§ 4.1 Fermat curve

For a positive integer n, let Fn denote the Fermat curve xⁿ+ yⁿ = 1 of degree n. There are some properties of Fermat curves.

Lemma 4.1. For n ≥ 1, the genus of Fn is (n − 1)(n − 2)/2, and for n ≥ 3, a basis for the space of holomorphic 1-form is

ω_i,j= xⁱdx

y^j+2, 0 ≤ i ≤ j ≤ n − 3.

As shown in [10], we have following two lammas.

Lemma 4.2. The Fermat curve Fn is the modular curve associated to the group Γn generated by

where Γ(2)⁰ denotes the commutator subgroup of Γ(2).

Moreover, let

Lemma 4.3. If n 6= 1, 2, 4, 8, then Γn is a noncongruence subgroup.

Let ζ = e^2πi/n and µn be the group of nth root of unity. The group G = µn× µn acts on Fn by (ζⁱ, ζ^j) : (x, y) 7→ (ζⁱx, ζ^jy). Let

σ : (x, y) 7→ (ζx, y), τ : (x, y) 7→ (x, ζy).

Assume that H is a subgroup of G. We consider the quotient curve Fn/H. The pullbacks of holomorphic 1-forms on Fn/H will be holomorphic 1-forms on Fn

that are invariant under the action of H. Say, ωi,j = xⁱdx/y^j+2 is invariant under the action of H. Using the parameterization given in Lemma 4.2, we get a cusp form

f_i,j= xⁱqdx/dq y^j+2 =

∞

X

k=1

a_kq^k/2n

on Γn. On the other hand, we may consider the L-function L(s, Fn/H), i.e., the L-function of the Galois representation ρ_Fn/H of Gal( ¯Q/Q) attached to the algebraic curve Fn/H. (We assume for the moment that Fn/H is always defined over Q for all H and n).

§ 4.2 Case x

+ y

= 1

Noticing that λ = 16q^1/2+ · · · , we slightly modify the Fermat curve and consider the curve

xⁿ+ 16yⁿ= 1

instead(so that the cusp form f_i,j = xⁱqdx/y^j+2dq has rational Fourier coeffi-cient). We shall still let Fn denote this curve. Also we let

σ : (x, y) 7→ (ζx, y), τ : (x, y) 7→ (x, ζy),

where ζ = e^2πi/n. Note that a differential form xⁱdx/y^j+2 is fixed by σ^aτ^b if and only if

(i + 1)a − (j + 2)b ≡ 0 mod 6.

The following table lists the subgroup H_i,j of G = µ₆ × µ6 that fixes ω_i,j. ω0,0 ω0,1 ω1,1 ω0,2 ω1,2 ω2,2 ω0,3 ω1,3 ω2,3 ω3,3

hσ²τ i hσ³τ i hσ³τ²i hσ⁴τ i hσ²τ, σ³i hσ²τ³i hσ⁵τ i hσ⁵τ²i hστ³i hστ²i We now work out the equations for the curves F6/Hi,j.

Lemma 4.4. We have

group differential forms equation hσ²τ i ω_0,0, ω_1,2 v²= u⁶+ 1 hσ³τ i ω0,1 v²= u³− 1 hσ⁴τ i ω0,2 v²= u³+ 4 hσ⁵τ i ω_0,3, ω_1,2 v²= u⁶− 1 hστ²i ω_1,2, ω_3,3 v²= u⁶+ 1 hσ³τ²i ω1,1 v²= u³+ 1 hσ⁵τ²i ω1,3 v²= u³+ 16 hσ²τ³i ω_2,2 v²= u³+ 4

hστ³i ω2,3 v²= u³− 16 hσ²τ, σ³i ω1,2 v²= u³+ 1

Proof. Here we prove the case hσ²τ i. Consider both of xy⁴ and y⁶ are fixed by hσ²τ i, and the mapping (x, y) 7→ (xy⁴, y⁶) is 6-to-1. Thus, xy⁴and y⁶generate the subfield of the function field of F6 that is fixed by hσ²τ i and an equation for F₆/hσ²τ i is given by the relation

U⁶= V⁴− 16V⁵

between U = xy⁴ and V = y⁶. Now the curve U⁶ = V⁴− 16V⁵ is birationally equivalent to v²= u⁶+ 1 with the birational maps

u = 2V

U , v = V²(8V − 1)

U³ , U = u²

4(u³− v), V = u³ 8(u³− v) This proves the case hσ²τ i.

Remark 4.5. We can compute the genus of F₆/H using the Riemann-Hurwitz formula. Taking H = hσ²τ i for example. For the affine part of F6, the covering F6→ F6/H is unramified at those points of F6where Pj(ζ^2jx, ζ^jy), j = 0, . . . , 5 are 6 distinct points. If y 6= 0, then the six points are distinct. At those points, the covering is unramified. On the other hand, if y = 0, then P0= P3, P1= P4, P2 = P5. The covering is ramified at those points with ramification index 2.

There are totally 6 such points (ζ^k, 0), k = 0, . . . , 5. Thus, the contribution from the affine part to the total branch number is 6. The infinity part of F6

consist of 6 points Q_j= (ζ^j+1/2: 1 : 0) We have

σ²τ (Q_j) = (ζ^j+5/2: ζ : 0) ∼ (ζ^j+3/2: 1 : 0) = Q_j+1

Therefore, the covering is unramified at the 6 infinity points, and the total branch number is 6. By the Riemann-Hurwitz formula, if g is the genus of F6/hσ²τ i, then

10 − 1 = 6(g − 1) +6 2.

Hence, we conclude that the genus of F6/hσ²τ i is 2 and the subspace of dif-ferential 1-forms on F6 that are invariant under hσ²τ i should have dimension 2.

Theorem 4.6. The genus of F_n/H for a cyclic subgroup H = hσ^aτ^bi of µ_n×µ_n with a, b are relative primes is

g = n − da− db− d_(a−b)

2 + 1

where dx is the greatest common divisor of x and n.

Proof. By the Riemann-Hurwitz formula, we only need to verify the total branch number B is n(da+ db+ d(a−b)− 3).

For the affine part of Fn, the covering Fn → Fn/H is unramified at those point of F_n where P_j = (ζ^ajx, ζ^bjy), j = 0, . . . , n − 1 are n distinct points. If x 6= 0 and y 6= 0, since a, b are relative primes, we know the n points are distinct.

At those points, the covering is unramified. On the other hand, if x = 0, then P₀ = P_n/db = . . . = P_(db−1)n/db, P₁ = P_n/db+1 = . . . = P_{(db−1)n/db+1}, . . ., P_n/db−1 = P_{n/db+n/db−1} = . . . = P_{(db−1)n/db+n/db−1}. The covering is ramified at those points with ramification index db. There are totally n such points.

Similarly, we can determine the case y = 0. Thus, the contribution from the affine part to the total branch number is n(da− 1) + n(db− 1). The infinity part of Fn consist of n points Qj = (ζ^j+1/2: 1 : 0) We have

σ^aτ^b(Qj) = (ζ^j+a+1/2: ζ^b: 0) ∼ (ζj+(a−b)+1/2: 1 : 0) = Q_j+(a−b) Replaces a − b by a − b mod n if necessary. Therefore, the ramification index of the covering is d_(a−b), and the total branch number of the infinity part is n(da−b − 1). Sum up the total branch numbers of the affine part and the infinity part, we have B = n(da+ db+ d_(a−b)− 3).

Lemma 4.7. The L-functions for the curves in Lemma 4.4 are equation L-function

v²= u³+ 16 L(s, f27) v²= u³+ 1 L(s, f36) v²= u³+ 4 L(s, f108) v²= u³− 1 L(s, f₃₆⊗ χ−4) v²= u³− 16 L(s, f27⊗ χ−4) v²= u⁶+ 1 L(s, f36)²

v²= u⁶− 1 L(s, f₃₆)L(s, f₃₆⊗ χ₋₄) Here

f27(τ ) = η(3τ )²η(9τ )², f36(τ ) = η(6τ )⁴

Remark 4.8. The modular forms f27, f36, f108 have the following description in terms of Hecke characters.

Let K = Q(√

−3) and ζ = e^2πi/6. The ring of integers OK is Z + Zζ. Let m = 3 and define χ as follows. If a + bζ ∈ O_K is not relatively prime to 3, we let χ(a + bζ) = 0. For each a + bζ in O_K relatively prime to m, there exists a unique integer j with 0 ≤ j < 6 such that a + bζ ≡ ζ^j mod m. We set χ(a + bζ) = ζ^−j(a + bζ). That is,

(a, b) mod 3 (0, 1) (0, 2) (1, 0) (1, 2) (2, 0) (2, 1)

χ(a + bζ)/(a + bζ) ζ⁵ ζ² 1 ζ −1 ζ⁴

Then

f₂₇(τ ) = 1 6

X

a+bζ∈O_K

χ(a + bζ)q^a2+ab+b2.

For f₃₆, we let m = 2√

−3 and define χ as follows. If a + bζ ∈ O_K is not relatively prime to m, we set χ(a + bζ) = 0. For each a + bζ in O_K that is relatively prime to 2√

−3, there exists a unique integer j with 0 ≤ j < 6 such that a + bζ ≡ ζ^j mod m. We set χ(a + bζ) = ζ^−j(a + bζ) Then

f36(τ ) = 1 6

X

a+bζ∈O_K

χ(a + bζ)q^a2+ab+b2.

Proof. The only parts that requires a proof are v² = u⁶+ 1 and v² = u⁶− 1.

Here we consider the case v² = u⁶− 1. Let x = u² and y = v. Then we have v² = u³− 1. In other words, we have a two-fold cover from v² = u⁶− 1 to y²= x³− 1. Likewise, let x = −1/u²and y = v/u³. We have y²= x³+ 1. Then L(s, v²− u⁶+ 1) = L(s, f36)L(s, f36⊗ χ−4).

Theorem 4.9. The cusp forms f_i,j= xⁱy^−j−2qdx/dq satisfy the ASD congru-ences with the following L-function.

fi,j L-function

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