Number fields and function fields: coalescences, contrasts and emerging applications

Area of Science:

• Number Theory
• Algebraic Geometry
• Arithmetic Geometry

Background:

• Analogies between prime numbers and irreducible polynomials over finite fields were noted by Gauss.
• Historically, distinct languages and techniques in analytic number theory and function field theory hindered interdisciplinary progress.
• Despite challenges, interaction between these fields has been ongoing for decades.

Purpose of the Study:

• To introduce recent developments at the intersection of number theory and function field arithmetic.
• To highlight how combining ideas from both fields stimulates progress on long-standing problems.
• To showcase advancements facilitated by bridging analytic number theory and function field settings.

Main Methods:

• Reviewing and synthesizing recent research at the interface of number theory and function fields.
• Explaining novel techniques that integrate methodologies from both analytic number theory and finite field arithmetic.
• Presenting case studies of problems solved by cross-pollinating ideas between number fields and function fields.

Main Results:

• Significant progress has been made on problems by integrating techniques from number theory and function field theory.
• The previously observed analogies between primes and irreducible polynomials are being further explored with new tools.
• Interchanges between analytic number theory and function field settings are becoming more fruitful.

Conclusions:

• The convergence of analytic number theory and function field arithmetic is a dynamic and productive area of research.
• Bringing together distinct mathematical languages and techniques is key to future breakthroughs.
• This collection of papers reflects a new era of synergy and advancement in arithmetic geometry.