Realization of modular Galois representations in the Jacobians of modular curves

• Published in: The Ramanujan journal Vol. 58; no. 2; pp. 389 - 405
• Main Author: Tian, Peng
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2022; Springer Nature B.V
• Subjects: Algorithms; Combinatorics; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Jacobians; Mathematics; Mathematics and Statistics; Number Theory; Representations; Modular forms; Primary 11Fxx; Modular Galois representations; Jacobians of modular curves; Secondary 11Y10; 11G30; 11G10
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-021-00546-0

In Tian (Acta Arith. 164:399–412, 2014 ), the author improved the algorithm proposed by Edixhoven and Couveignes for computing mod ℓ Galois representations associated to eigenforms f for the cases that ℓ ≥ k - 1 and f has level one, where k is the weight of f . In this paper, we generalize the results of Tian (Acta Arith. 164:399–412, 2014 ) and present a method to find the Jacobians of modular curves of minimal dimensions to realize the modular Galois representations. Our method works for the cases that ℓ ≥ 5 may be any prime without the assumption ℓ ≥ k - 1 and the eigenforms f have arbitrary levels prime to ℓ . Moreover, if k > 2 , we give criteria for realizing the mod ℓ Galois representations in the Jacobians J 0 of X 0 .