5. Summary and Outlook
genus going to infinity having a torsion divisor linear in the genus. This series is the characteristic zero analogon to a family of curves over finite fields given byscheidler. Furthermore, it is possible to useHensel’s Lemma to construct curves with aQ-rational torsion divisor in this case, too.
For hyperelliptic and superelliptic curves, we see that there is a close relation between the computation of a non-trivial unit in the coordinate ring of a curve and the order of a divisor supported only at infinity. For hyperelliptic curves such a unit can be computed by using continued fractions. For function fields which are degree three extensions of the rational function field, we are able to show a connection between the periodicity of the Voronoi algorithm and the order of a certain divisor. For this we need to assume a certain splitting behavior of the place at infinity namely to a rational place of degree one and a rational place of degree two. Using this approach, we are able to give a series of curves with a divisor of order quadratic in the genus.
In the last section we consider a special family of hyperelliptic curves and determine the specializations such that there exists point of given order on the specialization.
Further work: Since in this area of research still not much is known, there are a lot of questions yet to be answered. We have already seen in the introduction that there are a lot of open problems concerning the torsion subgroup of an abelian variety defined over a number field.
Problem. Given a positive integersN, g and a field K, is it possible to find an abelian variety A/K of dimension g such that
N|#Ators(K)?
In this thesis we construct curves of various genus with a rational point of prescribed order on the jacobian. Even in the dimension two case, it remains unclear whether a curve with a rational torsion point of orderN for any possible N is known. Furthermore, we just consider the case where we have one torsion point on the jacobian. It remains an open question whether we are able to construct a curve with a rational pointP of order N and a rational pointQof orderN0 on the jacobian. Another open question about this problem is whether it is possible to construct a three-dimensional family of curves with a rational point of orderN on the jacobian. This would be a rather large step towards a better understanding of the moduli space of these objects. Historically this would follow the path which is taken in the dimension one case. The examples found for genus two curves induce the expectation that the largest prime orderp of a rational point on the jacobian could bep= 29. Furthermore, it seems like the points of large order all lie on
156 Max Kronberg
the image of the curve under some embedding into the jacobian. It would be interesting to know whether this observation is at random or whether there is some reason behind it. For higher genus curves there are still only a few examples known for large torsion orders. We have tried to adapt the techniques of genus two curves to genus three curves, but we were not able to find any example for a large torsion point. A fruitful area of research is the development of techniques that apply for genus larger than two. Some of these questions perhaps can be tackled by similar techniques as presented in this thesis.
These presented questions about the explicit construction of torsion points on jacobians of curves are a good starting point for further research.
The next two open problems seem a lot harder than the first one.
Problem. Let A/K be an abelian variety of dimension g and P ∈Ators(K). Does there exist a number N0(K, g) such that
ord(P)≤N0(K, g)?
This problem, known as the Boundedness Conjecture, and its uniform version are at the moment the most prominent conjectures concerning the rational torsion subgroup of abelian varieties. A possible starting point is to prove the existence of such a constant restricting to jacobians of hyperelliptic curves of genus two over the rational numbers.
If we restrict to jacobians of hyperelliptic curves with complex multiplication, such a constant is already known. So restricting to hyperelliptic curves with real multiplication could be a possible next step.
Problem. Find explicit formulae for the moduli space describing abelian varieties of dimension g with a torsion point or a torsion subgroup of certain order.
This problem is as hard as the previous one. Only for smallN asN = 2,3andg= 2or for elliptic curves there are equations for the moduli space known. Questions concerning the moduli space of these objects can only be answered with a deep understanding of modular forms of higher dimension.
Summing up, we conclude that there is still a lot of interesting work to do in this area of research.
Max Kronberg 157
A. Magma Codes
A.1. 13-Torsion Following Leprévost
Lemma A.1. With the following Magma code we computed the family of hyperelliptic curves admitting a13-torsion point on its jacobian in Theorem 3.14.
//Setting up the rational function field K<p,t>:=RationalFunctionField(Rationals(),2);
R<x>:=PolynomialRing(K);
r:=0;
//Setting q:=-p^2 makes the first coefficient of a vanish q:=-p^2;
//Setting up the congruence modulo b
b:=x^4-(p^2-2*q)*x^3+(q^2-2*(p*r-t))*x^2-(r^2-2*q*t)*x+t^2;
R1:=x^3-(p^2-2*q)*x^2+(q^2-2*(p*r-t/2))*x-(r^2-q*t);
S:=quo<R|b>;
R1:=R1/(p*t);
//Lifting the congruence
R2:=R1-R!(S!((R!((R1^2-x)/b))/(2*R1)))*b;
S2:=quo<R|b^2>;
//Computing the defining polynomial a:=R!(S2!(R2^13));
f:=R!((a^2-x^13)/b^2);
//Computing the Igusa invariants IgusaClebschInvariants(f);
[
(2p^8t^3 - 2p^4t^4 - 2t^5)/p^12,
(1/4p^16t^6 - 23/4p^12t^7 + 25/2p^8t^8 + 23/4p^4t^9 + 1/4t^10)/p^24,
(1/8p^24t^9 173/64p^20t^10 + 75/16p^16t^11 559/32p^12t^12 -75/16p^8t^13 - 173/64p^4t^14 - 1/8t^15)/p^36,
(-1/16p^8t^19 + 349/256p^4t^20 + 1/16t^21)/p^44 ]
159
A. Magma Codes
A.2. 19-Torsion Family
Lemma A.2. The Magmacode for determining that there exist no isomorphism of our family and the family of Leprévost is the following,
//Constructing a rational function field in the indeterminates //a for our family
//and a2 for the family of Leprévost
K<a,a2>:=RationalFunctionField(Rationals(),2);
R<x>:=PolynomialRing(K);
//Setting up our family
t:=-4-(a^4 + 6*a^2+4*a+1)/a^3;
b:=-a-(4+t);
c:=6-a*b;
d:=-a*c-4;
A:=1/2*((a-b-1)*x^2-(a+c)*x-d);
f:=A^2-t*x^4*(x-1);
C:=HyperellipticCurve(f);
//Finished setting up our family //Compute absolute Igusa invariants ICI_c:=IgusaClebschInvariants(C);
absII_c:=[
ICI_c[2]/ICI_c[1]^2, ICI_c[3]/ICI_c[1]^3, ICI_c[4]/ICI_c[1]^5 ];
//Setting up Leprevost family
A2:=-(a2^10+6*a2^9+29*a2^8+168*a2^7+482*a2^6-348*a2^5+482*a2^4 +168*a2^3+29*a2^2+6*a2+1)*x^2
+2*(a2+1)^2*(a2^8+4*a2^7+20*a2^6+92*a2^5 +22*a2^4+92*a2^3+20*a2^2+4*a2+1)*x -(a2+1)^6*(a2^4+14*a2^2+1);
f2:=A2^2-1024*a2^4*(a2^2-1)^2*(a2^4+14*a2^2+1)^2*x^3*(x-1)^2;
C2:=HyperellipticCurve(f2);
//Finished setting up Lepre´vost family
//Compute absolute Igusa invariants for Leprévost family
160 Max Kronberg
A.3. 11-Torsion family
ICI_c2:=IgusaClebschInvariants(C2);
absII_c2:=[
ICI_c2[2]/ICI_c2[1]^2, ICI_c2[3]/ICI_c2[1]^3, ICI_c2[4]/ICI_c2[1]^5 ];
//taking the difference of the absolute Igusa invariants diff2:=[Numerator(absII_c[i]-absII_c2[i]): i in [1..3]];
//Setting up the scheme of parameters with //equal absolute Igusa invariants
AA:=AffineSpace(Parent(diff2[1]));
X:=Scheme(AA,diff2);
A.3. 11-Torsion family
Lemma A.3. With this Magma code we computed the family of hyperelliptic curves of genus two with a 11-torsion divisor on the image of the curve in its jacobian. The corresponding result in the thesis is Example 3.5.
//Constructing the rational function field
K<lambda,mu,eta>:=RationalFunctionField(Rationals(),3);
S<X>:=PolynomialRing(K);
//Setting the u-coordinate of the torsion divisor
xi:=eta^2+2*lambda*mu*eta-lambda+2*lambda^2*eta+lambda^2*mu^2 +2*lambda^3*mu +lambda^4;
u:=X+xi;
//Lifting the quadratic congruence with Hensel’s Lemma R:=X^2+mu*X+eta;
b:=X^3+(2*mu+lambda)*X^2+(2*eta+mu^2+2*lambda*mu+lambda^2)*X +2*mu*eta-1+2*lambda*eta+2*lambda^2*mu+lambda^3+lambda*mu^2;
S1:=quo<S|b>;
S2:=quo<S|b^2>;
lambda1:=X-lambda;
lambda2:=S!(S1!(-lambda1/(2*R)));
R2:=R+lambda2*b;
//Computing the defining polynomial with arbitrary u-coordinate F:=S!((S!(S2!(R2)^11)^2-u^11)/b^2);
R:=RingOfIntegers(K);
Max Kronberg 161
A. Magma Codes
//Computing the coefficients of F under the assumption that the //u-coordinate of the torsion divisor is equal to x
S:=quo<R|R!xi>;
R<x>:=PolynomialRing(S);
R!F;
A.4. 17-torsion over a Number Field
Lemma A.4. The Magma code for computing the defining equation for the 17-torsion example is the following.
//Setting up the rational function field
K<p,q,t,u>:=RationalFunctionField(Rationals(),4);
R<x>:=PolynomialRing(K);
//Setting up the congruence modulo Q r:=(-1/8*p^3 + 1/2*p*q + 2*u);
s:=(1/64*p^4 - 1/8*p^2*q + p*u + 1/4*q^2);
b:=x^6+p*x^5+q*x^4+r*x^3+s*x^2+t*x+u^2;
R1:=x^3+p/2*x^2+(q-1/4*p^2)/2*x+u;
S:=quo<R|b>;
//Lifting the congruence modulo Q to a congruence modulo Q^2 R2:=R1-R!(S!((R!((R1^2-(-1/4*p^2*u + q*u - t)*x)/b))/(2*R1)))*b;
S1:=quo<R|b^2>;
//Computing the defining polynomial a:=R!((S1!R2)^17);
F:=R!((a^2-((-1/4*p^2*u + q*u - t)*x)^17)/b^2);
162 Max Kronberg
A.5. 7-Torsion via Pells Equation
A.5. 7-Torsion via Pells Equation
Lemma A.5. The absolute Igusa invariants of Cλ in Theorem 3.37 are given by α(Cλ) = λ24+ 32λ22+ 436λ20+ 3431λ18+ 17663λ16+ 63280λ14+ 162608λ12
+302517λ10+807965
2 λ8+752535
2 λ6+3690073
16 λ4+ 82881λ2+ 13092
· 1
26(λ12+ 16λ10+ 90λ8+ 265λ6+ 432λ4+ 14034 λ2+ 102)2, β(Cλ) =
λ36+ 48λ34+ 1038λ32+27097
2 λ30+241237
2 λ28+6266553 8 λ26 + 7732161
2 λ24+29762419
2 λ22+363095039
8 λ20+884972175 8 λ18 +863944095
4 λ16+2694007649
8 λ14+3323900783
8 λ12+6387815919 16 λ10 +4661354371
16 λ8+9936719491 64 λ6 +453675089
8 λ4+ 12631657λ2+ 1289688
· 1
29(λ12+ 16λ10+ 90λ8+ 265λ6+ 432λ4+ 14034 λ2+ 102)3, γ(Cλ) = λ2+ 214
·
λ14+ 21λ12+ 160λ10+ 13012 λ8+31672 λ6+3733716 λ4+ 1921λ2+ 676 214(λ12+ 16λ10+ 90λ8+ 265λ6+ 432λ4+14034 λ2+ 102)5 .
Max Kronberg 163
B. Formulae for the Kummer Surface
Let P := (κ1 :. . .:κ4)∈ K(Jac(C))be a point on theKummersurface of the jacobian of the genus two hyperelliptic curveC. Thenδ2(P) = (a1(P) :. . .:a4(P)) is given by the coordinates
a1(P) =4κ1κ34+ 4κ24(κ21f2−κ2κ3f5−3κ23f6)
+ 4κ4(−4κ31f0f4+κ31f1f3−8κ21κ2f0f5−2κ21κ3f1f5−12κ1κ22f0f6−κ1κ22f1f5
−6κ1κ2κ3f1f6−2κ1κ2κ3f2f5−4κ1κ23f2f6−κ1κ23f3f5−2κ32f1f6−4κ22κ3f2f6
−6κ2κ23f3f6−8κ33f4f6+ 2κ33f52)
+ 4κ43f4f52+ 4κ41f12f4+ 4κ41f0f32−16κ43f2f62+ 4κ42f0f52−16κ43f42f6
−16κ2κ33f3f4f6−16κ41f02f6−16κ41f0f2f4−4κ21κ2κ3f1f3f5+ 4κ32κ3f1f52 + 4κ2κ33f3f52+ 4κ22κ23f2f52−16κ31κ2f0f1f6−32κ31κ2f0f2f5+ 8κ31κ2f12f5
+ 32κ31κ3f0f2f6−16κ31κ3f0f3f5−16κ31κ3f12f6−64κ21κ22f0f2f6−8κ21κ22f0f3f5
+ 12κ21κ22f12f6−16κ21κ2κ3f0f4f5−16κ21κ2κ3f1f2f6−48κ21κ23f0f4f6
+ 16κ21κ23f0f52+ 24κ21κ23f1f3f6−8κ21κ23f1f4f5−16κ21κ23f22f6−32κ1κ32f0f3f6
−16κ1κ22κ3f0f52−24κ1κ22κ3f1f3f6−16κ1κ2κ23f0f5f6
−8κ1κ2κ23f1f52−16κ1κ2κ23f2f3f6+ 32κ1κ33f0f62−16κ1κ33f1f5f6−8κ1κ33f32f6
−16κ42f0f4f6−16κ32κ3f0f5f6−16κ32κ3f1f4f6
−48κ22κ23f0f62−8κ22κ23f1f5f6−16κ22κ23f2f4f6−32κ2κ33f1f62+ 8κ43f3f5f6
a2(P) =4κ2κ34+κ24(4κ21f1+ 8κ1κ2f2−8κ1κ3f3+ 5κ22f3+ 8κ2κ3f4+ 4κ23f5) + 2κ4(2κ31f0f3+ 8κ21κ2f0f4+κ21κ2f1f3−8κ21κ3f0f5+ 8κ21κ3f1f4
−6κ21κ3f2f3+ 4κ1κ22f0f5+ 4κ1κ22f1f4−24κ1κ2κ3f0f6 + 8κ1κ2κ3f1f5+ 8κ1κ2κ3f2f4−5κ1κ2κ3f32
−8κ1κ23f1f6+ 8κ1κ23f2f5−6κ1κ23f3f4+ 2κ32f1f5+ 4κ22κ3f1f6+ 4κ22κ3f2f5 + 8κ2κ23f2f6+κ2κ23f3f5+ 2κ33f3f6)
+κ41f12f3+ 5κ21κ23f33+ 16κ43f1f62+κ43f3f52+ 16κ41f02f5−8κ1κ22κ3f1f3f5
+ 32κ21κ2κ3f12f6−20κ1κ2κ23f1f3f6−4κ41f0f2f3+ 32κ31κ2f02f6+ 16κ31κ2f0f1f5
165
B. Formulae for the Kummer Surface
−4κ31κ2f0f32−32κ31κ3f0f2f5+ 16κ31κ3f0f3f4+ 16κ31κ3f12f5−6κ31κ3f1f32 + 32κ21κ22f0f1f6−4κ21κ22f0f3f4+ 4κ21κ22f12f5−64κ21κ2κ3f0f2f6
−20κ21κ2κ3f0f3f5
+ 32κ21κ2κ3f0f42+ 16κ21κ2κ3f1f2f5−12κ21κ2κ3f1f3f4−20κ21κ23f0f3f6
−14κ21κ23f1f3f5+ 16κ21κ23f1f42+ 16κ21κ23f22f5−20κ21κ23f2f3f4−4κ1κ32f0f3f5 + 8κ1κ32f12f6−56κ1κ22κ3f0f3f6+ 32κ1κ22κ3f0f4f5+ 32κ1κ22κ3f1f2f6
−64κ1κ2κ23f0f4f6+ 32κ1κ2κ23f0f52+ 16κ1κ2κ23f1f4f5+ 32κ1κ2κ23f22f6
−12κ1κ2κ23f2f3f5−32κ1κ33f1f4f6+ 16κ1κ33f1f52+ 16κ1κ33f2f3f6−6κ1κ33f32f5
−4κ42f0f3f6+ 8κ32κ3f0f52−4κ32κ3f1f3f6+ 32κ22κ23f0f5f6+ 4κ22κ23f1f52
−4κ22κ23f2f3f6+ 32κ2κ33f0f62+ 16κ2κ33f1f5f6−4κ2κ33f32f6−4κ43f3f4f6 a3(P) =4κ3κ34+ 4κ24(−3κ21f0−κ1κ2f1+κ23f4)
+ 4κ4(−8κ31f0f2+ 2κ31f12−6κ21κ2f0f3−4κ21κ3f0f4−κ21κ3f1f3−4κ1κ22f0f4
−6κ1κ2κ3f0f5−2κ1κ2κ3f1f4−2κ1κ23f1f5−2κ32f0f5−12κ22κ3f0f6−κ22κ3f1f5
−8κ2κ23f1f6−4κ33f2f6+κ33f3f5)
+ 4κ42f12f6−16κ41f02f4−16κ41f0f22+ 4κ43f2f52+ 4κ41f12f2+ 4κ43f32f6−16κ43f0f62
−8κ21κ23f1f2f5+ 4κ31κ2f12f3+ 4κ21κ22f12f4+ 4κ1κ32f12f5+ 8κ41f0f1f3
−32κ31κ2f02f5−4κ1κ2κ23f1f3f5−16κ31κ2f0f2f3+ 32κ31κ3f02f6−16κ31κ3f0f1f5
−8κ31κ3f0f32−48κ21κ22f02f6−8κ21κ22f0f1f5−16κ21κ22f0f2f4−16κ21κ2κ3f0f1f6
−16κ21κ2κ3f0f3f4−8κ21κ2κ3f12f5−48κ21κ23f0f2f6+ 24κ21κ23f0f3f5
−16κ21κ23f0f42+ 16κ21κ23f12f6−16κ1κ32f0f1f6−16κ1κ32f0f2f5
−24κ1κ22κ3f0f3f5−16κ1κ22κ3f12f6−16κ1κ2κ23f0f4f5−16κ1κ2κ23f1f2f6
+ 32κ1κ33f0f4f6−16κ1κ33f0f52−16κ1κ33f1f3f6−16κ42f0f2f6−32κ32κ3f0f3f6
−64κ22κ23f0f4f6+ 12κ22κ23f0f52−8κ22κ23f1f3f6−16κ2κ33f0f5f6
−32κ2κ33f1f4f6+ 8κ2κ33f1f52−16κ43f2f4f6
a4(P) =κ44+ 2κ24(4κ21f0f4−κ21f1f3+ 4κ1κ2f0f5+ 8κ1κ3f0f6+ 4κ22f0f6 + 4κ2κ3f1f6+ 4κ23f2f6−κ23f3f5)
+ 4κ4(−2κ33f32f6−2κ31f0f32−2κ33f2f52+ 8κ21κ3f0f2f6−2κ31f12f4 + 16κ1κ22f0f2f6
−4κ1κ22f12f6+κ1κ2κ3f1f3f5+ 8κ31f0f2f4+ 12κ21κ2f0f2f5−3κ21κ2f12f5
+ 4κ21κ3f0f3f5−2κ21κ3f12f6+ 2κ1κ22f0f3f5+ 20κ1κ2κ3f0f3f6+ 8κ1κ23f0f4f6
−2κ1κ23f0f52+ 4κ1κ23f1f3f6+ 4κ32f0f3f6+ 16κ22κ3f0f4f6−4κ22κ3f0f52 + 2κ22κ3f1f3f6+ 12κ2κ23f1f4f6−3κ2κ23f1f52+ 8κ33f2f4f6)
166 Max Kronberg
+ 16κ43f22f62+κ43f32f52−2κ41f13f5+ 16κ42f02f62+κ41f12f32+ 16κ41f02f42
−4κ41f0f12f6+ 16κ41f0f22f4+ 16κ21κ2κ3f0f1f52−16κ43f1f3f62−8κ1κ2κ23f0f3f52
−2κ43f1f53+ 16κ43f0f4f62+ 16κ41f02f2f6+κ42f12f52−16κ41f02f3f5+ 8κ41f0f1f2f5
−8κ41f0f1f3f4−4κ41f0f2f32−4κ41f12f2f4−32κ31κ2f02f3f6+ 32κ31κ2f02f4f5 + 32κ31κ2f0f1f2f6−16κ31κ2f0f1f3f5+ 32κ31κ2f0f22f5−8κ31κ2f13f6
−8κ31κ2f12f2f5−64κ31κ3f02f4f6+ 32κ31κ3f02f52+ 16κ31κ3f0f1f3f6
+ 16κ31κ3f0f2f3f5−4κ31κ3f12f3f5+ 32κ21κ22f02f4f6+ 16κ21κ22f02f52
−24κ21κ22f0f1f3f6+ 64κ21κ22f0f22f6+ 8κ21κ22f0f2f3f5−16κ21κ22f12f2f6
−2κ21κ22f12f3f5+ 64κ21κ2κ3f02f5f6−32κ21κ2κ3f0f1f4f6 + 64κ21κ2κ3f0f2f3f6+ 8κ21κ2κ3f0f32f5−8κ21κ2κ3f12f3f6 + 64κ21κ23f02f62+ 96κ21κ23f0f2f4f6−32κ21κ23f0f2f52
−16κ21κ23f0f32f6+ 8κ21κ23f0f3f4f5−32κ21κ23f12f4f6 + 12κ21κ23f12f52+ 8κ21κ23f1f2f3f6+ 32κ1κ32f02f5f6
+ 32κ1κ32f0f2f3f6−8κ1κ32f12f3f6+ 64κ1κ22κ3f02f62
+ 32κ1κ22κ3f0f1f5f6+ 48κ1κ22κ3f0f32f6+ 64κ1κ2κ23f0f1f62
−32κ1κ2κ23f0f2f5f6+ 64κ1κ2κ23f0f3f4f6+ 16κ1κ2κ23f12f5f6+ 8κ1κ2κ23f1f32f6
−64κ1κ33f0f2f62+ 16κ1κ33f0f3f5f6+ 32κ1κ33f12f62+ 16κ1κ33f1f3f4f6
−4κ1κ33f1f3f52+ 16κ42f0f2f4f6−4κ42f0f2f52−4κ42f12f4f6+ 32κ32κ3f0f1f62 + 32κ32κ3f0f3f4f6−8κ32κ3f0f3f52+ 32κ22κ23f0f2f62−24κ22κ23f0f3f5f6
+ 64κ22κ23f0f42f6−16κ22κ23f0f4f52+ 16κ22κ23f12f62+ 8κ22κ23f1f3f4f6−2κ22κ23f1f3f52
−32κ2κ33f0f3f62+ 32κ2κ33f0f4f5f6−8κ2κ33f0f53+ 32κ2κ33f1f2f62
−16κ2κ33f1f3f5f6+ 32κ2κ33f1f42f6−8κ2κ33f1f4f52−4κ43f0f52f6+ 8κ43f1f4f5f6
−8κ43f2f3f5f6+ 16κ43f2f42f6−4κ43f2f4f52−4κ43f32f4f6.
Max Kronberg 167
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Selbstständigkeitserklärung
Hiermit versichere ich, Max Christian Kronberg, dass ich diese Dissertation selbstständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe.
Sie wurde weder in ihrer Gesamtheit noch in Teilen einer anderen wissenschaftlichen Hochschule zur Begutachtung in einem Promotionsverfahren vorgelegt.
Außerdem erkläre ich, dass ich die allgemeinen Prinzipien wissenschaftlicher Arbeit und Veröffentlichung, wie sie in den Leitlinien guter wissenschaftlicher Praxis der Carl von Ossietzky Universität Oldenburgfestgelegt sind, befolgt habe.
Oldenburg, 18.Mai 2015
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