发表机构
Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México (UNAM)（墨西哥国立自治大学数学研究所库埃纳瓦卡分部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从等边三角剖分曲面的剥皮几何出发，研究椭圆曲线的模性问题，构造了零次超奇异 Brandt 模中的非零类以显式化缺失的互反律，并获得了关于 Frobenius 判别式的新初等结果。
AI 中文摘要
本文从装饰等边三角剖分的剥皮的初等几何角度研究椭圆曲线的模性（modularity）问题。剥皮是一个有限的展开域，连同重建三角剖分曲面的边界识别。在亏格为一的情形下，它恢复了万有覆盖及其秩二覆盖格。对于固定的素数 $\ell$，有限非分歧覆盖 $[\ell^n]:E\to E$ 构成一个相容的塔，其覆盖群为 $E[\ell^n]$；它们的逆向极限是 Tate 模 $T_\ell(E)$，在 $\Q$ 上相容的 Galois 作用给出通常的 $\ell$-adic 表示。相比之下，循环同源给出指数为 $q$ 的格邻域，它们构成 Hecke 与 Brandt 对应的局部模型。对于素数导子 $p$ 的半稳定椭圆曲线 $E/\Q$，缺失的互反律是在不使用模性的情况下，构造零次超奇异 Brandt 模中的一个非零类 $c_E$，使得在每个好素数处 $B_qc_E=a_q(E)c_E$。我将这一障碍明确化，并研究附属于 Frobenius 判别式 $D_q=a_q(E)^2-4q$ 的有向 CM 包。新的初等结果表明：当且仅当剩余 Frobenius 多项式不可约时，$D_q$ 模 $p$ 是非平方；任何包含这样一个元素的剩余像给出一个正密度的素数集合，对于这些素数，$p$ 在由 $D_q$ 确定的二次域中惰性；并且相应的判别式必然无界。
英文摘要
This paper studies the modularity problem for elliptic curves from the elementary geometry of peels of decorated equilateral triangulations. A peel is a finite developing domain together with the boundary identifications that reconstruct the triangulated surface. In genus one it recovers the universal cover and its rank-two deck lattice. For a fixed prime $\ell$, the finite unramified coverings $[\ell^n]:E\to E$ form a compatible tower whose deck groups are $E[\ell^n]$; their inverse limit is the Tate module $T_\ell(E)$, and over $\Q$ the compatible Galois action gives the usual $\ell$-adic representation. Cyclic isogenies, by contrast, give the index-$q$ lattice neighbors that form the local model for Hecke and Brandt correspondences. For a semistable elliptic curve $E/\Q$ of prime conductor $p$, the missing reciprocity is the construction, without using modularity, of a nonzero class $c_E$ in the degree-zero supersingular Brandt module such that $B_qc_E=a_q(E)c_E$ at every good prime. I make this obstruction explicit and study oriented CM packets attached to the Frobenius discriminants $D_q=a_q(E)^2-4q$. New elementary results show that $D_q$ is a nonsquare modulo $p$ exactly when the residual Frobenius polynomial is irreducible, that any residual image containing such an element gives a positive-density set of primes for which $p$ is inert in the quadratic field determined by $D_q$, and that the corresponding discriminants are necessarily unbounded.
