arXiv subjects

Peter Chocian

Publications and source records attributed to Peter Chocian.

4 recordsLinked to original sources

arXiv2026

The Primitive Generalized Fermat Equation x^3+y^5=z^7: A computer-assisted proof

We prove that the generalized Fermat equation $x^3+y^5=z^7$ has no solution in nonzero coprime integers, and we make explicit how the proof extends the prior work cited below. Dahmen-Siksek had established the signed local descent and eliminated the three cases in which the associated degree-seven algebra is reducible. Putz had then proved that the remaining irreducible case can involve only seven septic fields: six pure fields and one exceptional field. The missing step was to exclude those seven fields. For the six pure fields we construct an explicit Fano resolvent, descend to a smooth plane quartic of genus three over $\mathbb{Q}(\sqrt{-7})$, and prove that its rational-parameter locus consists only of five points above the branch values. For the exceptional field we combine the modularity and conductor results of Pacetti-Villagra Torcomian with level lowering over $\mathbb{Q}(\sqrt{5})$ and a finite Hecke comparison at $29$. We also reconstruct the three reducible-sector arguments of Dahmen-Siksek as independently replayable calculations, including their database-free identification of the quadratic field, and add an intrinsic formal-group treatment at the ramified prime. The paper labels every major step as a literature input, reconstructed input, or new argument. All project-specific finite calculations are supplied as exact certificates, programs, inputs, and authenticated logs.

math.NT↗

arXiv2026

Twisted Bernoulli Zeros in Quasi-Linear Time: Distribution, Depth, and Explicit Hilbert Class Components

The divided generalized Bernoulli values $b_{χ,j} = fB_{1,χω^{-j}} \bmod p$, for $χ$ an odd primitive Dirichlet character of conductor $f$ and order $d$ with $d \mid p-1$, control (for $p \nmid φ(f)$) the odd isotypic components of the $p$-class group of $Q(ζ_{fp})$ through the characterwise abelian Main Conjecture; a zero is a twisted irregular pair, a branch with positive Iwasawa lambda-invariant, in the tradition studied by Ernvall, Holden, Delbourgo-Knospe and Knospe. We survey these zeros in the regime complementary to existing tabulations: fixed small conductor and large $p$ ($f = 3, 5$ to $p < 10^5$; all odd primitive characters of conductor at most 20 to $p < 2 \cdot 10^4$), computing each spectrum by a residue-class weight formula and one Bluestein convolution over $F_p$, a direct finite-field alternative of the same quasi-linear order as the standard power-series method. The survey records 27,508 zero lines over 55,121 character-prime pairs, each verified by two independent code paths with an exact order of vanishing; the counts and digits are consistent with the random model, and eight lines are non-simple, including one of depth three at $(f,p,j) = (19,37,16)$, giving class components of order exactly $p^2$ and $37^3$. The main contribution converts zeros into explicit certified generators: the conductor-three catalogue of arXiv:2607.23177 is extended from $p < 500$ to $p < 10^5$, all 2,441 zero lines simple, each projected circular unit proven to generate its complete order-$p$ Hilbert class field component by a fresh split-prime Artin certificate -- the largest in the degree-199,980 field $Q(ζ_{299973})$. Ancillary files contain all tables, certificates, and a verification program.

math.NT↗

arXiv2026

Character Fourier Spectra of Circular Units and Twisted Bernoulli Class Components

Let $χ$ be a primitive odd Dirichlet character of conductor $f$. For the universal projector polynomials $P_m$ introduced in arXiv:2607.23177, defined by $\sum_m P_m(X)Y^m = -\log(1-X(1-e^{-Y}))$, we evaluate the character Fourier spectrum at $h_t = ζ_f^t/(ζ_f^t-1)$: for every odd $m$, $\sum_t \barχ(t)P_m(h_t) = τ(\barχ)B_{m,χ}/(m\,m!)$. After reduction at any prime above $p \nmid f$, the case $m = p-j$ identifies this spectrum, including its exact nonzero scalar, with the divided generalized Bernoulli value attached to $χω^{-j}$; the spectral-zero and Bernoulli-zero criteria therefore agree over every residue field, with no splitting hypothesis on the coefficient field. We connect the identity with the local Kummer spectrum of the circular unit $1-ζ_fζ_p$ and carry the programme through in the first non-real case: for the two primitive quartic characters modulo 5 and primes $p < 500$, $p \equiv 1 \pmod{20}$, exactly eleven zero lines occur. On each line an integral character projection of $1-ζ_5ζ_p$ is everywhere locally unramified, a finite split-prime Artin computation proves it is not a global $p$-th power, and the character-wise Main Conjecture shows the radical generates the complete order-$p$ Hilbert class component. Six of the eleven components occur at classically regular primes. At $p = 61$ the two conjugate characters contribute on different indices. A deterministic integer-arithmetic program (ancillary file) verifies the enumeration, the divided digits, and every certificate.

math.GM↗

arXiv2026

Explicit Twisted Hilbert Class Components Beyond Classical Irregularity

Let $p \equiv 1 \pmod 6$ be prime and $K_p = \mathbf{Q}(ζ_{3p})$. We study the reflected circular unit $μ_p = (1+zζ_p)/(1+\bar zζ_p)$, $z = -ζ_3^2$, and its character projections. A universal Stirling polynomial $P_m$ gives an exact identity between the anti-spectrum of $μ_p$ and the primitive divided $χ_{-3}$-twisted Stickelberger spectrum: $P_{p-j}(h) - P_{p-j}(1-h) = -(2h-1)(j-1)!\,b_j$, $h = z/(1+z)$. Thus the locally blind lines of the reflected unit are precisely the zeros of the corresponding divided twisted Bernoulli eigenvalues. For every $p < 500$ we enumerate these zeros. Exactly twelve character lines occur. On each line an explicit integral idempotent product of $μ_p$ is a local $p$-th power at the conductor primes but not a global $p$-th power. Small completely split primes provide finite Artin certificates. The generalized Bernoulli number has exact $p$-adic valuation one in every case; the character-wise Main Conjecture therefore proves that each radical generates the complete Hilbert-class-field component, which has order $p$. Seven of the twelve lines occur at classically regular primes, so twisted degeneracy below 500 is more often invisible to ordinary irregularity than aligned with it. The first case, $p = 67$, is worked out in full, and a deterministic integer-arithmetic program (included as an ancillary file) reproduces the enumeration and every certificate.

math.NT↗