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

2025/06/25 by K. Lakshmanan, Lakshmanan, K.
Computer Science · Mathematics · Social Sciences · #11G05 #11G40 #Advanced Mathematical Modeling in Engineering #Analytic Number Theory Research #European Linguistics and Anthropology #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.20175

openalex publication_date 2025/06/25 · openalex created_date 2025/10/09 · openalex updated_date 2026/08/01

Abstract

We investigate the role of the conductor in analytic rank bounds for elliptic curves over \(ℚ\). Let \(E/ℚ\) 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 \(Φ(E)\), and yield rank bounds of the form rank(E)≪ log Φ(E). Using the Modularity Theorem, we show that any such invariant must satisfy Φ(E)≥ 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.

Citations

• On the rank of elliptic curves

Related

• A heuristic for boundedness of ranks of elliptic curves
• Conditionally bounding analytic ranks of elliptic curves
• Elliptic curves of large rank and small conductor
• Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zero
• Growth of the analytic rank of modular elliptic curves over quintic extensions