What question did this study set out to answer?

To establish mathematical connections between E8 lattice theta functions, modular forms, interpolated Apéry sequences, and critical central L-values within the Langlands framework.

August 16, 20260 citationsOpen Access

Modular Forms, E8 Lattice Theta Functions, and Interpolated Critical L-Values — E8 Intelligence Research

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ACAndrew Stewart Caldin

Key Points

To establish mathematical connections between E8 lattice theta functions, modular forms, interpolated Apéry sequences, and critical central L-values within the Langlands framework.
Formulated the E8 lattice theta function as a modular form of weight 4 for the modular group SL(2, Z) using divisor sums.
Evaluated interpolated Apéry numbers using hypergeometric series to investigate central critical L-values.
Examined Ramanujan's weight 12 cusp form and Eichler–Shimura congruences to relate Fourier coefficients to Galois representations.
Represented the E8 lattice theta function as a weight 4 modular form with Fourier expansion governed by the sum of cubes of divisors.
Connected the generating series of interpolated Apéry sequences to generalized hypergeometric functions representing critical L-values at central points.
Demonstrated that Fourier coefficients of the Ramanujan tau function satisfy congruence relations with Galois representations modulo primes.

Abstract

FINDING: Modular forms count points on E8 lattice via theta functions, linking number theory, complex analysis, and Langlands program; new interpolated sequences connect to critical L-values. MATH: - E8 lattice theta function: \ (₄₈ () = ₗ ₄₈ q^x x = 1 + 240 ₍=₁^ ₃ (n) qⁿ\) (with \ (q = e^2 i \), \ (₃ (n) \) sum of cubes of divisors). This is a modular form of weight 4 for \ (SL₂ (Z) \). - Critical L-values: For a modular form \ (f\) of weight \ (k\), \ (L (f, s) \) at \ (s = k/2\) (central point). Interpolated Apéry numbers \ (Aₙ\) satisfy \ (₍=₀^ Aₙ tⁿ = hypergeometric \; ₄F₃\), linked to \ (L (f, k/2) \). - Ramanujan's 1916 discovery: \ ( () = q ₍=₁^ (1-qⁿ) ^24\), a cusp form of weight 12, with Fourier coefficients \ ( (n) \) (Ramanujan tau function). Eichler–Shimura: \ ( (p) 1 + p^11 p\) for prime \ (p\), linking to Galois representations. CONN Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com

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Modular Forms, E8 Lattice Theta Functions, and Interpolated Critical L-Values — E8 Intelligence Research

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