On the Minimality of the Conductor for Elliptic Curve $L$-Functions

Mathematics > Number Theory [Submitted on 25 Jun 2025 (v1), last revised 16 Jun 2026 (this version, v2)] Title:On the Minimality of the Conductor for Elliptic Curve $L$-Functions View PDF HTML (experimental)Abstract:We investigate the role of the conductor in analytic rank bounds for elliptic curves over \(\mathbb{Q}\). Let \(E/\mathbb{Q}\) be an elliptic curve with conductor \(N_E\). We consider hypothetical degree-two \(L\)-functions associated to (E) that satisfy analytic continuation, a functional equation involving an arithmetic invariant \(\Phi(E)\), and yield rank bounds of the form \[ \operatorname{rank}(E)\ll \log \Phi(E). \] Using the Modularity Theorem, we show that any such invariant must satisfy \[ \Phi(E)\ge N_E. \] Thus the conductor is minimal among arithmetic invariants that can appear in this analytic framework. In particular, the standard logarithmic rank bounds arising from the conductor cannot be improved by replacing \(N_E\) with a strictly smaller invariant while preserving the same degree-two functional equation structure. These results provide a structural explanation for the distinguished role of the conductor in analytic approaches to the rank problem. Submission history From: K Lakshmanan [view email][v1] Wed, 25 Jun 2025 07:03:00 UTC (7 KB) [v2] Tue, 16 Jun 2026 07:48:55 UTC (7 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.