The Iwasawa Main Conjectures for ${\rm GL}_2$ and derivatives of $p$-adic $L$-functions (Francesc Castella, Xin Wan)


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.



Advances in Mathematics, Volume 400, 14 May 2022.



Francesc Castella

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



Xin Wan

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