从图的L-函数中恢复拉普拉斯格

中文摘要 AI 辅助

我们引入与有限图的雅可比行列式特征相关的L-函数，作为代数曲线非分歧覆盖产生的L-函数的图论类似物。这些L-函数利用图上的黎曼-罗赫结构定义，扩展了洛伦齐尼（Lorenzini）的双变量zeta函数。我们证明，若两个无桥图具有同构的雅可比行列式，且其L-函数在特征的诱导对应下一致，则它们的拉普拉斯格重合。还证明洛伦齐尼zeta函数在桥收缩下不变，解释了主定理中无桥假设的必要性。最后给出例子表明，仅雅可比行列式或洛伦齐尼zeta函数均无法确定拉普拉斯格。

英文摘要

We introduce $L$-functions associated with characters of the Jacobian of a finite graph, as a graph-theoretic analogue of the $L$-functions arising from unramified coverings of algebraic curves. These $L$-functions are defined using the Riemann--Roch structure on the graph and extend Lorenzini's two-variable zeta function. We show that if two graphs without bridges have isomorphic Jacobians and their $L$-functions agree under the induced correspondence of characters, then their Laplacian lattices coincide. We also show that Lorenzini's zeta function is invariant under contraction of bridges, explaining the necessity of the bridge-free hypothesis in the main theorem. Finally, we give examples showing that neither the Jacobian nor the Lorenzini zeta function alone determine the Laplacian lattice.

发表机构

• University of Ottawa（渥太华大学）

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