We prove under mild hypotheses the three-variable Iwasawa main conjecture for $p$-ordinary modular forms in the indefinite setting. Our result is in a setting complementary to that in the work of Skinner-Urban, and it has applications to Greenberg's nonvanishing conjecture for the first derivatives at the center of $p$-adic $L$-functions of cusp forms in Hida families with root number $-1$ and to Howard's horizontal nonvanishing conjecture.

　　Publication: 

　　Advances in Mathematics, Volume 400, 14 May 2022. 

　　Author: 

　　Francesc Castella 

　　Department of Mathematics, University of California, Santa Barbara, CA 93106, USA 

　　Email: castella@ucsb.edu  

　　Xin Wan 

　　Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Science, No. 55 Zhongguancun East Road, Beijing, 100190, China 

　　Email: xwan@math.ac.cn