# Two-Selmer Set of a Curve Let $C: y^2=f(x)$ be a hyperelliptic curve of genus $g$ over a number field $k$. We say $\pi_\delta: D_\delta\rightarrow C$ is a *two-cover* of $C$ if, over an algebraic closure, $\pi_\delta$ is isomorphic as a cover to the pull-back of an embedding of $C$ into its Jacobian along multiplication by $2$. The two-Selmer set classifies the $k$-isomorphism classes of two-covers of $C$ that have points everywhere locally. The hyperelliptic involution acts on such covers by pull-back: $\iota^*(\pi_\delta)=\pi_\delta\circ\iota$ (apply $\iota$ after $\pi_\delta$). The *fake* two-Selmer set is the set $$ \{\delta: D_\delta(k_v)\neq \emptyset \text{ for all places }v\text{ of }k\}/\iota^* $$ If this set is empty, then $C$ has no $k$-rational points. This can happen even if $C$ itself does have points everywhere locally. If the set is non-empty, it gives information about rational points on $C$. See [[Bruin and Stoll, 2009](../../references.md#cite-brusto-twocovdesc)] for the underlying theory as well as a description of an algorithm to compute the set. ## `TwoCoverDescent(C): CrvHyp -> SetEnum, Map, [Map, SeqEnum]` ```magma Bound : RngIntElt Default: -1 Fields : SetEnum Default: {} Raw : BoolElt Default: false PrimeBound : RngIntElt Default: 0 PrimeCutoff: RngIntElt Default: 0 ``` Computes the fake $2$-Selmer set as an abstract set. The map returned as second value can be used to obtain a representation of these abstract elements as elements in an algebra, which allows explicit construction of the corresponding cover. The optional parameters `Bound`, `Fields` and `Raw` perform the same function as for [`TwoSelmerGroup`](jacobians-number-fields.md#function-twoselmergroup) and we refer to it for their description. The remaining optional parameters are specific to this routine and we describe them here. `PrimeBound`: Two covers are of very high genus. Hence, according to the Weil bounds, they can have local obstructions at very large good primes. For instance, for genus $2$ one should check all primes up to norm $1153$ and for genus $3$ one should check all primes up to norm $66553$. This is very time consuming. One can use `PrimeBound` to restrict the good primes to be considered to only those whose norm do not exceed the given bound. In principle, this can result in a larger set being returned than the proper two-selmer set. `PrimeCutoff`: If the curve has bad reduction at some large prime, it can be prohibitively expensive to check the local conditions at this prime. This bound allows a restriction on the norm of bad primes where local conditions are considered. Setting this bound can result in a larger set being returned than the proper two-selmer set. ## `Example: Two-cover descent (ex-4cc57e)` As an illustration of `TwoCoverDescent`, we give the Magma-code to perform the computations related to the examples in [[Bruin and Stoll, 2009](../../references.md#cite-brusto-twocovdesc)]. First we give some examples of genus $2$ curves that have points everywhere locally, but have an empty $2$-Selmer set and hence have no rational points. ```magma > Q:=Rationals(); > Qx:=PolynomialRing( Q ); > C:=HyperellipticCurve(2*x^6+x+2); > Hk,AtoHk:=TwoCoverDescent(C); > #Hk; 0 > C:=HyperellipticCurve(-x^6+2*x^5+3*x^4-x^3+x^2+x-3); > Hk,AtoHk:=TwoCoverDescent(C); > #Hk; 0 ``` In the following we consider a curve of genus $2$ that does have rational points. In fact, its Jacobian has Mordell–Weil rank $2$. We compute its two-covers with points everywhere locally. There are two. They both cover an elliptic curve over a quadratic extension. We use `Chabauty` following [[Bruin, 2002](../../references.md#cite-bruin-phdthesis)] to determine the rational points on $C$. First we check that we can represent the fake two-Selmer set of the curve with some nice elements. ```magma > f:=2*x^6+x^4+3*x^2-2; > C:=HyperellipticCurve(f); > Hk,AtoHk:=TwoCoverDescent(C:PrimeBound:=30); > A:=Domain(AtoHk); > deltas:={-1-theta,1-theta}; > {AtoHk(d): d in deltas} eq Hk; true ``` Next, we determine a factorisation of $f$ into a quartic and a quadratic polynomial. ```magma > L:=NumberField(x^2+x+2); > LX:=PolynomialRing(L); > g:=(X^2-1/2)*(X^2-alpha); > h:=2*(X^2+alpha+1); > g*h eq Evaluate(f,X); true ``` For some $\gamma=\gamma(\delta)$, we have that a $2$-cover $D_\delta$ covers the elliptic curve $$ E:y^2=\gamma\, g(x). $$ This allows us to translate the question about rational points on $D_\delta$ into a question about L-rational points on E with the additional property that x is rational. We verify that the two values of $\delta$ we found above, correspond to $\gamma=(1-\alpha)/2$. ```magma > LTHETA:=quo; > j:=homLTHETA|THETA>; > gamma:=1/2*(-alpha + 1); > {Norm(j(delta)):delta in deltas} eq {gamma}; true > E:=HyperellipticCurve( gamma * g ); > P1:=ProjectiveSpace(Rationals(),1); > EtoP1:=mapP1|[E.1,E.3]>; ``` We present $E$ explicitly as an elliptic curve to Magma ```magma > P0:=E![1,(1-alpha)/2]; > Eprime,EtoEprime:=EllipticCurve(E,P0); > Etilde:=EllipticCurve(X^3+(1-alpha)*X^2+(2-9*alpha)*X+(16-2*alpha)); > EprimeToEtilde:=Isomorphism(Eprime,Etilde); > EtoEtilde:=EtoEprime*EprimeToEtilde; > EtildeToP1:=Expand(Inverse(EtoEtilde)*EtoP1); ``` We determine a group of finite odd index in $E(L)$. ```magma > success,MWgrp,MWmap:=PseudoMordellWeilGroup(Etilde); > success; true > MWgrp; Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*MWgrp.1 = 0 ``` We determine the set of $L$-rational points on $E$ that have a ${\mathbb{Q}}$-rational image under `EtildeToP1`. ```magma > V,R:=Chabauty(MWmap,EtildeToP1:IndexBound:=2); > V; { 0, MWgrp.1 - MWgrp.2, MWgrp.1, -MWgrp.2 } > R; 4 ``` Since we found earlier that in this case, there is only one value of $\gamma$ to consider, we know that any rational point on $C$ must correspond to one of these points on $E$. The correspondence is given by the fact that $E$ and $C$ both cover the $x$-line. We determine these points explicitly. ```magma > CtoP1 := map< C -> P1 | [C.1,C.3] >; > pi := Extend( EtildeToP1 ); > { pi( MWmap( v ) ) : v in V }; { (1 : 1), (-1 : 1) } > [ RationalPoints( p@@CtoP1 ): p in { P1(Q)| pi( MWmap( v ) ) : v in V } ]; [ {@ (1 : -2 : 1), (1 : 2 : 1) @}, {@ (-1 : -2 : 1), (-1 : 2 : 1) @} ] ```