Jeffrey Manning

Contact Info:

Papers:

1. Mod l multiplicities in certain U(4) Shimura varieties
   ArXiV version
2. Congruence modules in higher codimension and zeta lines in Galois cohomology, (with Srikanth Iyengar, Chandrashekhar Khare and Eric Urban), Proceedings of the National Academy of Sciences 121 (2024), no. 17, e2320608121
   ArXiV version
3. Freeness of Hecke modules at non-minimal levels, (with Srikanth Iyengar and Chandrashekhar Khare), Math Research Letters to appear
   ArXiV version
4. Congruence modules and the Wiles-Lenstra-Diamond numerical criterion in higher codimensions, (with Srikanth Iyengar and Chandrashekhar Khare), Inventiones Mathematicae 238, 769-864 (2024), https://doi.org/10.1007/s00222-024-01292-y
   ArXiV version
5. Wiles defect of Hecke algebras via local-global arguments: With an appendix by Najmuddin Fakhruddin and Chandrashekhar B. Khare, (with Gebhard Böckle and Chandrashekhar Khare), Journal of the Institute of Mathematics of Jussieu 23(6) (2024), pp. 2461-2541
   ArXiV version
6. Wiles defect for Hecke algebras that are not complete intersections, (with Gebhard Böckle and Chandrashekhar Khare), Compositio Mathematica 157 (2021), no. 9, 2046-2088
   ArXiV version
7. Ihara's lemma for Shimura curves over totally real number fields via patching. (with Jack Shotton), Mathematische Annalen 379 (2021), 187-234
   ArXiV version
8. Patching and Multiplicity 2^k for Shimura Curves, Algebra and Number Theory Vol. 15 (2021), No. 2, 387-434
   ArXiV version
9. EZADS inputs which produce half-factorial block monoids.
   Semigroup Forum 90.3 (2015), pp. 775-799.

Expository writing:

This is an overview of the construction (due to Eichler and Shimura) of a two dimensional Galois representation associated to a weight two modular form (or more precisely, to a cuspform which is an eigenform for all of the Hecke operators). It is (a somewhat expanded version of) my topic proposal, which I wrote during the second year of my Ph.D. at UChicago.

This (current version: December 2019) is an expanded version of the appendix to my thesis. It's roughly my attempt to present the commutative algebra behind the Taylor-Wiles-Kisin patching argument (one of the main tools used in the proof of Fermat's Last Theorem) as cleanly and systematically as possible, using the "ultrapatching" approach introduced by Scholze. It is very much a work in progress, and I hope to eventually expand it to include some of the more sophisticated uses of patching (e.g. patching functors, Ihara avoidance, the Calegari-Geraghty method, derived deformation rings and Hecke algebra, etc.).

Teaching:

At UCLA: (websites on CCLE)

At UChicago: