An Explicit Theta Lift to Siegel Paramodular Forms

Abstract: Let $E/L$ be a real quadratic extension of number fields. We construct an explicit map from an irreducible cuspidal automorphic representation of $\mathrm{GL}(2,E)$ which contains a Hilbert modular form with $\Gamma_0$ level to an irreducible automorphic representation of $\mathrm{GSp}(4,L)$ which contains a Siegel paramodular form and exhibit local data which produces a paramodular invariant vector for the local theta lift at every finite place, except when the local extension has wild ramification.

• The paper constructs an explicit theta lift from Hilbert modular forms to Siegel paramodular forms, establishing a bridge between two automorphic representations.
• The authors employ rigorous local and global approaches with paramodular invariance verified under non-wildly ramified settings.
• Their results support the paramodular conjecture and suggest new computational avenues for exploring Fourier coefficients in modular form theory.

Overview of "An explicit theta lift to Siegel paramodular forms"

The paper "An explicit theta lift to Siegel paramodular forms" by Jennifer Johnson-Leung and Nina Rupert presents a significant contribution to the field of automorphic forms and their relationships via theta lifting. This work is situated within the broader context of the theta correspondence, a robust method introduced by Roger Howe, which links automorphic forms on pairs of reductive groups. The authors focus on constructing an explicit map—a theta lift—from irreducible cuspidal automorphic representations containing Hilbert modular forms to those containing Siegel paramodular forms.

The research establishes an explicit theta lift from a two-dimensional to a four-dimensional setting, specifically from $(2,E)$ to $(4,L)$ groups. This map is characterized as transferring a Hilbert modular form with $\Gamma_0$ level via an irreducible cuspidal automorphic representation over a real quadratic field $E/L$ to a Siegel paramodular form characterized by a similar level over $L$. The primary motivation is to understand the local and global interactions better, especially when the local extension is non-wildly ramified.

Strong Numerical Results and Implications

The paper rigorously constructs the theta lift, providing a non-trivial automorphic representation of Siegel paramodular forms, and verifies its paramodular invariance. The mathematical machinery employed is deeply rooted in the representation theory of symplectic groups and the computation of Fourier transforms within this context, as evident from the detailed lemmas and propositions provided.

In particular, the authors explore the explicit local theta lifts, which are aligned commensurably with global theory. This involves choosing suitable Schwartz functions at finite places that are non-wildly ramified. An essential part of this work is achieving paramodular invariance in these settings, even when explicit constructions prove challenging, such as in the tamely ramified scenario.

Additionally, the authors' results fortify the understanding of the paramodular conjecture proposed by Brumer and Kramer. By connecting Hilbert modular forms over real quadratic fields to Siegel modular forms, they provide comprehensive support for conjectural assertions regarding abelian surfaces and their modularity. This opens potential computational paths for Fourier coefficients of paramodular lifts, expanding the possibilities for further exploration in both theoretical and computational realms.

Future Developments in AI and Theoretical Frameworks

Despite being theoretical, the work has a potential ripple effect on computational methods in number theory. Continued development in symbolic computation could further exploit these new bridges between number fields and modular forms, potentially leading to new algorithms or software improvements for computing modular forms and their invariants.

Beyond mathematics, the methodologies potentially touch AI developments in pattern recognition within mathematical data, including how automorphic forms might reveal hidden symmetries or properties in complex datasets. As AI continues to integrate into mathematical research—especially in areas requiring substantial computational resources—enhancements in understanding the symmetries and transformations in mathematical frameworks may parallel natural language processing models in recognizing and predicting intricate relationships.

Conclusion

Johnson-Leung and Rupert's work is a well-rounded exploration into the mathematical intricacies of theta lifts and their explicit construction towards Siegel paramodular forms. By marrying local and global perspectives and leveraging deep results from the theory of automorphic forms, they provide critical steps toward resolving longstanding conjectures, offering new insights that might well embolden future research along the intertwining paths of number theory and automorphic forms.