The Geometry of Saito-Kurokawa lifts on small parabolic Siegel eigenvarieties

Understanding the behaviour of L-functions of modular forms is a very classical and yet open problem. The Bloch-Kato conjecture predicts that the order of vanishing of the L-function of a modular form should be given by the rank of certain Bloch-Kato Selmer groups. In order to give a lower bound to these ranks in certain cases where the L-function vanishes, Bellaiche and Chenevier developed a clever strategy where they construct classes in the Selmer group via the geometry of points corresponding to certain lifts of modular forms on higher dimensional eigenvarieties. This strategy was successfully adapted for ordinary modular forms by Berger and Betina to give a lower bound in terms of the smoothness of Saito-Kurokawa points on a genus 2 Siegel eigenvariety. We generalise this work to finite slope and crucially infinite slope forms which are not seen on the Coleman-Mazur eigencurve - here we must develop the machinery of small parabolic eigenvarieties for the problem to be well defined. As a result we get new results towards the Bloch—Kato conjecture for infinite slope forms.