The algorithmic conductor of the Frey–Hellegouarch curve

Let O be a Dedekind domain with fraction field F, and let A and B be coprime elements of O with A ≡ -1 (mod 4) and 32 ∣ B, such that the Frey curve y² = x (x - A) (x + B) is elliptic over F. The curve is then semistable, with minimal discriminant ideal generated by d = A² B² (A + B)² / 2⁸ (TauCeti.isSemistable_freyCurve, TauCeti.minimalDiscriminantIdeal_freyCurve), so its algorithmic conductor is the radical of (d). Because 32 ∣ B, every prime dividing 2 still divides d = (A (B / 16) (A + B))², so (d) has the same prime factors as (A B (A + B)). Hence the algorithmic conductor is the radical of (A B (A + B)): the product of the primes dividing A, B or A + B, each to the first power.

For a solution aᵖ + bᵖ + cᵖ = 0 of Fermat's equation with p ≥ 5, normalised so that b is even and aᵖ ≡ -1 (mod 4), this says that the Frey curve of (aᵖ, bᵖ) has algorithmic conductor (rad (a b c)) over ℤ. Together with the minimal discriminant (a b c)^(2 p) / 2⁸, this is the input the Frey curve gives to Szpiro's conjecture.

The hypothesis 32 ∣ B cannot be weakened to the 16 ∣ B that semistability needs: over ℤ, if B = 16 β with β odd, then d is odd, so the curve has good reduction at 2 and its conductor omits 2. For Fermat's equation, 32 ∣ bᵖ holds as soon as b is even and p ≥ 5.

When 16 ∤ A B (A + B) the Frey curve over ℤ need not be semistable at 2. Its reduction at every odd prime is still good or multiplicative, and the exponent of 2 in its algorithmic conductor is at most the valuation of its minimal discriminant 16 (A B (A + B))², which is less than 12. So the algorithmic conductor divides 2¹¹ rad (A B (A + B)). This is the case that Szpiro's conjecture needs besides the semistable one, to bound every coprime triple a + b = c.

Main results

• TauCeti.algorithmicConductor_freyCurve: over a Dedekind domain, if A and B are coprime, A ≡ -1 (mod 4) and 32 ∣ B, the Frey curve has algorithmic conductor radical (A B (A + B)).
• TauCeti.algorithmicConductor_freyCurve_pow: for a normalised solution of aᵖ + bᵖ + cᵖ = 0 with p ≥ 5, the Frey curve of (aᵖ, bᵖ) has algorithmic conductor (rad (a b c)) over ℤ.
• TauCeti.algorithmicConductor_freyCurve_dvd: over ℤ, if A and B are coprime and 16 ∤ A B (A + B), the algorithmic conductor of the Frey curve divides 2¹¹ rad (A B (A + B)).

References

• G. Frey, Links between stable elliptic curves and certain Diophantine equations, Ann. Univ. Sarav. Ser. Math. 1 (1986), 1–40.
• H. Darmon, F. Diamond, R. Taylor, Fermat's Last Theorem, in Current Developments in Mathematics 1995, International Press (1995), 1–154.

theorem TauCeti.algorithmicConductor_freyCurve {O : Type u_1} [CommRing O] [IsDedekindDomain O] [∀ (v : IsDedekindDomain.HeightOneSpectrum O), PerfectField (IsLocalRing.ResidueField (Localization.AtPrime v.asIdeal))] {F : Type u_2} [Field F] [Algebra O F] [IsFractionRing O F] {A B : O} (hA : 4 ∣ A + 1) (hB : 32 ∣ B) (hAB : IsCoprime A B) [((freyCurve A B).baseChange F).IsElliptic] : WeierstrassCurve.algorithmicConductor O ((freyCurve A B).baseChange F) = UniqueFactorizationMonoid.radical (Ideal.span {A * B * (A + B)})

The algorithmic conductor of the Frey curve is the radical of (A B (A + B)) when it is elliptic over the fraction field F, A and B are coprime, A ≡ -1 (mod 4) and 32 ∣ B. The curve is then semistable, and its primes of bad reduction are exactly the primes dividing A, B or A + B.

theorem TauCeti.algorithmicConductor_freyCurve_pow {p : ℕ} (hp : 5 ≤ p) {a b c : ℤ} (hab : IsCoprime a b) (h : a ^ p + b ^ p + c ^ p = 0) (hb : 2 ∣ b) (ha : a ^ p ≡ -1 [ZMOD 4]) [((freyCurve (a ^ p) (b ^ p)).baseChange ℚ).IsElliptic] : WeierstrassCurve.algorithmicConductor ℤ ((freyCurve (a ^ p) (b ^ p)).baseChange ℚ) = Ideal.span {UniqueFactorizationMonoid.radical (a * b * c)}

The algorithmic conductor of the Frey curve of a Fermat triple is (rad (a b c)). If aᵖ + bᵖ + cᵖ = 0 with a and b coprime, b even, aᵖ ≡ -1 (mod 4) and p ≥ 5, and y² = x (x - aᵖ) (x + bᵖ) is elliptic over ℚ, its algorithmic conductor over ℤ is generated by the radical of a b c.

The Frey curve over ℤ when 16 ∤ A B (A + B)

theorem TauCeti.algorithmicConductor_freyCurve_dvd {A B : ℤ} (hAB : IsCoprime A B) (h16 : ¬16 ∣ A * B * (A + B)) [((freyCurve A B).baseChange ℚ).IsElliptic] : WeierstrassCurve.algorithmicConductor ℤ ((freyCurve A B).baseChange ℚ) ∣ Ideal.span {2 ^ 11 * UniqueFactorizationMonoid.radical (A * B * (A + B))}

The algorithmic conductor of the Frey curve divides 2¹¹ rad (A B (A + B)) when 16 ∤ A B (A + B), for coprime A and B. The curve need not be semistable at 2, but at an odd prime its reduction is good or multiplicative, with exponent at most 1, and bad only at the primes dividing A B (A + B). At 2 the exponent is at most the valuation of the minimal discriminant 16 (A B (A + B))², which is less than 12.