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Eungang Cho

Publications and source records attributed to Eungang Cho.

2 recordsLinked to original sources

arXiv2026

Petersson-Rigid Lattices in a Census of 100 Rank-Three Root Bases

The first paper worked out one family of four lattices in full -- $|\det| = 12, 24, 36$ and $72$. This paper generalizes it. We enumerate the 44 symmetrizable rank-three hyperbolic Cartan matrices and their 56 depth-one edits, 100 root bases in all, and compute the obstruction space $S_{5/2}(\rho_L)$ of the 98 within our weight-$5/2$ budget: 10 vacuous, 34 unobstructed, 54 obstructed. The vacuous ten are the lattices Bruinier, Ehlen and Freitag call simple, read in signature $(2,3)$. Of their fifteen, our ten realize five; of the rest, five need more than three generators and cannot be the discriminant form of a rank-three lattice at all, three fail $|\det L| = 2k^2$, and two are simply not root bases. Within the rank-three hyperbolic world the simple lattices are exactly the Feingold-Frenkel neighbours of index $k \le 4$. The discriminant group $L'/L$ carries a finite quadratic form, and the finite group of its isometries acts on the weight-$3/2$ cusp forms for $\rho_L$, the bottom antisymmetric rung. We survey the lattices on which that action is absolutely irreducible of dimension at least two, so that the Petersson pairing there is pinned down up to a single scalar. The condition alone cuts the census to three: $L_4$ of the first paper, and two new ones at $|\det| = 40$ and $88$. The quaternion discriminants that occur are 6, 10 and 22, the three for which the Shimura curve $X^D$ has genus zero. Rigidity uses no quaternion input, so we record it as an observation, not a characterisation. Both invariants of the first paper -- the shadow norm $\|\Xi\|^2$ and the Petersson scalar $t$ -- were single points there. Three rigid lattices instead of one are where their special faces come off: $\|\Xi\|^2$ generalizes against $L(f,1)$ where the first paper read $L(f,2)$, and $t$ against an elliptic curve's imaginary period where it read $\Gamma(1/3)$. Both numerically, to 26-58 digits.

math.NT↗

arXiv2026

Forced Shadows of an Obstructed Hyperbolic Kac-Moody Denominator

The four orders of the quaternion algebra B6 carry four reflective wall data on lattices of signature (3,2); three integrate to Borcherds denominators and one is obstructed. The failed denominator survives as a weakly harmonic Maass form, and we prove its shadow is a Hecke eigenform on the line of the newform 6.4.a.a, with zero twist component. The mechanism is invariance selection: the obstruction functional is invariant under the discriminant isometry group, whose invariants in S_{5/2} are one-dimensional; the same mechanism, verified at quaternion discriminants 10 and 22, places the shadows there on 10.4.a.a and 22.4.a.c. On the weight-1/2 layer we prove a determination theorem: the canonical form exists and is unique precisely when the obstruction space vanishes, and among the 71 discriminants below 230 this happens exactly for D in {6, 10, 22}, the genus-zero compact Shimura curves, whose maximal-order ternaries are reflective with integral Weyl chambers of ranks 3, 4, 4; completeness beyond that range is reduced to an estimate on a quadratic Dedekind-type sum, given a bound on the Gauss-sum term. Parity confines this layer to odd channels; on the obstructed orientation the section layer is obstructed outside an explicit 40-element locus of orientations (a double shadow: CM, 36.2.a.a, at weight 3/2 and newform at weight 5/2), while the deck-symmetric directions instead carry a unique canonical weight-1/2 form. The defect invariant satisfies ||Xi||^2 = 144 exactly and equals L(f,2)/48 pi^2 to 31 digits. On the section layer the Petersson geometry is rigid: the Gram matrix of S_{3/2}(rho_4) is a single transcendental multiple of an exact rational form, and that transcendental is identified, to 40 digits, as 3 Gamma(1/3)^3 / (2^{7/3} pi^2): the weight-3/2 shadow norms lie in the Chowla-Selberg ring; the weight-5/2 norm is numerically excluded from it.

math.NT↗