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Drinfeld Modular Forms Mirror Classical Hecke Eigenvalues and q-Expansion Arithmetic — E8 Intelligence Research

  • Andrew Stewart Caldin1
  1. 1Independent Researcher, United Kingdom
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Abstract

FINDING: Drinfeld modular forms connect Hecke eigenvalues to q-expansion coefficients over function fields, mirroring classical modular form arithmetic. | MATH: Classical: Tn f = λn f, f(q)=∑ an qn, amn=am an for (m,n)=1, apr+1=ap apr−pk−1apr−1. Drinfeld analogue: Hecke operators Tp act on Drinfeld cusp forms; eigenvalues λp satisfy ap = λp for good primes, with q-expansion over 𝔽q(T). Armana's work: bounds/relations for coefficients an in Drinfeld setting, including an = λp an/p − |p|k−1 an/p2 for p | n. | CONNECTION: The Hecke algebra is a commutative ring generated by operators indexed by ideals — a lattice/root-system-like structure (the divisor lattice of 𝔽q[T]). Eigenvalues are algebraic integers, and the constant qk−1 (where \(

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Cite

BibTeXRIS
BibTeX
@misc{caldin2026drinfeld,
  title = {{Drinfeld Modular Forms Mirror Classical Hecke Eigenvalues and q-Expansion Arithmetic — E8 Intelligence Research}},
  author = {Andrew Stewart Caldin},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23658203},
  url = {https://zenodo.org/records/23658203},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Drinfeld Modular Forms Mirror Classical Hecke Eigenvalues and q-Expansion Arithmetic — E8 Intelligence Research
AU  - Andrew Stewart Caldin
PY  - 2026
DA  - 2026/10/26
PB  - Zenodo
DO  - 10.5281/zenodo.23658203
UR  - https://zenodo.org/records/23658203
AB  - FINDING: Drinfeld modular forms connect Hecke eigenvalues to q-expansion coefficients over function fields, mirroring classical modular form arithmetic. | MATH: Classical: T_n f = λ_n f, f(q)=∑ a_n q^n, a_mn=a_m a_n for (m,n)=1, a_p^r+1=a_p a_p^r−p^k−1a_p^r−1. Drinfeld analogue: Hecke operators T_p act on Drinfeld cusp forms; eigenvalues λ_p satisfy a_p = λ_p for good primes, with q-expansion over 𝔽_q(T). Armana's work: bounds/relations for coefficients a_n in Drinfeld setting, including a_n = λ_p a_n/p − |p|^k−1 a_n/p^2 for p | n. | CONNECTION: The Hecke algebra is a commutative ring generated by operators indexed by ideals — a lattice/root-system-like structure (the divisor lattice of 𝔽_q[T]). Eigenvalues are algebraic integers, and the constant q^k−1 (where \(  Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER  -