Abstract

Let $A/K$ be an ordinary semistable elliptic curve over a global function field of characteristic , and let $L/K$ be a ${\mathbf{Z}}_p^d$-extension ramified at only finitely many places where $A$ has ordinary or multiplicative reduction. We study the Iwasawa main conjecture for such ramified extensions, relating the characteristic ideal of the dual Selmer group over $L$ to a multivariable $p$-adic $L$-function. The main new ingredient is a $\chi$-formula comparing $\chi$-isotypic characteristic ideals with suitable specializations of the $p$-adic $L$-function. Combined with functional equations and specialization and restriction formulae, this yields the main conjecture under a natural $\mu$-minimality hypothesis detectable on the unramified line. 

Organizers

Ming-Lun Hsieh (NTU), Kwok Wing Tsoi (NTU)