Iwasawa theory for Rankin-Selberg product at an Eisenstein prime

• Published in: Journal of number theory Vol. 279; pp. 348 - 410
• Main Authors: Jha, Somnath, Shekhar, Sudhanshu, Vangala, Ravitheja
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.02.2026
• Subjects: Congruence of modular forms; Iwasawa main conjecture; p-adic L-functions; Rankin-Selberg product; Selmer group; p-adic L-functions; Selmer group; Rankin-Selberg product; Iwasawa main conjecture; Congruence of modular forms; primary
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2025.06.007

Let p be an odd prime, f be a p-ordinary newform of weight k and h be a normalized cuspidal p-ordinary Hecke eigenform of weight l<k. In this article, we study the p-adic L-function and p∞-Selmer group of the Rankin-Selberg product of f and h under the assumption that p is an Eisenstein prime for h i.e. the residual Galois representation of h at p is reducible. We show that the p-adic L-function and the characteristic ideal of the p∞-Selmer group of the Rankin-Selberg product of f,h generate the same ideal modulo p in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for f⊗h holds mod p. As an application to our results, we explicitly describe a few examples where the above congruence holds.