BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Syed Waqar Ali Shah (Bilkent) DTSTART:20251128T124000Z DTEND:20251128T134000Z DTSTAMP:20260423T052709Z UID:OBAGS/75 DESCRIPTION:Title: E uler systems for exterior square motives\nby Syed Waqar Ali Shah (Bilk ent) as part of ODTU-Bilkent Algebraic Geometry Seminars\n\n\nAbstract\nTh e Birch–Swinnerton-Dyer conjecture relates the behavior of the L-functio n of an elliptic curve at its central point to the rank of its group of ra tional points. The Bloch–Kato conjecture generalizes this principle to a broad family of motivic Galois representations\, predicting a precise rel ationship between the order of vanishing of motivic L-functions at integer values and the structure of the associated Selmer groups. Since the found ational work of Kolyvagin in the nineties\, Euler systems have played a ce ntral role in approaching these conjectures\, and in recent years their sc ope has expanded significantly within the automorphic setting of Shimura v arieties.\n\nIn this talk\, I will focus on unitary Shimura varieties GU(2 \,2)\, whose middle-degree cohomology realizes the exterior square of the four-dimensional Galois representations attached to certain automorphic re presentations of GL_4. The period integral formula of Pollack–Shah for e xterior square L-functions has a natural motivic interpretation\, suggesti ng the feasibility of constructing a nontrivial Euler system. A key obstac le to this construction is the failure of a suitable multiplicity-one prop erty\, which has long prevented the verification of the certain norm relat ions required for Euler system methods. I will present a new approach that overcomes this difficulty. The resulting Euler system in the middle-degre e cohomology of GU(2\,2) provides the first nontrivial evidence toward the Bloch–Kato conjecture for exterior square motives and opens several pro mising avenues for further arithmetic applications. This is joint work wit h Andrew Graham and Antonio Cauchi.\n LOCATION:https://researchseminars.org/talk/OBAGS/75/ END:VEVENT END:VCALENDAR