August ℤ/5ℤ
@ModalMetamodel
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In logic, there are no morals.
L(ℝ)
Joined October 2024
Unterlogiker: ‘In logic, there are morals.’ Überlogiker: ‘Hail Gödelian immoralism!’
Arithmetisation of syntax is usually realised through Gödel-encoding with prime factorisation, but it’s more complicated within PA. Try to think of a way to express ‘the exponent of prime p in factorisation of n is k’ as a predicate with three variables (p,n,k) in PA’s language.
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When we ain’t busy chasin’ new hoes to fill up the roster, we chasin’ diagrams (go tell ya department’s professors that we be catchin’ more’n’em, yeh).
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ζ ∈ 𝒪_K is a root of unity, since u_N and its image have the same x-coordinates, and ζ = 1 can be assumed via replacement of α with ζ⁻¹α (note that we still have 𝔭 = (ζ⁻¹α)). Then α fixes u_N and hence all of A_N (which is to say that α ≡ 1 (mod N)). That’s all for now.
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Still, let’s assume that we can do this (it’s doable if we alter the analytic parametrisation A ≃ ℂ/𝔞). We have this commutative rectangle; if α ≡ 1 (mod N), σ_𝔓 fixes all of A_N and h(A_N), but if σ_𝔓 fixes h(A_N), it must map a K-generator u_N ∈ (K/𝔞)_N of A_N to ζu_N.
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Choose a prime 𝔓 of L_N above 𝔭 so that B = A^(L/K, 𝔓). Our previous isogeny λ : A → B corresponds to σ_𝔓 = (L/K, 𝔓) : A → A. Now I’m glossing over a technicality here, since a priori we only know that this holds after reduction mod 𝔓 up to an automorphism of A (mod 𝔓).
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My claim is class-field-theoretically tantamount to showing that a prime 𝔭 of K splits in L_N if and only if 𝔭 = (α) is principal and such an α ≡ 1 (mod N) can be found. 𝔭 = (α) may be assumed, since any prime that splits in L_N also splits in the Hilbert class field L.
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To be clear on the notation, A_N is the set of N-torsion points of A ≃ ℂ/𝔞 and h denotes the Weber h-function, which basically just returns the x-coordinate of a point (x, y) on the elliptic curve in short Weierstrass form.
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The full classification of K’s abelian extensions is a similar case. By class field theory, it suffices to describe the ray class field for an arbitrarily divisible modulus. Let N ∈ ℕ be an integer. I claim that the ray class field of modulus N is L_N = L(h(A_N)).
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We also find that a prime of K splits in L if and only if it’s principal: L is K’s Hilbert field class. We’re going to prove the inseparability of λ (mod 𝔓) right now, so pay close attention (unless you spedded out too hard right there).
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Particularly, L/K is abelian, and since any ideal class is representable by a degree-1 prime, we obtain a surjective morphism C_K → Gal(L/K), which must be an isomorphism since the degree of j(𝔞) is at most h_K. (L/K, 𝔭)’s description is actually valid for every 𝔭.
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Alright, we’re back with some more #VL5AlgebraicNT. So under the assumption of this claim, λ (mod 𝔓) must be isomorphic to the Frobenius since it has degree p, and hence we find j(A) ≡ j(B)ᵖ (mod 𝔓) by taking j-invariants, which exactly means that (L/K, 𝔓)j(𝔞) = j(𝔭⁻¹𝔞).
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Jus’ pull up to the Kenyattan category theory seminar real quick, boy. 🥷🏿🥷🏿
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There are more conceptual ways of proving that J isn’t trivial, but I hold the assumption that this can be accomplished simply by demonstrating a cusp form such that its L-function doesn’t vanish at 1 when ℓ is greater than 13 (I haven’t gone about doing that).
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This is indeed Hecke-stable, and furthermore it has rank 0 by the work of Gross-Zagier and Kolyvagin establishing part of the Birch-Swinnerton-Dyer conjecture in rank 1 (precisely the implication L(f, 1) ≠ 0 ⇒ |A_f(ℚ)| < ∞).
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Tensoring this with J₀, we end up getting a decomposition up to isogeny J₀ ~ ∏_f A_f, where A_f = J₀/(I_f J₀) is the abelian variety that corresponds to f. Then, by this decomposition, the winding quotient corresponds to J ~ ∏_{L(f, 1) ≠ 0} A_f.
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Recall that the Hecke algebra 𝕋 is diagonalisable as the product of K_f over all (Galois orbits of) normalised newforms of some level that’s a divisor of N, where K_f = 𝕋/I_f and I_f is the kernel of the character 𝕋 → ℂ corresponding to f (so that its image is K_f).
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Subsequent mathematical developments allowed for a more simple construction with the winding quotient though, which is (assuming the Birch-Swinnerton-Dyer conjecture) the largest possible quotient that satisfies the aforementioned conditions.
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All we have to do now is just construct a J that satisfies the three conditions mentioned in the very first slide. Historically, Mazur made use of the Eisenstein quotient Jᴱ = J₀/IᴱJ₀ (where Iᴱ is the ideal generated by Hecke operators Tₚ − p − 1).
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This next lemma right here implies that x = ∞ (recall that an unramified morphism’s diagonal is an open immersion, so the set of points at which x and ∞ coincide must be an open set, and they must be equal since it isn’t empty). At this point, we’re almost finished.
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We’ll suppose otherwise; up to replacing x with w(x), where w is X₀(ℓ)’s Atkin-Lehner involution (which swaps 0 and ∞), xₚ = ∞ may be assumed. Given this, it follows that we have two sections x and ∞ of the S-scheme X₀(ℓ) which have the same image by η : X₀(ℓ) → J.
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