Alternative Proofs and Refutations: Exploring the Boundaries of Modular Form Conjectures

While most works (e.g., Lozano-Robledo 2011; Guitart & Molina 2025) focus on proving or applying classical conjectures (like Oda, Prasanna-Venkatesh), this idea takes a “question the norm” approach: What happens if we deliberately seek exceptions, alternative proofs, or unexplained phenomena—especially in less-explored classes like quasi-modular, meromorphic, or non-holomorphic modular forms? By building formal models that allow for “controlled violation” or “extension” of these conjectures, and by leveraging computational experiments, we might discover not only new proofs but also new conjectures or phenomena (e.g., analogues for vector-valued or higher-weight forms). The novelty lies in the systematic and constructive search for “exceptions that prove the rule,” which could significantly deepen our understanding of the modular-elliptic interface.

References:

1. Elliptic Curves, Modular Forms, and Their L-functions. Álvaro Lozano‐Robledo (2011).
2. Periods of modular forms and applications to the conjectures of Oda and of Prasanna-Venkatesh. Xavier Guitart, Santiago Molina (2025).