STNB2018 (32nd edition)

On the exceptional specializations of Big Heegner points

Presenters

In this talk, our aim will be to explain Castella's approach to obtain a derivative formula for the specializations of Howard's big Heegner points at exceptional primes in the Hida family (in other words, we will consider a weight two newform with split multiplicative reduction at a prime $p$). This has many resemblances with the setting in which special values of $p$-adic $L$-functions vanish and the arithmetic information they usually encode is given by their derivatives (exceptional zero formulas à la MTT). Here, we will prove an explicit reciprocity law relating a derived system of cohomology classes obtained from big Heegner points with a $p$-adic $L$-function. Then, it will turn out that this $p$-adic $L$-function encodes information about images of Heegner cycles in étale cohomology via an extension of the formula of Bertolini, Darmon and Prasanna to the case of multiplicative reduction.