Behavior of partition values modulo powers of primes

<p> Let <italic>n</italic> be a positive integer. A partition of <italic>n</italic> is a sequence of non-increasing positive integers whose sum is <italic>n</italic>. The partition function, <italic>p(n)</italic>, counts the number of partitions of <italic>n</italic>. Although <italic>p(n)</italic> is easy to define, questions on the arithmetic of its values lie much deeper. Ninety years ago, Ramanujan proved the celebrated congruences for <italic>p(n)</italic> which bear his name. For example, he proved, for all <italic>n</italic>, that <italic>p(5n + 4) == 0 mod 5</italic>. Work of Ono and others over the last ten years reveals the extent to which <italic>p(n)</italic> satisfies congruence phenomena of this type. The unifying framework that explains such properties of <italic>p(n)</italic> comes from the fact that its values arise naturally as theFourier coefficients of a modular form. Using the theory of modular forms, we study the behavior of <italic>p(n)</italic> and related partition statistics and relate partition values to other objects of interest in the theory of automorphic forms, the central critical values of <italic>L</italic>-functions. </p> <p>Recent work of Folsom, Kent, and Ono reveals, for primes <italic>l > 3</italic>, that <italic>p(n)</italic> is <italic>l</italic>-adically fractal. They show that the generating functions for partition values on particular arithmetic progressions of <italic>n</italic> are eventually self-similar when viewed reduced modulo powers of <italic>l</italic>, in effect generalizing Ramanujan's congruences for every power of every prime <italic>l &ge 5</italic>. Boylan and the author extend this work by closely examining the structure of modules of modular forms associated to these generating functions. A sharp refinement of the "zoom rate", the rate at which the generating functions become self-similar, is proven. The generating functions are also shown to be periodic. New examples of congruences for all primes <italic>13 &le l < 1300</italic> are given in the appendix. </p> <p>For <italic>l</italic> in <italic>{5,7,11}</italic>, we prove congruences modulo <italic>l</italic> between ratios of partition values and ratios of central critical values of <italic>L</italic>-functions associated to certain modular forms. The proof uses deep theorems of Shimura and Waldspurger which give a connection between the Fourier coefficients of a half-integer weight newforms and the central critical values of <italic>L</italic>-functions attached to twists of an integer weight Hecke eigenform. The result can be used to show that these central critical values are non-zero if <italic>p(n)</italic> is not divisible by <italic>l<super>2</super></italic> for a specific <italic>n</italic>. </p> <p> Let <italic>k</italic> be a positive integer. We say that a partition of <italic>n</italic> is <italic>k</italic>-regular if none of its parts is divisible by <italic>k</italic>. Let <italic>b₁₃(n)</italic> denote the number of 13-regular partitions of <italic>n</italic>. Calkin, et al. conjectured that for all integers <italic>n</italic> and <italic>t</italic> with <italic>&ge 0</italic> and <italic>t &ge 2</italic>, we have<italic>b₁₃( 3<super>t</super> n + (5* 3<super>t -1</super> - 1)/2) == 0 mod 3 .</italic> We confirm this conjecture by relating the generating function for <italic>b₁₃(3n+1)</italic> to a modular form and then studying its image under certain operators. </p> <p> We say that a partition of <italic>n</italic> is <italic>t</italic>-core if none of the hook lengths in the Ferrers-Young diagram are divisible by <italic>t</italic>. We prove a wealth of new congruences for <italic>2<super>t</super></italic>-core partitions modulo 2 by examining the nilpotency of Hecke operators on level 1 cusp forms. Based on extensive calculations we give a conjecture for the structure of <italic>T_p</italic>-cyclic subspaces for these forms.</p>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Campus Access Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2011

Author dc:creator

• Webb, John J. B.

Contributors dc:contributor

• Matthew G Boylan

• Mathematics
• Physical Sciences and Mathematics
• modular forms
• Partition function
• Ramanujan congruences

Statement dc:rights

• © 2011, John J. B. Webb

Repository record dc:identifier

https://scholarcommons.sc.edu/etd/1618

OAI identifier oai:identifier

oai:scholarcommons.sc.edu:etd-2619