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Counting primes, groups, and manifolds.

Dorian Goldfeld1, Alexander Lubotzky, Nikolay Nikolov

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Proceedings of the National Academy of Sciences of the United States of America
|September 9, 2004
PubMed
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The study analyzes the number of congruence subgroups for the modular group SL(2,Z), providing a precise asymptotic estimate. This research also extends to subgroup growth in higher-rank semisimple Lie groups, revealing surprising independence from specific lattice structures.

Area of Science:

  • Number Theory
  • Group Theory
  • Combinatorics

Background:

  • Understanding the structure and growth of subgroups within mathematical groups is fundamental.
  • The modular group SL(2,Z) and its congruence subgroups are key objects in number theory and geometry.
  • Estimating the number of subgroups of a given index is a challenging problem with implications for group growth theory.

Purpose of the Study:

  • To determine the precise asymptotic behavior of the number of congruence subgroups of the modular group SL(2,Z).
  • To investigate and establish general results on subgroup growth for lattices in higher-rank semisimple Lie groups.
  • To explore the relationship between subgroup growth and the algebraic properties of Lie groups and their lattices.

Main Methods:

  • Utilizing the Bombieri-Vinogradov theorem (Riemann hypothesis on average).

Related Experiment Videos

  • Solving a novel extremal problem in combinatorial number theory.
  • Applying techniques from algebraic groups, finite group theory, and advanced number theory.

    • Main Results:

    • Established a sharp asymptotic formula for the number of congruence subgroups of SL(2,Z) of index n.
    • Demonstrated that subgroup growth in irreducible lattices of higher-rank semisimple Lie groups is independent of the specific lattice.
    • Showed that this growth depends only on the Lie type of the group's direct factors and can be derived from its root system.

    Conclusions:

    • The subgroup growth of lattices in higher-rank semisimple Lie groups exhibits a universal behavior determined by the group's algebraic structure.
    • The findings provide deep insights into the quantitative aspects of group theory and its connections to number theory and geometry.
    • The results offer a framework for calculating subgroup growth based on fundamental properties like root systems, with implications for both conditional and unconditional cases (Generalized Riemann Hypothesis).