsΛ /K ( b, ‘ ) y = ‘ ( g ( x )) P ij := ( t ij , h ( g ( t ij ))) i = 1 , . . . , d, j = 1 , /K ( b, ‘ ) y = x‘ ( g ( x )) Q ij := ( t ij , z ij h ( g ( t ij ))) i = 1 , . . . , d, j = 1 , e Λ /K ( b, ‘ ) y = ‘ ( g ( x )) e P ± ij := ( ± z ij , h ( g ( z ij ))) i = 1 , . . . , d, j = 1 , Table 13.3: Curves with K -pointsAs in (10.8), these curves fit together into a diagram of K ( b, ‘ )-schemes: e Λ ( x, y )Λ Λ ( x , y ) ( x , xy ) , ϕ ϕ (13.4) so by Proposition 10.10, we have a K ( b, ‘ )-isogeny J e Λ −→ J Λ × J Λ . Now, we determine lower bounds for rank J Λ / K and rank J Λ / K ; then, we will applyProposition 10.10 to get lower bounds for rank J e Λ / K . Theorem 13.5.
Recall from Table 13.3 the nice curve Λ /K ( b, ‘ ) .(i) We have ( K ) (cid:62) d = 8 g + 16 , where g = d − is the genus of Λ .(ii) If d = 3 , then Λ is of genus , and rank Λ / K (cid:62) .(iii) If d (cid:62) , then rank Λ / K (cid:62) d − g + 7 .Proof. Part (i) follows from the observation that Λ has the distinct K -rational points P ij , ι ( P ij ), for i = 1 , . . . , d and j = 1 ,
2. For parts (ii) and (iii), we first note that thepoints P ij are actually F -rational points of Λ , where F = K ( b, t , . . . , t d, ) (cf. proof ofProposition 12.9 (i)), so it suffices to replace K with F in the statements of (ii) and (iii). Let O denote the local ring at the generic point of the hyperplane { b = 0 } of Spec K [ b, t , . . . , t d, ],and let n denote its maximal ideal. Each t i = b − t i reduces modulo n to − t i , so each u i = − t i t i = − t i ( b − t i ) reduces modulo n to t i (cf. (12.5)); thus, we have an inclusion offields K ( m ) = K ( m , . . . , m d − ) , −→ O / n = K ( t , . . . , t d, ), defined by m j ( − d − j s d − j ( u , . . . , u d ) = ( − d − j s d − j ( t i , . . . , t d, ) , j = 0 , . . . , d − . On the other hand, recall the field K ( t ) = K ( t , . . . , t d ) appearing in Theorem 10.15; again,we have an inclusion K ( m ) , −→ K ( t ) defined by m j −→ ( − d − j s d − j ( t , . . . , t d ) , j =0 , . . . , d − t i −→ t i , i = 1 , . . . , d, defines a K ( m )-isomorphismof fields O / n ∼−−→ K ( t ). Now, let F ( x, z ) ∈ O [ x, z ] denote the homogenization of ‘ ( g ( x )) = ‘ ( x − bx ), let O [ x, y, z ] denote the graded O -algebra in which deg x = deg z = 1 anddeg y = g + 1 = d −
1, and consider the O -scheme X := Proj O [ x, y, z ] / ( y − F ( x, z )). Thegeneric fiber of X / Spec O is isomorphic to Λ /F , and the special fiber is isomorphic to( e Γ) K ( t ) /K ( t ), where we recall from Table 10.3 that e Γ /K ( ‘ ) is a smooth proper model of theaffine curve y = ‘ ( x ). Note that X is covered by two open affine subsets isomorphic toSpec R and Spec R , where R = O [ x, y ] / ( y − F ( x, R = O [ x, y ] / ( y − F (1 , z )) areboth relative global complete intersections over O (cf. [Sta18, Definition 00SP]). [Sta18, Lemma00SW] then implies that X −→ Spec O is flat, so X −→ Spec O is smooth and projective.Under the reduction map ρ : Λ ( F ) −→ e Γ( K ( t )), we have ρ ( P i ) = ρ ( t i , h ( t i − bt i )) = ( t i , h ( t i )) = e P + i ∈ e Γ( K ( t )) , i = 1 , . . . , d, and similarly ρ ( P i ) = e P − i for i = 1 , . . . , d (cf. Table 10.18). Abusing notation to let ρ : J Λ ( F ) −→ J e Γ ( K ( t )) denote the reduction map on Jacobians, we have ρ ( (cid:15) ( P i )) = (cid:15) ( e P + i ) and ρ ( (cid:15) ( P i )) = (cid:15) ( e P − i ) , i = 1 , . . . , d, so under the homomorphism ρ , the subgroup of J Λ ( F ) generated by the (cid:15) ( P ij )’s is mappedsurjectively onto the subgroup of J e Γ ( K ( t )) generated by the (cid:15) ( e P ± i )’s. Now, part (ii) (resp.part (iii)) follows from part (iii) (resp. (iv)) of Theorem 10.15, so we are done. (cid:3) Theorem 13.6.
Recall from Table 13.3 the nice curve Λ /K ( b, ‘ ) . Let D denote the closedpoint (0 , of Λ .(i) The curve Λ is of genus g = d − .(ii) We have ( K ) (cid:62) d + 2 = 8 g + 10 . (iii) The points r ( Q ij ) := [ Q ij − D ] , i = 1 , . . . , d, j = 1 , , are linearly independent in J Λ ( K ) , so they generate a subgroup of rank d = 4 g + 4 .Proof. See 1.3 for the first part. The second part follows by noting that Λ has the distinct K -points Q ij , ι ( Q ij ) for i = 1 , . . . , d, j = 1 ,
2, and two Weierstrass points ( D and a point atinfinity). For the third part, assume for a contradiction that we have a linear dependencerelation d X i =1 a i r ( Q i ) + a i r ( Q i ) = 0 ∈ J Λ ( K ) , (13.7)with a pk = 0 for some 1 (cid:54) p (cid:54) d and 1 (cid:54) k (cid:54)
2. Let σ ∈ Gal( K /F ) be the transpositionthat sends z pk to − z pk and fixes all the other z ij ’s (cf. the proof of Proposition 12.9 (ii)).Then, σ fixes Q ij ∈ Λ ( K ) if ( i, j ) = ( p, k ), and σ ( Q pk ) = ( σ ( t pk ) , σ ( z pk h ( g ( t pk )))) = ( t pk , − z pk h ( g ( t pk ))) = ι ( Q pk ) ∈ Λ ( K ) . Thus, σ fixes r ( Q ij ) ∈ J Λ ( K ) if ( i, j ) = ( p, k ), and we have σ ( r ( Q pk )) = σ (cid:16) [ Q pk − D ] (cid:17) = [ ι ( Q pk ) − D ] = − r ( Q pk ) ∈ J Λ ( K ) . Applying σ to the identity (13.7) and subtracting the result from the original, we find that2 a pk r ( Q pk ) = 0, i.e. r ( Q pk ) is a torsion point of J Λ ( K ). Lemma 9.3 then implies that Q pk = ( t pk , z pk h ( g ( t pk ))) is algebraic over K ( b, ‘ ), which implies that t pk , and hence also u p = t pk ( b − t pk ) is algebraic over K ( b, ‘ ). By the Fundamental Theorem of symmetricfunctions (cf. [Hun80, Page 254, Theorem 2.18]), the extension K ( b, u ) /K ( b, m ) is Galois,with Gal( K ( b, u ) /K ( b, m )) ∼ = S d acting by permutation on u , . . . , u d , so every u i is algebraicover K ( b, ‘ ), which implies that each m j = ( − d − j s d − j ( u , . . . , u d ) is algebraic over K ( b, ‘ ).But this implies that K ( b, m ) /K ( b, ‘ ) is an algebraic extension, which is a contradictionbecause trdeg K ( b, ‘ ) K ( b, m ) = d > (cid:3) Corollary 13.8.
Recall from Table 13.3 the nice curve e Λ /K ( b, ‘ ) .(i) We have e Λ( K ) (cid:62) d = 8 e g + 24 , where e g = 2 d − denotes the genus of e Λ .(ii) If d = 2 , then e g = 1 and rank e Λ / K (cid:62) .(iii) If d = 3 , then e g = 3 , and rank e Λ / K (cid:62) .(iv) If d (cid:62) , then rank e Λ / K (cid:62) d − e g + 11 .Proof. For (i), we note that e Λ has the distinct K -points e P + ij , e P − ij , ι ( e P + ij ) , ι ( e P − ij ) for i = 1 , . . . , d and j = 1 ,
2. The identity e g = 2 d − e g + 2 = deg ‘ ( g ( x )) = 4( d − d = 2, Λ is of genus 0 (cf. Theorem 13.5 (i)) and rank Λ (cid:62) d = 3, then rank Λ / K (cid:62) / K (cid:62)
12, so part (iii) follows from Proposition 10.10.When d (cid:62)
4, then rank Λ / K (cid:62) d − / K (cid:62) d (cf.Theorem 13.6 (iii)), so by Proposition 10.10, we have rank e Λ / K (cid:62) d − e g + 11, whichestablishes part (iv). (cid:3) Lemma 13.9.
The curve e Λ /K ( b, ‘ ) is non-constant.Proof. If d = 2, then e Λ is a smooth, proper model of the affine curve y = l ( x + bx ) + l ,and the reader can easily verify that this is a non-constant genus 1 curve.Assume now that d (cid:62)
3. If we set b = 0, then the equation y = ‘ ( g ( x )) specializes tothe equation y = ‘ ( x ), where ‘ ( x ) = l d − x d − + · · · + l x + l ∈ K ( ‘ )[ x ]. If we further specialize by setting l i = 1 for i = 0 , d −
1, and l i = 0 for i = 2 , . . . , d − , d −
2, then theequation above becomes the equation y = ( x ) d − + l x + 1. [LL18, Lemma 6.3.1] guaranteesthat this equation defines a non-constant curve over K ( l ), so we conclude that e Λ /K ( b, ‘ ) isnon-constant. (cid:3) Theorem 13.10.
Let e g be an odd positive integer. Then(i) N ( K, e g ) (cid:62) e g + 24 .(ii) If e g (cid:62) , then R ( K, e g ) (cid:62) e g + 11 .Proof. Let d (cid:62) e Λ / K from Corollary 13.8, where K = K ( Z d ). Since the genus e g of e Λ is given by e g = 2 d −
3, we see that as d ranges over thepositive integers, e g ranges over the positive odd integers.Now, Proposition 12.3 tells us that Z d is a K -rational variety, and Lemma 13.9 aboveimplies that e Λ / K is non-constant (since it is non-constant over K ( b, ‘ )). Thus, we can applyCorollary 2.5 to Corollary 13.8 (i) to conclude that N ( K, e g ) (cid:62) e g + 24 if e g is odd. When d ranges over the integers { , , . . . , } in the setting of Corollary 13.8, then e g ranges over the odd integers { , , , . . . } ; applying Corollary 2.5 to Corollary 13.8 (iv), we conclude that if e g (cid:62) R ( K, e g ) (cid:62) e g + 11. (cid:3) We conclude this section with the proof of Theorem 0.2.
Proof of Theorem 0.2.
Set K = Q , and let g be a positive integer not divisible by 6.(1) If g ≡ , g ≡ N ( K, g ) (cid:62) g + 32 fromTheorem 11.2 (i). If moreover g (cid:62)
8, then we get R ( K, g ) (cid:62) g +15 from Theorem 11.2(iv).(2) If g ≡ , g is odd, so we get N ( K, g ) (cid:62) g + 24 from Theorem 13.10(i). If moreover g (cid:62)
7, then we get R ( K, g ) (cid:62) g + 11 from Theorem 13.10 (ii).(3) If g ≡ g ≡ N ( K, g ) (cid:62) g + 17 from Theorem 11.3(i). If moreover g (cid:62)
4, then we get R ( K, g ) (cid:62) g + 8 from Theorem 11.3 (ii). (cid:3) Curves of high rank over cyclotomic fields
In this section, we consider ( g +1 , m ( x g +1 ) ∈ K [ x ] with deg m = 6; under the assumption that g + 1 is an odd prime, we obtaingenus g curves of the form y = ‘ ( x g +1 ), with deg ‘ ( x ) = 2, having high rank over numberfields that contain the g + 1-th roots of unity. Now, and for the rest of this section, we fix an integer g (cid:62) g + 1 isa prime number . We set e = g + 1 and n = 6, and we let K be a number field that containsa primitive g + 1-th root of unity ζ .Let ‘ , h , m , u , t denote the tuples of indeterminates ( l , l , l ) , ( h , h , h ) , ( m , . . . , m ) , ( u , . . . , u ) , and ( t , . . . , t ) respectively. Consider the inclusion of function fields K ( ‘ ) ⊂ K ( ‘ , h ) ∼ = K ( m ) ⊂ K ( u ) ⊂ K ( t ) , where, letting m ( x ) = x + m x + · · · + m , h ( x ) = x + h x + h x + h , and ‘ ( x ) = l x + l x + l ,we define: u i = t g +1 i , for i = 1 , . . . , ,m ( x ) = ( x − u ) · · · ( x − u ) ,m ( x ) = h ( x ) − ‘ ( x ) , and the last identity gives the isomorphism K ( ‘ , h ) ∼ = K ( m ) (cf. Lemma 3.1). Note that m ( x g +1 ) = ( x g +1 − t g +11 ) · · · ( x g +1 − t g +16 ) = Y (cid:54) i (cid:54) Y (cid:54) j (cid:54) g ( x − ζ j t i ) ∈ K [ t ][ x ] , so A ( t ) parametrizes ( g + 1 , X /K ( ‘ ) be the affine curve defined by the equation y = ‘ ( x g +1 ), and let X/K ( ‘ )denote its smooth, proper model (as in 1.3). Note that ‘ ( x g +1 ) is square-free, so the genus of X is g . The curve X /K intersects the curve y = h ( x g +1 ) in the 6( g + 1) K ( t )-points P ij := ( ζ j t i , h ( u i )) , i = 1 , . . . , , j = 0 , . . . , g. Counting also the hyperelliptic conjugates, we have X ( K ( t )) (cid:62) g + 1). As in 1.3, for P ∈ X ( K ( t )), we let (cid:15) ( P ) := [2 P − D ∞ ] ∈ J ( K ( t )), where J/K ( ‘ ) denotes the Jacobian of X/K ( ‘ ). The curve X/K ( ‘ ) admits an automorphism θ defined by ( x, y ) ( ζx, y ), andclearly h θ i ∼ = Z / ( g + 1) Z . We write ζ ∈ Aut( J ) to denote the automorphism induced by θ . Lemma 14.2.
The homomorphism f : Z [ z ] −→ End( J ) defined by z ζ induces aninjection of rings Z [ ζ ] , −→ End( J ) .Proof. Since θ is of order g + 1, so is ζ . Thus, the kernel of f contains z g +1 −
1. Since g + 1 is prime, this polynomial factors into irreducibles as z g +1 − z − z ), whereΦ( z ) = 1 + z + · · · + z g ∈ Z [ z ] is the ( g + 1)-th cyclotomic polynomial. We want to showthat ker f is generated by Φ( z ), for which it suffices to show that Φ( z ) ∈ ker f (since Φ( z )irreducible).Consider the nice curve C/K ( ‘ ) defined by the equation y = ‘ ( x ), which is of genus 0since ‘ ( x ) is quadratic. We have a natural map α : X −→ C defined by ( x, y ) ( x g +1 , y ).By abuse of notation, let ζ ∈ Aut(Div( X )) denote the automorphism induced by θ ∈ Aut( X ).Then, for a divisor D ∈ Div( X ), we have α ∗ α ∗ D = D + ζ ( D ) + · · · + ζ g D = Φ( ζ )( D ) ∈ Div( X ) . Passing to Pic’s, this implies that the endomorphism f (Φ( z )) = Φ( ζ ) ∈ End( J ) coincideswith the composition Pic ( X ) α ∗ −−→ Pic ( C ) α ∗ −−→ Pic ( X ); the desired result now follows sincePic ( C ) = 0. (cid:3) The above lemma allows us to abuse notation and denote by ζ (instead of ζ ) the automor-phism of J induced by θ ∈ Aut( X ). Theorem 14.3.
The points P i := (cid:15) ( P i ) , i = 1 , . . . , generate a free Z [ ζ ] -submodule H (cid:54) J ( K ( t )) of rank . Thus, we have rank X/K ( t ) (cid:62) g .Proof. The claim about H immediately implies the claim about rank X ( K ( t )), since[ H : Z ] = [ H : Z [ ζ ]] · [ Z [ ζ ] : Z ] = 6 g. Assume for a contradiction, then, that we have a linear dependence relation a P + · · · + a P = 0 ∈ J ( K ( t )) , (14.4)with each a i ∈ Z [ ζ ], and with some a p = 0. For each i = 1 , . . . , j = 0 , . . . , g , let σ ij ∈ Aut( K ( t )) denote the automorphism defined by σ ij ( t k ) = ζ j t k , if k = it k otherwise. The group G := h σ ij i ∼ = ( Z / ( g +1) Z ) is nothing but the Galois group Gal( K ( t ) /K ( t g +11 , . . . , t g +16 )).In particular, G (cid:54) Aut K ( ‘ ) K ( t ), so G acts on X ( K ( t )). We have σ ij ( P k ) = θ j ( P k ) , if k = iP k otherwise.The action of G on X ( K ( t )) induces an action of G on J ( K ( t )); if σ ij ∈ Aut( J ( K ( t )))denotes the automorphism induced by σ ij ∈ G , we have σ ij ( P k ) = ζ j P k , if k = i P k otherwise.Now, for each i = 1 , . . . ,
6, consider the endomorphism τ i := σ i + · · · + σ ig of J ( K ( t )), whichis clearly also an endomorphism of H . We have τ i ( P k ) = , if k = i ( g + 1) P k , otherwise.Thus, applying τ k to a P + · · · + a P kills off the term a k P k , and multiplies all the otherterms by g + 1. By successively applying τ i for each i ∈ { , . . . , } \ { p } , we end up with arelation a · P p = 0, where a = a p ( g + 1) = 0 ∈ Z [ ζ ]. This implies that P p must be torsion ;indeed, letting N denote the norm N Z [ ζ ] / Z ( a ) ∈ Z , we have N · P p = 0. Lemma 9.3 then impliesthat the point P p = ( t p , h ( t p )) ∈ X ( K ( t )) is algebraic over K ( ‘ ), which implies that u p = t g +1 p is algebraic over K ( ‘ ). Since Gal( K ( u ) /K ( m )) ∼ = S acts by permutation on u , . . . , u , every u i is algebraic over K ( ‘ ), which implies that each m j := ( − − j s − j ( u , . . . , u ) isalgebraic over K ( ‘ ). But this is absurd, since trdeg K ( ‘ ) K ( m ) = 3 > (cid:3) Proof of Theorem 0.5.
Note that X K ( t ) /K ( t ) is a curve of positive genus g , whichsatisfies rank X K ( t ) /K ( t ) (cid:62) g . In order to apply Corollary 2.5 and conclude R ( K, g ) (cid:62) g , itremains only to show X K ( t ) /K ( t ) is non-constant, for which it suffices to show that X/K ( ‘ ) isnon-constant. This follows, for instance, from [LL18, Lemma 6.3.1], which says that the curve X /K ( l ) defined by y = x g +2 + l x g +1 + 1 is non-constant. Consequently, we have infinitelymany specializations (that are nice genus g curves over K ) of the form ( l , l , l ) = (1 , a, a ∈ K , so we are done. Remark 14.6.
Recall from 5.1 the Q -variety of ( e, n )-composite tuples G ( e, n ) ⊂ A en Q =Spec Q [ t , . . . , t n ]. The Kummer locus is the closed subscheme K ( e, n ) := { s ( t ) = · · · = s e − ( t ) = 0 } ⊂ G ( e, n ) . The reader can easily verify that for any field K ⊃ Q , we have K ( e, n )( K ) = { ( a ij ) ∈ G ( e, n )( K ) | Y i,j ( x − a ij ) = M ( x e ) for some M ( x ) ∈ K [ x ] } . If K contains the e -th roots of unity, then K ( e, n ) K is the union of ( e − n copies of A nK . Remark 14.7.
Consider the curve X /K ( ‘ ) above, given by the equation y = l x g +2 + l x g +1 + l , which has a subgroup H (cid:54) Aut( X ) of order 2( g + 1). Above, we constructed six “full” orbits (under H ) of K ( t )-points, for a total of 6 · g + 1) = 12( g + 1) K ( t )-points. Weremark here that the assumption that g + 1 be prime is used only in bounding the rank ofthe Jacobian.Brumer [Cap95, Section 5] considered curves C a,b /K of the form y = ax g +2 + bx g +1 + a ,parametrized by ( a, b ) ∈ K ; here, g (cid:62) g + 1 need not be prime), and K contains the primitive g + 1-th roots of unity. These curves admit the extra involution( x, y ) (1 /x, y/x g +1 ), so there is a subgroup H (cid:54) Aut( C a,b ) with H = 4( g + 1).Brumer showed that there are infinitely many choices of ( a, b ) ∈ K that give rise to aninfinite set of curves C a,b /K , each having at least four full orbits of K -points under H (so C a,b ( K ) (cid:62) · g + 1) = 16( g + 1)). This yields the current record N ( K, g ) (cid:62) g + 1) fornumber fields K containing the g + 1-th roots of unity.It is natural to wonder, then, if our curve X /K ( t ) admits a specialization of the form C a,b /K . More precisely, if P is a K -point of the locally closed subset W := { l = l } \ (cid:54) i,j (cid:54) { (( t i t j ) g +1 − t g +1 i − t g +1 j ) = 0 } \ { disc ‘ ( x g +1 ) = 0 } ⊂ A ( t ) , then the specialization of X at P would be a curve of the form C a,b /K having six full orbits(under H ) of K -points, giving C a,b ( K ) (cid:62) · g + 1) = 24( g + 1). So, we ask: Does W have a K -point, or even better, does W contain a K -rational variety ? At present, we do notknow if W ( K ) is non-empty. References [BLR90] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud,
Néron models , Ergebnisse derMathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21,Springer-Verlag, Berlin, 1990. MR1045822 ↑
5, 13[Cap95] Lucia Caporaso,
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