Morningside Center of Mathematics, CAS.
The goal of this series of talks is to explain the recent work [LTXZ1] and [LTXZ2] on slopes of modular forms and its various arithmetic applications.
(4/7) Liang Xiao: Talk 1 – Introduction to ghost conjecture and its local version. (notes)
Review the ghost conjecture raised by Bergdall and Pollack in [BP16+, BP17+]. Explain the formulation of local ghost conjecture following [LTXZ1, §2].
(4/7) Liang Xiao: Talk 2 – Application of ghost conjectures. (notes)
Give an overview of the proof of ghost conjecture and introduce its applications following [LTXZ2, §1].
(4/7) Ruochuan Liu: Talk 3 – Abstract p-adic modular forms and corank theorem. (notes)
Introduce the notions of abstract classical, overconvergent and p-adic forms and the power basis on spaces of abstract over-convergent forms. Explain the theta map and Atkin-Lehner involution under the abstract setup following [LTXZ1, §3]. Prove the (general) corank theorem and explain the intuition behind the ghost multiplicities in the definition of ghost series, following [LTXZ2, §3].
(4/14) Bin Zhao: Talk 4 – Basic properties of ghost series. (notes)
Give the dimension formulas for the space of abstract classical forms. Following [LTXZ1, §4], explain the implications below of the dimension formulas: (a) a formula on the degrees of the coefficients of ghost series; (b) compatibility of the ghost series with theta maps, the Atkin–Lehner involutions and the p-stabilization process; (c) ghost duality of ghost series; (d) an estimate of “old form slopes”.
(4/14) Bin Zhao: Talk 5 – Further properties of the ghost Newton polygon. (notes)
Following [LTXZ1, §5], introduce the notion of “near Steinberg range”, and use it to give a criterion to detect vertices of the Newton polygon of ghost series.
(4/14) Bin Zhao: Talk 6 – Integrality of slopes of ghost series at classical weights. (notes)
Show the integrality property of slopes of ghost series at classical weights.
(4/21) Ruochuan Liu: Talk 7 – Rigidity of power series of ghost series type. (notes)
Follow [LTXZ2, §9] to prove the finiteness of irreducible components of the zero locus of a power series of ghost series type, and hint at the later application to finiteness of irreducible components of eigencurves.
(4/21) Ruochuan Liu: Talk 8 – On triangulline deformation space and Paškūnas modules.(notes)
Recall triangulline deformation space of Breuil–Hellmann–Schraen and discuss the crystalline slopes on it. Give a brief introduction to Paškūnas modules and local Langlands correspondence for GL(2,Qp).
(4/21) Liang Xiao: Talk 9 – Crystalline slopes on triangulline deformation spaces. (notes)
Translate local ghost theorem to results of crystalline slopes on triangulline deformation spaces following [LTXZ2, §7]. Follow [LTXZ2, §8] to deduce the ghost conjecture of Bergdall–Pollack from the local ghost conjecture. Then explain some arithmetic consequences, e.g. Breuil-Buzzard–Emeerton conjecture, Gouvêa conjecture, Gouvêa-Mazur conjecture, etc.
(4/21) Bin Zhao: Talk 10 – Proof of local ghost conjecture I. (notes)
Follow [LTXZ2, §4] to explain how to explain how Lagrange interpolation can reduces the proof of local ghost conjecture to an estimate on the determinant of minors of the Up-operator on the power basis.
(4/28) Liang Xiao: Talk 11 – Proof of local ghost conjecture II. (notes)
Follow [LTXZ2, §6] to explain how to inductively prove the needed estimate, especially the cofactor expansion argument that reduces to an estimate that will follow from inductive hypothesis.
(4/28) Bin Zhao: Talk 12 – Proof of local ghost conjecture III. (notes)
Discuss the improved halo estimate and modified Mahler basis; follow [LTXZ2, §3]. Follow [LTXZ2, §5] to conclude the proof of local ghost conjecture, where we explain how to use the improved halo estimates to complete the inductive proof.
See also a pdf version of the orginal syllabus.