CoCalc -- 2022-05.ipynb

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ubuntu2004

Kernel: SageMath 9.5

In [0]:

g.base_extend(GF(3))

Dear Professor Ribet:

I am a senior student and I am reading your notes "Lectures on Serre's Conjectures". Now I meet some problem while trying to confirming the properties of the representation mentioned in section 2.4.2. You mentioned the elliptic curve $y^2+y=x^3+x^2-12x+2$ and the mod 3 representation it induces, and by Serre's conjecture it arises from an eigenform of level 47 and weight 4. I have checked the $a_p(E)$ of the elliptic curve on LMFDB: https://www.lmfdb.org/EllipticCurve/Q/141/a/1.

We hope the coefficients are congruent mod 3 at all primes other than 3 and 47, since they should give the same mod 3 representation. However, just check the $q^2$ coefficients, the elliptic curve is 1 mod 3 and the eigenform is 0 or 2 mod 3 (since the minimal polynomial of a decomposes to $a(a+1)^2$ mod 3), which are unequal.

At first I was afraid of that there is a typo for the expression and the polynomial for g and a (in the note), so I found the eigenform on LMFDB: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/47/4/a/a/, and I again found that the $q^2$ should have coefficient congruent to 0 or 2. I also checked the Fourier coefficients of another newform in $S_4(\Gamma_0(47))$ , https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/47/4/a/b/, and it couldn't solve the problem either.

So what is wrong here?

In [8]:

E = EllipticCurve([0,1,1,-12,2]); show(E)

Out[8]:

\renewcommand{\Bold}[1]{\mathbf{#1}}y^2 + y = x^{3} + x^{2} - 12 x + 2

In [15]:

show(factor(E.conductor()))

Out[15]:

\renewcommand{\Bold}[1]{\mathbf{#1}}3 \cdot 47

In [9]:

show(E.aplist(20))

Out[9]:

\renewcommand{\Bold}[1]{\mathbf{#1}}\left[-2, 1, -3, -3, -5, 2, -6, -6\right]

In [10]:

rho = E.galois_representation(); rho

Out[10]:

Compatible family of Galois representations associated to the Elliptic Curve defined by y^2 + y = x^3 + x^2 - 12*x + 2 over Rational Field

In [12]:

# all mod p reps are surjective:
rho.non_surjective()

Out[12]:

[]

In [18]:

M = ModularSymbols(Gamma0(47),4)
f2 = M.hecke_operator(2).charpoly('x')
show(factor(f2))

Out[18]:

\renewcommand{\Bold}[1]{\mathbf{#1}}(x - 9)^{2} \cdot (x^{3} + 5 x^{2} - 2 x - 12)^{2} \cdot (x^{8} - 3 x^{7} - 50 x^{6} + 124 x^{5} + 844 x^{4} - 1549 x^{3} - 5393 x^{2} + 5418 x + 10316)^{2}

In [19]:

F2 = factor(f2)
F2[1][0](-2)

Out[19]:

4

In [20]:

F2[2][0](-2)

Out[20]:

-2724

In [21]:

F2[2][0](-2) % 3

Out[21]:

0

The above confirms what Ken wrote in the email, namely that the degree 8 eigenform has a mod-3 reduction that is consistent with it giving rise to the Galois representation mod 3 attached to $E$ .

Your mistake I think is you say "I found the eigenform in LMFDB". However, there are two entries with trivial character in the LMFDB in this space. Here's the other one: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/47/4/a/b/

In [22]:

# It's also possible to use Sage to directly list the mod-3 newforms associated to a space of cusp forms...

In [30]:

S = CuspForms(Gamma0(47),4); show(S)

Out[30]:

\renewcommand{\Bold}[1]{\mathbf{#1}}\verb|Cuspidal|\verb| |\verb|subspace|\verb| |\verb|of|\verb| |\verb|dimension|\verb| |\verb|11|\verb| |\verb|of|\verb| |\verb|Modular|\verb| |\verb|Forms|\verb| |\verb|space|\verb| |\verb|of|\verb| |\verb|dimension|\verb| |\verb|13|\verb| |\verb|for|\verb| |\verb|Congruence|\verb| |\verb|Subgroup|\verb| |\verb|Gamma0(47)|\verb| |\verb|of|\verb| |\verb|weight|\verb| |\verb|4|\verb| |\verb|over|\verb| |\verb|Rational|\verb| |\verb|Field|

In [31]:

N = S.newforms('a'); N

Out[31]:

[q + a0*q^2 + (-1/2*a0^2 - 5/2*a0 - 1)*q^3 + (a0^2 - 8)*q^4 + (a0^2 + a0 - 10)*q^5 + O(q^6),
 q + a1*q^2 + (2575/589584*a1^7 - 2899/147396*a1^6 - 36941/294792*a1^5 + 185647/294792*a1^4 + 170587/294792*a1^3 - 1116547/196528*a1^2 + 1034621/294792*a1 + 2162681/147396)*q^3 + (a1^2 - 8)*q^4 + (-7159/1179168*a1^7 - 3875/294792*a1^6 + 162005/589584*a1^5 + 298865/589584*a1^4 - 1925443/589584*a1^3 - 2464661/393056*a1^2 + 4843039/589584*a1 + 8400595/294792)*q^5 + O(q^6)]

In [32]:

g = N[1]; g

Out[32]:

q + a1*q^2 + (2575/589584*a1^7 - 2899/147396*a1^6 - 36941/294792*a1^5 + 185647/294792*a1^4 + 170587/294792*a1^3 - 1116547/196528*a1^2 + 1034621/294792*a1 + 2162681/147396)*q^3 + (a1^2 - 8)*q^4 + (-7159/1179168*a1^7 - 3875/294792*a1^6 + 162005/589584*a1^5 + 298865/589584*a1^4 - 1925443/589584*a1^3 - 2464661/393056*a1^2 + 4843039/589584*a1 + 8400595/294792)*q^5 + O(q^6)

In [34]:

MS = g.modular_symbols(); MS

Out[34]:

Modular Symbols subspace of dimension 16 of Modular Symbols space of dimension 24 for Gamma_0(47) of weight 4 with sign 0 over Rational Field

In [41]:

M3 = ModularSymbols(Gamma0(47), 4, base_ring=GF(3), sign=1); M3

Out[41]:

Modular Symbols space of dimension 13 for Gamma_0(47) of weight 4 with sign 1 over Finite Field of size 3

In [42]:

D3 = M3.decomposition()

In [43]:

D3

Out[43]:

[
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 13 for Gamma_0(47) of weight 4 with sign 1 over Finite Field of size 3,
Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 13 for Gamma_0(47) of weight 4 with sign 1 over Finite Field of size 3,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 13 for Gamma_0(47) of weight 4 with sign 1 over Finite Field of size 3,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 13 for Gamma_0(47) of weight 4 with sign 1 over Finite Field of size 3,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 13 for Gamma_0(47) of weight 4 with sign 1 over Finite Field of size 3,
Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 13 for Gamma_0(47) of weight 4 with sign 1 over Finite Field of size 3,
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 13 for Gamma_0(47) of weight 4 with sign 1 over Finite Field of size 3
]

In [44]:

for D in D3:
    print(D.hecke_matrix(2).charpoly('x')(1))

Out[44]:

In [49]:

for p in primes(100):
    print(p, D3[1].hecke_matrix(p), E.ap(p) % 3)

Out[49]:

[1] 1
[1] 1
[0] 0
[0] 0
[1] 1
[2] 2
[0] 0
[0] 0
[0] 0
[1] 1
[1] 1
[1] 1
[0] 0
[2] 2
[1] 1
[0] 0
[0] 0
[1] 1
[2] 2
[1] 1
[1] 1
[0] 0
[2] 2
[1] 1
[1] 1

In [0]:

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