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Dimensions of paramodular forms and compact twist modular forms with involutions

Tomoyoshi Ibukiyama

Abstract

We give an explicit dimension formula for paramodular forms of degree two of prime level with plus or minus sign of the Atkin--Lehner involution of weight $\det^k\operatorname{Sym}(j)$ with $k\geq 3$, as well as a dimension formula for algebraic modular forms of any weight associated with the binary quaternion hermitian maximal lattices in non-principal genus of prime discriminant with fixed sign of the involution. These two formulas are essentially equivalent by a recent result of N. Dummigan, A. Pacetti. G. Rama and G. Tornaría on correspondence between algebraic modular forms and paramodular forms with signs. So we give the formula by calculating the latter. When $p$ is odd, our formula for the latter is based on a class number formula of some quinary lattices by T. Asai and its interpretation to the type number of quaternion hermitian forms given in our previous works. On paramodular forms, we also give a dimensional bias between plus and minus eigenspaces, some list of palindromic Hilbert series, numerical examples for small $p$ and $k$, and the complete list of primes $p$ such that there is no paramodular cusp form of level $p$ of weight 3 with plus sign. This last result has geometric meaning on moduli of Kummer surface with $(1,p)$ polarization.

Dimensions of paramodular forms and compact twist modular forms with involutions

Abstract

We give an explicit dimension formula for paramodular forms of degree two of prime level with plus or minus sign of the Atkin--Lehner involution of weight with , as well as a dimension formula for algebraic modular forms of any weight associated with the binary quaternion hermitian maximal lattices in non-principal genus of prime discriminant with fixed sign of the involution. These two formulas are essentially equivalent by a recent result of N. Dummigan, A. Pacetti. G. Rama and G. Tornaría on correspondence between algebraic modular forms and paramodular forms with signs. So we give the formula by calculating the latter. When is odd, our formula for the latter is based on a class number formula of some quinary lattices by T. Asai and its interpretation to the type number of quaternion hermitian forms given in our previous works. On paramodular forms, we also give a dimensional bias between plus and minus eigenspaces, some list of palindromic Hilbert series, numerical examples for small and , and the complete list of primes such that there is no paramodular cusp form of level of weight 3 with plus sign. This last result has geometric meaning on moduli of Kummer surface with polarization.
Paper Structure (12 sections, 15 theorems, 229 equations, 11 tables)

This paper contains 12 sections, 15 theorems, 229 equations, 11 tables.

Table of Contents

  1. Introduction
  2. Main Theorems
  3. Proof of Theorem \ref{['paramodular']}
  4. Proof of Theorem \ref{['compactdim']}
  5. The case $p=2$ and $3$
  6. Appendix on dimensions and characters
  7. Bias of the dimensions of plus and minus
  8. Numerical Examples
  9. The case $p=5$ and $p=7$
  10. Small primes $p$ and palindromic Hilbert series
  11. The case of $j>0$
  12. The case of small $k$

Key Result

Theorem 2.1

We assume that $n=2$. Then an explicit formula for $Tr(R_{f_1,f_2}(\pi))$ is given for any $f_1\geq f_2\geq 0$ with $f_1\equiv f_2 \bmod 2$ as follows. For $p\equiv 1 \bmod 4$, we have For $p\equiv 3 \bmod 4$ and $p>3$, we have For $p=2$, we have For $p=3$, we have Here we have $\chi_i=Tr(\rho_{f_1,f_2}(g_i))$ for any $g_i \in G^1$ whose principal polynomials are given by $\phi_i(\pm x)$ for $

Theorems & Definitions (22)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • Lemma 4.3: ibuquinary Lemma 4.1 and Corollary 4.4
  • Lemma 4.4
  • Lemma 4.5
  • proof
  • proof : Proof of Theorem \ref{['compactdim']}
  • ...and 12 more