Sage Days 22 Final Presentation
system:sage


<p><span style="font-size: x-large;"><strong>Background:&nbsp;</strong></span></p>
<p><span style="font-size: x-large;">Let $G_\QQ=Gal(\QQbar/\QQ), \ell$ be a prime and $E/\QQ$ be an elliptic curve.</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">There is a natural action of $G_\QQ$ on $E[n]\cong (\ZZ/n\ZZ)^2$. </span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">So we have a mod $n$ representation $\rho: G_\QQ\rightarrow Aut(E[n])\cong GL_2(\ZZ/n\ZZ)$.</span></p>
<p><span style="font-size: medium;"><span style="font-size: x-large;"><br /></span></span></p>
<p><span style="font-size: x-large;">The action of $G_\QQ$ commutes with the multiplication by $\ell$ map, so we get an action of $G_\QQ$ on $T_\ell(E)=\varprojlim&nbsp;E[\ell^n]\cong \ZZ_\ell\times\ZZ_\ell$.</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">Then we have an $\ell$-adic representation $\rho: G_\QQ\rightarrow Aut(T_\ell(E)) \cong GL_2(\ZZ_\ell)$.</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><strong><span style="font-size: x-large;">Motivation:</span></strong></p>
<p>&nbsp;</p>
<p><strong><span style="font-size: x-large;">Theorem (Serre)&nbsp;</span></strong><span style="font-size: x-large;">Let $\ell\geq 5$ and G be a closed subgroup of $SL_2(\ZZ_\ell)$ which surjects onto $SL_2(\mathbb{F}_\ell)$. Then $G=SL_2(\ZZ_\ell)$.</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;"><strong><span style="font-size: x-large;">Consequence</span></strong><span style="font-size: x-large;">: </span></span></p>
<p><span style="font-size: x-large;">Let $\ell\geq 5$ and $\rho_\ell: G_\QQ\rightarrow GL_2(\ZZ_\ell)$ be the $\ell$-adic representation of an elliptic curve $E/\QQ$. If $\rho_\ell$ surjects onto $GL_2(\ZZ/\ell\ZZ)$, then it surjects onto $GL_2(\ZZ/\ell^n\ZZ)$ for all $n$.</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p>&nbsp;</p>
<p><span style="font-size: medium;"><strong><span style="font-size: x-large;">Note</span><span style="font-weight: normal;"><span style="font-size: x-large;">: </span></span></strong></span></p>
<p>&nbsp;</p>
<p><span style="font-size: medium;"><strong><span style="font-weight: normal;"><span style="font-size: x-large;">The above is false for $\ell=2,3$. For the case of $\ell=3$, Elkies showed that there are infinitely many $E/\QQ$ with surjective mod $3$ representation but non-surjective mod $9$ representation.</span></span></strong></span></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p><strong><span style="font-size: xx-large;"><span style="font-size: x-large;">Problem</span></span></strong><span style="font-size: xx-large;"><span style="font-size: x-large;">: </span></span></p>
<p><span style="font-size: xx-large;"><span style="font-size: x-large;">&nbsp;</span><span style="font-size: x-large;"><span style="font-size: x-large;">Find examples of elliptic curves $E/\QQ$ such that:</span></span></span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: xx-large;"><span style="font-size: x-large;"><span style="font-size: x-large;">1. The mod 2 representation $\rho: G_\QQ\rightarrow Aut(E[2])\cong GL_2(\ZZ/2\ZZ)$ is surjective.</span></span></span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: xx-large;"><span style="font-size: x-large;"><span style="font-size: x-large;">2. The mod 4 representation&nbsp;</span></span></span><span style="font-size: x-large;">$\rho: G_\QQ\rightarrow Aut(E[4])\cong GL_2(\ZZ/4\ZZ)$ is not surjective.</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><strong><span style="font-size: xx-large;"><span style="font-size: x-large;">Equivalently</span></span></strong><span style="font-size: xx-large;"><span style="font-size: x-large;">:&nbsp;</span></span></p>
<p><span style="font-size: xx-large;"><span style="font-size: x-large;">&nbsp;</span><span style="font-size: x-large;"><span style="font-size: x-large;">Find examples of elliptic curves $E$ such that $Gal(\QQ(E[2])/\QQ)\cong GL_2(\ZZ/2\ZZ)$ but $Gal(\QQ(E[4])/\QQ)\not\cong GL_2(\ZZ/4\ZZ)$.</span></span></span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: xx-large;"><span style="font-size: x-large;"><span style="font-size: x-large;"><br /></span></span></span></p>
<p><span style="font-size: x-large;">Sage already knows when the mod 2 representation of $E$ is surjective, so the non-trivial part is the mod 4 representation.</span></p>

{{{id=1|
E=EllipticCurve([-1,1])
print E
rho=E.galois_representation()
print rho.is_surjective(2)
///
Elliptic Curve defined by y^2 = x^3 - x + 1 over Rational Field
True
}}}

{{{id=19|

///
}}}

<p><span style="font-size: x-large;"><strong>Approach 1</strong><span style="font-size: x-large;">: </span></span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">The order of $GL_2(\ZZ/4\ZZ)$ is $96$, so it suffices to find examples of elliptic curves $E$ such that $\#Gal(\QQ(E[4])/\QQ)&lt;96$. </span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">Let $f(x)$ generate the extension $\QQ(E[4])$ over $\QQ$. </span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">The Chebotarev density theorem tells us that the density of primes $p$ for which $f(x)$ splits completely in $\mathbb{F}_p[x]$ is $1/\#Gal(\QQ(E[4])/\QQ)$.</span></p>
<p><span style="font-size: large;"><br /></span></p>
<p><span style="font-size: x-large;">So we can guess the size of $Gal(\QQ(E[4])/\QQ)$ by factoring $f(x)$ in $\mathbb{F}_p[x]$ for many $p$.</span></p>
<p>&nbsp;</p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">Were we more ambitious, we would examine the factorization of $f(x)$ in various $\mathbb{F}_p[x]$ to determine the frequency of the various cycle types &nbsp;of $Gal(\QQ(E[4])/\QQ)$ and identify the group explicitly.</span></p>

{{{id=3|
for E in cremona_curves([120..220]):
    rho=E.galois_representation();
    if rho.is_surjective(2) and E.has_cm()==False:   
        g=E.torsion_polynomial(4);
        g1=E.torsion_polynomial(2);
        g=g/g1
        split=0
        for p in primes(5*10**3):
            R.<x>=GF(p)[]
            gg=R(g)
            if gg!=0: 
                if len(gg.factor())==gg.degree(): split+=1
        ratio=split/prime_pi(3*10**3)  
        if abs(RR(ratio))>.036: print E.label();
///
121a1
121a2
121c1
121c2
162a1
162a2
162d1
162d2
185a1
}}}

<p><span style="font-size: x-large;">We can check to make sure that these curves do, in fact, have surjective mod 4 representations.</span></p>

{{{id=5|
R.<x>=QQ[]
E=EllipticCurve('121a1')
g=E.torsion_polynomial(4);
g1=E.torsion_polynomial(2);
g=g/g1
g=R(g(x/2)*2^5)
F.<y>=NumberField(g)
L.<t>=F.galois_closure()
print 'The degree of the 4-torsion field is at most twice', L.degree()
///
The degree of the 4-torsion field is at most twice 24
}}}

{{{id=20|

///
}}}

<p><strong><span style="font-size: x-large;">Approach 2</span></strong><span style="font-size: x-large;">:</span></p>
<p>&nbsp;</p>
<p><span style="font-size: x-large;">Suppose that $Gal(\QQ(E[4])/\QQ)\cong GL_2(\ZZ/4\ZZ)$. </span></p>
<p>&nbsp;</p>
<p><span style="font-size: x-large;">It was proven by Holden that $b_E=x^4-4\Delta x-12a_4\Delta$ generates the unique $S_4$ subfield of $\QQ(E[4])$. </span></p>
<p>&nbsp;</p>
<p><span style="font-size: x-large;">So if we find an elliptic curve $E$ such that $b_E$ is reducible, then the mod 4 representation must be non-surjective.</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><strong><span style="font-size: x-large;">Remark</span></strong><span style="font-size: x-large;">: Although this approach will not find all examples with conductor $&lt;N$, it provides a computationally inexpensive proof for those curves that it does catch.</span></p>

{{{id=6|
R.<x>=QQ[]

for E in cremona_curves([10..700]):   
    rho=E.galois_representation();
    if rho.is_surjective(2) and E.a1()==0 and E.a2()==0 and E.a3()==0 and E.has_cm()==False:
        a=E.a4()
        delta=E.discriminant()
        b=x^4-4*delta*x-12*delta*a
                          
        if b.is_irreducible()==False:
            print E.label()
///
216a1
216d1
432g1
432h1
648a1
648c1
}}}

<p><strong><span style="font-size: x-large;">Note: </span></strong><span style="font-size: x-large;">Suppose that $E$ has a surjective mod 2 representation but non-surjective mod 4 representation. In this case we'd like to know what the image of the mod 4 representation is.&nbsp;</span></p>
<p><span style="font-size: x-large;">For $n=3$ (i.e. mod 3 but not mod 9) there is only one proper subgroup of $GL_2(\ZZ/9\ZZ)$ which surjects onto $GL_2(\ZZ/2\ZZ)$ under the reduction mod 2 homomorphism and whose determinant surjects onto $(\ZZ/9\ZZ)^*$ (cyclotomic determinant).&nbsp;</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">In our case, there are multiple subgroups which are potential images. It turns out that for each of these subgroups, there is an elliptic curve whose mod 4 image whose image is the subgroup.&nbsp;</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">A relatively fast way to see this is to study the distribution of the $a_p\pmod{4}$ associated to an elliptic curve and compare these numbers to the distributions of traces of elements in the subgroup of $GL_2(\ZZ/4\ZZ)$.&nbsp;</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">Gagan will (or already has) talk more about this.</span></p>

{{{id=11|

///
}}}

<p><strong><span style="font-size: x-large;">Parameterizing the examples.</span></strong></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">We work in analogy to Elkies' handling of the $\ell=3$ case.</span></p>
<p>&nbsp;</p>
<p><span style="font-size: x-large;">We'd like to take a subgroup $G$ of $SL_2(\ZZ/4\ZZ)$ which can be extended to a proper subgroup of $GL_2(\ZZ/4\ZZ)$ (which will be the image of our mod $4$ representations) and:</span></p>
<p><span style="font-size: x-large;">1. Show that the modular curve $\mathcal{X}_4=X(4)/(G/\{1,-1\})$ is defined over $\QQ$.</span></p>
<p>&nbsp;</p>
<p><span style="font-size: x-large;">2. Find a rational function $f$ realizing the cover $\mathcal{X}_4\rightarrow X(1)$.</span></p>
<p>&nbsp;</p>
<p><span style="font-size: large;"><br /></span></p>
<p><strong><span style="font-size: x-large;">Progress:</span></strong></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">We have Galois covers $X(4)\rightarrow \mathcal{X}_4 \rightarrow X(1)$.</span></p>
<p><span style="font-size: x-large;"><br /></span></p>
<p><span style="font-size: x-large;">The outer Galois group is $PSL_2(\ZZ/4\ZZ)$ and the cover $X(4)\rightarrow \mathcal{X}_4$ has Galois group $G/\{1,-1\}$.</span></p>
<p>&nbsp;</p>
<p><span style="font-size: x-large;">All of these curves are of genus $0$. </span></p>
<p>&nbsp;</p>
<p><span style="font-size: x-large;">The Hauptmodul of $X(1)$ is the $j$-invariant.</span></p>
<p>&nbsp;</p>

{{{id=9|
# Define the modular j invariant as a laurent series

R.<q>=LaurentSeriesRing(QQ,'q')
jq=j_invariant_qexp(30)
jq=jq.coefficients()
jq.remove(1)
j=q^(-1)+sum([q^(n)*jq[n] for n in range(30)])
j
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{q} + 744 + 196884q + 21493760q^{2} + 864299970q^{3} + 20245856256q^{4} + 333202640600q^{5} + 4252023300096q^{6} + 44656994071935q^{7} + 401490886656000q^{8} + 3176440229784420q^{9} + 22567393309593600q^{10} + 146211911499519294q^{11} + 874313719685775360q^{12} + 4872010111798142520q^{13} + 25497827389410525184q^{14} + 126142916465781843075q^{15} + 593121772421445058560q^{16} + 2662842413150775245160q^{17} + 11459912788444786513920q^{18} + 47438786801234168813250q^{19} + 189449976248893390028800q^{20} + 731811377318137519245696q^{21} + 2740630712513624654929920q^{22} + 9971041659937182693533820q^{23} + 35307453186561427099877376q^{24} + 121883284330422510433351500q^{25} + 410789960190307909157638144q^{26} + 1353563541518646878675077500q^{27} + 4365689224858876634610401280q^{28} + 13798375834642999925542288376q^{29}</span></html>
}}}

{{{id=10|

///
}}}

<p><span style="font-size: x-large;">The Hauptmodul of $X(4)$, denoted $j_4$, can be expressed as a quotient of Jacobi theta series.</span></p>

{{{id=16|
#Define the hauptmodul j4 of X(4)

R.<q>=LaurentSeriesRing(QQ,'q')
x=sum([q^(n^2) for n in range(-30,30)])
y=sum([(-1)^(n)*q^(n^2) for n in range(-30,30)])
x=R(x)
y=R(y)
j4=x/y
j4
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}1 + 4q + 8q^{2} + 16q^{3} + 32q^{4} + 56q^{5} + 96q^{6} + 160q^{7} + 256q^{8} + 404q^{9} + 624q^{10} + 944q^{11} + 1408q^{12} + 2072q^{13} + 3008q^{14} + 4320q^{15} + 6144q^{16} + 8648q^{17} + 12072q^{18} + 16720q^{19} + O(q^{20})</span></html>
}}}

<p><span style="font-size: x-large;">We can relate $j_4$ to $j$ via: $$j=\frac{1}{108}\cdot\frac{(j4^8+14*j4^4+1)^3}{(j4^5-j4)^4}$$</span></p>

{{{id=18|

///
}}}