Background:

Let $G_\QQ=Gal(\QQbar/\QQ), \ell$ be a prime and $E/\QQ$ be an elliptic curve without complex multiplication.

There is a natural action of $G_\QQ$ on $E[n]\cong (\ZZ/n\ZZ)^2$.

So we have a mod $n$ representation $\rho: G_\QQ\rightarrow Aut(E[n])\cong GL_2(\ZZ/n\ZZ)$.

The action of $G_\QQ$ commutes with the multiplication by $\ell$ map, so we get an action of $G_\QQ$ on $T_\ell(E)=\varprojlim E[\ell^n]\cong \ZZ_\ell\times\ZZ_\ell$.

Then we have an $\ell$-adic representation $\rho: G_\QQ\rightarrow Aut(T_\ell(E)) \cong GL_2(\ZZ_\ell)$.

Motivation:

Theorem (Serre) Let $\ell\geq 5$ and G be a closed subgroup of $SL_2(\ZZ_\ell)$ which surjects onto $SL_2(\mathbb{F}_\ell)$. Then $G=SL_2(\ZZ_\ell)$.

Consequence:

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 $n$.

Note:

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.

Problem:

Find examples of elliptic curves $E/\QQ$ such that:

1. The mod 2 representation $\rho: G_\QQ\rightarrow Aut(E[2])\cong GL_2(\ZZ/2\ZZ)$ is surjective.

2. The mod 4 representation $\rho: G_\QQ\rightarrow Aut(E[4])\cong GL_2(\ZZ/4\ZZ)$ is not surjective.

Equivalently:

Find examples of elliptic curves $E$ such that $Gal(\QQ(E[2])/\QQ)\cong GL_2(\ZZ/2\ZZ)$ but $Gal(\QQ(E[4])/\QQ)\not\cong GL_2(\ZZ/4\ZZ)$.

Sage already knows when the mod 2 representation of $E$ is surjective, so the non-trivial part is the mod 4 representation.

{{{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| /// }}}

Approach 1:

The order of $GL_2(\ZZ/4\ZZ)$ is $96$, so it suffices to find examples of elliptic curves $E$ such that $\#Gal(\QQ(E[4])/\QQ)<96$.

Let $f(x)$ generate the extension $\QQ(E[4])$ over $\QQ$.

The Chebotarev density theorem tells us that the density of primes $p$ for which $f(x)$ splits completely in $\mathbb{F}_p[x]$ is $1/\#Gal(\QQ(E[4])/\QQ)$.

So we can guess the size of $Gal(\QQ(E[4])/\QQ)$ by factoring $f(x)$ in $\mathbb{F}_p[x]$ for many $p$.

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 of $Gal(\QQ(E[4])/\QQ)$ and identify the group explicitly.

{{{id=3| # Search through Cremona's database to find elliptic curves whose 4 torsion field has Galois group of order less than 96 by factoring # the minimal polynomial of a primitive element in F_p[x] for E in cremona_curves([120..130]): 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.=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 }}}

We can check to make sure that these curves do, in fact, have surjective mod 4 representations.

{{{id=5| # Double check the curves listed above by explicitly computing the degree of the 4 torsion field R.=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.=NumberField(g) L.=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| /// }}}

Approach 2:

Suppose that $Gal(\QQ(E[4])/\QQ)\cong GL_2(\ZZ/4\ZZ)$.

It was proven by Holden that $b_E=x^4-4\Delta x-12a_4\Delta$ generates the unique $S_4$ subfield of $\QQ(E[4])$.

So if we find an elliptic curve $E$ such that $b_E$ is reducible, then the mod 4 representation must be non-surjective.

Remark: Although this approach will not find all examples with conductor $<N$, it provides a computationally inexpensive proof for those curves that it does catch.

{{{id=6| # Search through Cremona's database and look for curves for which b_E is reducible. R.=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 }}} {{{id=21| /// }}}

Note: 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.

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).

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.

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 $GL_2(\ZZ/4\ZZ)$.

Gagan will (or already has) talk more about this.

{{{id=11| /// }}}

Parameterizing the examples.

We work in analogy to Elkies' handling of the $\ell=3$ case.

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:

1. Show that the modular curve $\mathcal{X}_4=X(4)/(G/\{1,-1\})$ is defined over $\QQ$.

2. Find a rational function $f$ realizing the cover $\mathcal{X}_4\rightarrow X(1)$.

Progress:

We have Galois covers $X(4)\rightarrow \mathcal{X}_4 \rightarrow X(1)$.

The outer Galois group is $PSL_2(\ZZ/4\ZZ)$ and the cover $X(4)\rightarrow \mathcal{X}_4$ has Galois group $G/\{1,-1\}$.

All of these curves are of genus $0$.

The Hauptmodul of $X(1)$ is the $j$-invariant.

{{{id=9| # Define the modular j invariant as a laurent series R.=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 /// \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} }}} {{{id=10| /// }}}

The Hauptmodul of $X(4)$, denoted $j_4$, can be expressed as a quotient of Jacobi theta series.

{{{id=16| #Define the hauptmodul j4 of X(4) R.=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 /// \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}) }}}

We can relate $j_4$ to $j$ via: $$j=\frac{1}{108}\cdot\frac{(j_4^8+14*j_4^4+1)^3}{(j_4^5-j_4)^4}$$

{{{id=18| /// }}}