Constructing Arithmetic Siegel Modular Forms: Theta Lifting and Explicit Methods for Real Multiplication Abelian Surfaces
Robin Jackson
TL;DR
The paper develops an explicit, computational framework to realize vector-valued Siegel modular forms associated with abelian surfaces with real multiplication by leveraging the theta correspondence for the unitary dual pair $(\mathrm{U}(2,2), \mathrm{Sp}_4)$. Central to the construction are precisely chosen local Schwartz functions encoding Hodge structure at the archimedean place and arithmetic data at non-archimedean places, including distinguished test vectors for ramified primes, enabling an explicit theta lift to $\mathrm{Sp}_4$. The authors demonstrate, via the doubling method and explicit local zeta integrals, that the resulting L-function $L(s, \Theta(\pi_f))$ matches the Hasse-Weil L-function $L(A/K, s)$, thereby realizing a concrete instance of Langlands functoriality and outlining a full computational pipeline. The framework is designed to be numerically implementable, with complexity analyses and feasibility discussions, and it connects the automorphic output to the geometry of the RM moduli space $\mathcal{A}_{2,K}$ and arithmetic invariants of RM abelian surfaces.
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.