Overview
This study investigates the murmuration density for families of characters over function fields. Specifically, it computes this density for quadratic characters and also for primitive cubic characters within a Kummer setting. The research establishes a relationship between the expectation of the murmuration density for quadratic characters and a correction term found in the traces of high powers of the Frobenius class within this family.
Research Context
The research is situated within the study of number theory, particularly concerning function fields and character theory. It involves concepts such as Frobenius classes, which are fundamental in the study of arithmetic over function fields, and $L$-functions, which are central objects in analytic number theory. The investigation of murmuration density for these character families contributes to understanding their statistical properties.
Approach
The study primarily involved computational methods to determine the murmuration density for the specified character families. For quadratic characters over function fields, the approach involved:
- Computing the murmuration density.
- Showing that the expectation of this density aligns with a correction term observed in the traces of high powers of the Frobenius class within the family.
- Determining large-genus asymptotics for this correction term.
- Describing this correction term in relation to the stable homology of the moduli space of hyperelliptic curves of genus $g$ with one marked Weierstrass point.
- Identifying this correction as a lower-order term within the one-level density.
- Obtaining a non-vanishing result at the central point for quadratic $L$-functions over function fields through an alternative method to the ratios conjecture.
For the thin family of primitive cubic characters in the Kummer setting, the approach involved computing their murmuration density.
Findings
The research yielded several key findings:
- The murmuration density for the family of quadratic characters over function fields was computed.
- The expectation of this murmuration density was shown to coincide with a correction term arising in the traces of high powers of the Frobenius class in this family.
- Large-genus asymptotics for this correction term were determined.
- This correction term was found to admit a description in terms of the stable homology of the moduli space of hyperelliptic curves of genus $g$ with one marked Weierstrass point.
- The correction was identified as a lower-order term in the one-level density.
- A non-vanishing result at the central point for quadratic $L$-functions over function fields was obtained via an alternative approach to using the ratios conjecture.
- The murmuration density for the thin family of primitive cubic characters in the Kummer setting was computed.