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Product of Eisenstein series with multiplicative power series

Boyuan Xiong

Abstract

We say a power series $a_0+a_1q+a_2q^2+\cdots$ is \emph{multiplicative} if $n\mapsto a_n/a_1$ for positive integers $n$ is a multiplicative function. Given the Eisenstein series $E_{2k}(q)$, we consider formal multiplicative power series $g(q)$ such that the product $E_{2k}(q)g(q)$ is also multiplicative. For fixed $k$, this requirement leads to an infinite system of polynomial equations in the coefficients of $g(q)$. The initial coefficients can be analyzed using elimination theory. Using the theory of modular forms, we prove that each solution for the initial coefficients of $g(q)$ leads to one and only one solution for the whole power series, which is always a quasimodular form. In this way, we determine all solutions of the system for $k \le 20$. For general $k$, we can regard the system of polynomial equations as living over a symbolic ring. Although this system is beyond the reach of computer algebra packages, we can use a specialization argument to prove it is generically inconsistent. This is delicate because resultants commute with specialization only when the leading coefficients do not specialize to $0$. Using a Newton polygon argument, we are able to compute the relevant degrees and justify the claim that for $k$ sufficiently large, there are no solutions. These results support the conjecture that $E_{2k}(q)g(q)$ can be multiplicative only for $k = 2, 3, 4, 5, 7$.

Product of Eisenstein series with multiplicative power series

Abstract

We say a power series is \emph{multiplicative} if for positive integers is a multiplicative function. Given the Eisenstein series , we consider formal multiplicative power series such that the product is also multiplicative. For fixed , this requirement leads to an infinite system of polynomial equations in the coefficients of . The initial coefficients can be analyzed using elimination theory. Using the theory of modular forms, we prove that each solution for the initial coefficients of leads to one and only one solution for the whole power series, which is always a quasimodular form. In this way, we determine all solutions of the system for . For general , we can regard the system of polynomial equations as living over a symbolic ring. Although this system is beyond the reach of computer algebra packages, we can use a specialization argument to prove it is generically inconsistent. This is delicate because resultants commute with specialization only when the leading coefficients do not specialize to . Using a Newton polygon argument, we are able to compute the relevant degrees and justify the claim that for sufficiently large, there are no solutions. These results support the conjecture that can be multiplicative only for .

Paper Structure

This paper contains 4 sections, 7 theorems, 111 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Solutions for small values of k
  3. Extreme monomials and Newton polygons
  4. Solutions for k large

Key Result

Lemma 2.1

Suppose $l\in\mathbb{P}$ and $n\notin\mathbb{P}$. If $\tfrac{n}{2}<l<n$, then $b_l$ appears linearly in $E_{k,n}$ with In particular, if $p\in\mathbb{P}$ and $p+1\notin\mathbb{P}$, then $b_p$ appears linearly in $E_{k,p+1}$ with $\operatorname{coef}_{b_p}(E_{k,p+1})=1$.

Figures (1)

  • Figure 1: The support of $f(x,y)$ and its Newton polygon, with lattice points labeled by coordinates.

Theorems & Definitions (14)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm1']}
  • Theorem 2.4
  • Definition 3.1: Extreme monomials of a multivariate polynomial
  • Definition 3.2: Newton polygon / polytope
  • Definition 3.3: Minkowski sum
  • ...and 4 more