Signed Selmer groups over ramified extensions

• Applicant: Dr. Bence Forrás
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 559516518

In the Iwasawa theory of elliptic curves over number fields, one studies the arithmetic of elliptic curves by considering the colimit of their Selmer groups along an infinite tower of field extensions, for instance, the cyclotomic ℤₚ-extension, where p ≥ 5 is a prime number. In the case of elliptic curves with good ordinary reduction at all p-adic places, one may use the p-primary Selmer group. If there are places with good supersingular reduction, the colimit fails to be cotorsion over the Iwasawa algebra, which led Kobayashi to define plus/minus Selmer groups over the rationals by modifying the local condition at p appropriately. The corresponding Iwasawa main conjecture has important consequences regarding the Birch and Swinnerton-Dyer conjecture for elliptic curves with supersingular reduction. Many results on signed Selmer groups have since been generalised to number fields that are at most weakly ramified at supersingular places, thereby leading to arithmetically interesting results such as a Kida formula and an integrality result for characteristic elements. Using methods from the theory of Selmer complexes and from integral p-adic Hodge theory, as well as the work of Lei–Loeffler–Zerbes on supersingular modular forms over unramified base fields, we want to study the arithmetic of supersingular elliptic curves and modular forms in new contexts: (1) signed Selmer complexes of supersingular elliptic curves over number fields with arbitrary ramification; (2) signed Selmer groups of supersingular modular forms over base fields with some ramification; (3) signed Selmer complexes of supersingular modular forms.

• DFG Programme: WBP Fellowship
• International Connection: Canada
• Host: Professor Antonio Lei, Ph.D.