Constructing Arithmetic Siegel Modular Forms: Theta Lifting and Explicit Methods for Real Multiplication Abelian Surfaces

TL;DR

This paper develops explicit methods and computational tools for constructing vector-valued Siegel modular forms associated with real multiplication abelian surfaces, using theta correspondence and local zeta integrals.

Contribution

It provides a detailed, explicit blueprint for constructing these modular forms, including local data, test vectors, and a computational pipeline, advancing practical realization of Langlands functoriality.

Findings

Explicit local Schwartz functions for archimedean and non-archimedean places

Concrete examples of ramified principal series representations

A computational pipeline with complexity analysis

Abstract

We present an explicit and computationally actionable blueprint for constructing vector-valued Siegel modular forms associated to real multiplication (RM) abelian surfaces, leveraging the theta correspondence for the unitary dual pair $(\U(2,2), \Sp_4)$. Starting from the modularity theorem, we furnish explicit local Schwartz functions: Gaussian functions modulated by harmonic polynomials at archimedean places and characteristic functions of lattices at non-archimedean places, with a significantly enhanced focus on constructing distinguished test vectors at ramified primes. We provide detailed, concrete examples for ramified principal series representations, illustrating adapted lattice construction and local zeta integral computation using Rankin-Selberg methods. A computational pipeline is outlined, detailing the interdependencies of each step, and a computational complexity assessment provides a realistic feasibility analysis. The congruence of $L$-functions is theoretically demonstrated via the doubling method, and strategies for explicit evaluations of local zeta integrals, even in ramified settings, are discussed. This work provides a roadmap for realizing a concrete instance of Langlands functoriality, paving the way for computational exploration of arithmetic invariants and bridging the gap between abstract theory and practical verification.