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# Marking, normalization and character data for T665
Write C_N: X^N+Y^N=Z^N, with x=X/Z, y=Y/Z, and zeta_N=exp(2*pi*i/N).
The plus-plus-minus projective equation is fixed even when N is even.
Let alpha(x,y)=(zeta_N*x,y), beta(x,y)=(x,zeta_N*y). The path
δ(t)=(t^(1/N),(1-t)^(1/N)) uses nonnegative roots and increasing t from
0 to 1. As an oriented chain,
κ = δ - beta_*δ + (alpha beta)_*δ - alpha_*δ.
The four segments concatenate in that order into a loop at (0,1).
The endpoint contributions cancel exactly: alpha fixes (0,1) and beta
fixes (1,0). Thus κ is an integral cycle, with no division by N in its
homology class. The differential is
ω_N^(a,b) = x^(a-1) y^(b-N) dx, a,b>=1, a+b<N.
This is Otsubo 2010's unnormalized ω_0, rather than his beta-normalized ω.
It is Otsubo 2014's ω_N. Substitution along δ gives
δ*ω = (1/N) t^(a/N-1) (1-t)^(b/N-1) dt.
Pullback by alpha^r beta^s multiplies ω by zeta_N^(ar+bs). Consequently
P_N(a,b) = ((1-zeta_N^a)(1-zeta_N^b)/N) B(a/N,b/N).
This establishes the sign, phase and factor 1/N directly, including N=3.
Exchanging a,b leaves the scalar value unchanged, not the basis vector.
For k>=1 the pullback under (x,y)->(x^k,y^k) from C_(kN) to C_N sends
ω_N^(a,b) to k ω_(kN)^(ka,kb); its pushforward sends κ_(kN) to κ_N.
Thus P_(kN)(ka,kb)=P_N(a,b)/k.
The trigonometric factorization is
P_N(a,b) = -4 exp(pi*i*(a+b)/N) sin(pi*a/N) sin(pi*b/N)
B(a/N,b/N)/N.
The imaginary part is strictly negative on this holomorphic domain. The
real part vanishes exactly when 2(a+b)=N, so that component is stored as
exact zero rather than a rounded tiny value. There is no modulus or
absolute-value convention in the table. For the quartic, the beta duplication
formula gives B(1/4,1/4)=4 K(m=1/2), hence P_4(1,1)=-2i K(1/2).
Here K uses the parameter m, not the modulus sqrt(m). T25's live parameter
identity is `1/2`. The equality is also verified against an AGM computation.
## Cycle orbit and homology relations
The N^2 cycles κ_(r,s)=(alpha^r beta^s)_*κ, 0<=r,s<N, generate H_1(C_N,Z)
as a Z-module (Otsubo 2010 Section 4.1; Otsubo 2014 Section 5.1).
They form an overcomplete orbit, not an integral symplectic basis.
The rank is 2g_N=(N-1)(N-2). In homology they satisfy periodicity in each
index and the relations
sum_(j=0)^(N-1) κ_(r+j,s) = 0,
sum_(j=0)^(N-1) κ_(r,s+j) = 0,
sum_(j=0)^(N-1) κ_(r+j,s+j) = 0.
These are stated relations, not a claim of a complete integral presentation.
For each character (a,b) in full H^1, a,b,a+b are nonzero modulo N.
Pairing any displayed relation with that eigenform gives a vanishing
geometric sum. Since those eigenforms span H^1 and H_1 has no torsion,
the integral relations follow. checks.py verifies the exponent sums exactly
for all full-H^1 characters in the covered degrees, not just the
holomorphic half. Complex conjugation sends κ to (alpha beta)_*^(-1)κ.
The translated periods are zeta_N^(ar+bs) P_N(a,b). No period matrix,
intersection matrix or determinant is tabulated. The Jacobian is understood
with its canonical principal polarization; no polarization-dependent scalar
is computed here.
## Character and Hodge metadata
character-data.json records every entry's degree, ordered pair, genus, gcd
h=gcd(N,a,b), conductor index d=N/h, reduced pair (a/h,b/h), and orbit under
units modulo d. The cyclotomic embedding is zeta_d=exp(2*pi*i/d). For every
unit u modulo d, the embedding sends zeta_d to zeta_d^u. Its holomorphic
membership is exactly (u*a/h mod d)+(u*b/h mod d)<d, where residues are in
1,...,d-1. Both the embedding labels and the orbit pairs are retained.
This is character/Hodge data and does not assert that each isotypic factor
is simple or that the whole Jacobian has one CM field. Rational projectors
can define submotives without selecting an integral symplectic basis.
Otsubo 2014 Sections 2.1–2.2 supplies this decomposition. Arithmetic Jacobi
sums in T143 are a different realization with their own positive-sum
convention; none is asserted to equal a complex period.
## Independent numerical check
For x,y in (0,1], split Euler's integral at 1/2. On the first half expand
(1-t)^(y-1) by the binomial theorem. Its integrated positive terms are
u_k = (1-y)_k / k! * 2^(-x-k)/(x+k),
u_(k+1)/u_k = (k+1-y)/(k+1) * (x+k)/(x+k+1) / 2 <= 1/2.
After the first 450 terms, the omitted sum is bounded by 2*u_450.
The second half exchanges x and y. checks.py evaluates both sums in
500-bit ball arithmetic and adds these rigorous tail bounds. No beta or
gamma evaluation enters that calculation. Trigonometric real and imaginary
parts supply the independent phase factor. Controls return B(1,1)=1,
B(1/2,1/2)=pi, and K(0)=pi/2 before any period is compared. A deliberate
10^-90 perturbation must fail on every row. Both overlapping enclosures
must be finite with small relative radii; the check does not accept NaNs or
vacuously wide balls. Readback checks contain the full independent enclosure
inside the exact interval claimed by each stored decimal component.