论文标题
关于$ q $ series和模块化表格的系数
On the coefficients of $q$-series and modular forms
论文作者
论文摘要
在此博士学位论文（弗吉尼亚大学，2022年）中，我们证明了有关分区理论生成函数系数和整数重量模块化形式的系数的结果。使用各种形式的圆方法，我们证明了有关残基类别中的分区统计信息模型$ t $的结果。例如，我们证明分区部分分为不同部分的部分是均衡的模型$ t $（但某些偏见仍然存在），并且分区中的$ t $ hooks的数量通常不是等均等的modulo prime。我们还使用模块化转换定律获得了$ t $ hook计数函数的精确公式。我们还采用了圆形方法来证明Coll，Mayers和Mayers的猜想，即$ Q $ series $（q，-q^3; q^4）_ \ infty^{ - 1} $具有非阴性系数。这些主题涵盖了第3-6章。第7章使用格里芬，ono，rolen和Zagier的标准，将分区渐近学用于证明詹森多项式的双曲线。第8章提供了一种求解$ a_f（n）=α$的方程式的新方法，其中$α\ in \ mathbb {z} $是奇数，而$ a_f（n）$是具有Trivial mod 2 Galois表示的标准化Hecke eigenform的系数。该方法基于Bilu，Hanrot和Voutier的原始质原理以及有效代数几何形状的方法，用于椭圆形曲线，虚拟曲线和THUE方程。
In this Ph.D dissertation (University of Virginia, 2022), we prove results about the coefficients of partition-theoretic generating functions and of coefficients of integer weight modular forms. Using various forms of the circle method, we prove results about the distribution of partition statistics in residue classes modulo $t$. For example, we prove that the parts of partitions into distinct parts are equidistributed modulo $t$ (but that certain biases occur nonetheless) and that the number of $t$-hooks in a partition is generally not equidistributed modulo primes. We also obtain exact formulas for the $t$-hook counting functions using modular transformation laws. We also employ the circle method to prove a conjecture of Coll, Mayers and Mayers that the $q$-series $(q, -q^3; q^4)_\infty^{-1}$ has non-negative coefficients. These topics cover Chapters 3-6. Chapter 7 gives an application of partition asymptotics for proving hyperbolicity of Jensen polynomials using the criterion of Griffin, Ono, Rolen and Zagier. Chapter 8 gives a new method for solving equations of the form $a_f(n) = α$, where $α\in \mathbb{Z}$ is odd and $a_f(n)$ are the coefficients of a normalized Hecke eigenform with trivial mod 2 Galois representation. The method is based on the primitive prime divisor theorem of Bilu, Hanrot, and Voutier along with methods in effective algebraic geometry for elliptic curves, hyperelliptic curves, and Thue equations.
