BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Sho Tanimoto (Nagoya University) DTSTART:20250930T203000Z DTEND:20250930T213000Z DTSTAMP:20260423T130519Z UID:MITNT/117 DESCRIPTION:Title: Homological sieve and Manin's conjecture\nby Sho Tanimoto (Nagoya Univ ersity) as part of MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nI present our proofs fo r a version of Manin's conjecture over $\\mathbb F_q$ for $q$ large and Co hen—Jones—Segal conjecture over $\\mathbb C$ for rational curves on sp lit quartic del Pezzo surfaces. The proofs share a common method which bui lds upon prior work of Das—Tosteson. We call this method as homological sieve method. The main ingredients of this method are (i) the construction of bar complexes formalizing the inclusion-exclusion principle and its po int counting estimates\, (ii) dimension estimates for spaces of rational c urves using conic bundle structures\, (iii) estimates of error terms using arguments of Sawin—Shusterman based on Katz's results\, and (iv) a cert ain virtual height zeta function revealing the compatibility of bar comple xes and Peyre's constant. Our argument verifies the heuristic approach to Manin's conjecture over global function fields given by Batyrev and Ellenb erg-Venkatesh\, and it is a nice combination of various tools from algebra ic geometry (birational geometry of moduli spaces of rational curves)\, ar ithmetic geometry (simplicial schemes\, their homotopy theory\, and Grothe ndieck—Lefschetz trace formula)\, algebraic topology (the inclusion-excl usion principle and Vassiliev type method of the bar complexes) and some e lementary analytic number theory. This is joint work with Ronno Das\, Bria n Lehmann\, and Phil Tosteson with a help by Will Sawin and Mark Shusterma n.\n LOCATION:https://researchseminars.org/talk/MITNT/117/ END:VEVENT END:VCALENDAR