It used to be the case that the Langlands programme could be used to say something interesting about arithmetic 3-manifolds qua hyperbolic manifolds. Now, after the work of Agol, Wise, and others has blown the subject to smithereens, this gravy train appears to be over. It seems to me, however, that the great advance in our knowledge of hyperbolic 3-manifolds has precious little to say about arithmetic 3-manifolds qua lattices in semi-simple groups. As a basic example, suppose that \(X\) is a maximal compact arithmetic three orbifold associated to a quaternion algebra \(Q/F\) for some field \(F\) (with the appropriate behavior at the infinite primes). Then one may ask whether \(X\) has positive Betti number after some finite congruence cover \(\widetilde{X} \rightarrow X\). Let’s call this the virtual congruence positive Betti number conjecture. (This conjecture should be true – it is a consequence of Langland’s conjectural base change for \(\mathrm{SL}(2)\), which everyone believes but is probably very difficult.) AFAIK, there’s not really much one can say about this problem from the geometric group theory/RAAG/LERF/etc perspective, where the arithmetic structure of the tautological \(\mathrm{SL}(2)\)-representation does not seem to play so much of a role. A related question is the extent to which arithmetic 3-manifolds are intrinsically different from their non-arithmetic hyperbolic brethren. Is the virtual congruence Betti number conjecture (for arithmetic manifolds) something that could plausibly answered using geometric group theory?
-
Recent Posts
Recent Comments
- Persiflage on Persiflage, 2012-2024
- Shubhrajit Bhattacharya on Persiflage, 2012-2024
- Persiflage on “Fields of definition”
- Jordan Ellenberg on “Fields of definition”
- Persiflage on “Fields of definition”
Blogroll
Categories
Tags
- Akshay Venkatesh
- Ana Caraiani
- Andrew Wiles
- Bach
- Bao Le Hung
- Barry Mazur
- Class Field Theory
- Coffee
- completed cohomology
- David Geraghty
- David Helm
- Dick Gross
- Galois Representations
- Gauss
- George Boxer
- Gowers
- Grothendieck
- Hilbert modular forms
- Inverse Galois Problem
- Jack Thorne
- James Newton
- Joel Specter
- John Voight
- Jordan Ellenberg
- Ken Ribet
- Kevin Buzzard
- Langlands
- Laurent Clozel
- Mark Kisin
- Matthew Emerton
- Michael Harris
- modular forms
- Patrick Allen
- Peter Scholze
- Richard Moy
- Richard Taylor
- RLT
- Robert Coleman
- Ruochuan Liu
- Serre
- Shiva Chidambaram
- The Hawk
- Toby Gee
- torsion
- Vincent Pilloni
Archives
- December 2024 (2)
- November 2024 (1)
- October 2024 (1)
- September 2024 (2)
- August 2024 (1)
- July 2024 (2)
- June 2024 (2)
- May 2024 (1)
- February 2024 (1)
- October 2023 (2)
- September 2023 (2)
- June 2023 (2)
- May 2023 (2)
- April 2023 (1)
- March 2023 (1)
- February 2023 (4)
- November 2022 (2)
- July 2022 (2)
- June 2022 (2)
- April 2022 (3)
- March 2022 (1)
- February 2022 (1)
- January 2022 (1)
- December 2021 (1)
- November 2021 (1)
- August 2021 (2)
- June 2021 (1)
- April 2021 (2)
- March 2021 (2)
- February 2021 (2)
- November 2020 (2)
- October 2020 (3)
- June 2020 (2)
- May 2020 (2)
- April 2020 (5)
- March 2020 (8)
- February 2020 (2)
- January 2020 (3)
- December 2019 (2)
- November 2019 (1)
- October 2019 (4)
- September 2019 (4)
- August 2019 (3)
- July 2019 (2)
- June 2019 (2)
- May 2019 (1)
- April 2019 (2)
- March 2019 (3)
- February 2019 (1)
- January 2019 (5)
- December 2018 (3)
- November 2018 (2)
- October 2018 (3)
- September 2018 (1)
- August 2018 (2)
- July 2018 (1)
- June 2018 (3)
- May 2018 (2)
- April 2018 (2)
- March 2018 (1)
- February 2018 (2)
- January 2018 (3)
- December 2017 (2)
- November 2017 (3)
- October 2017 (4)
- September 2017 (2)
- August 2017 (1)
- July 2017 (2)
- June 2017 (4)
- May 2017 (1)
- April 2017 (3)
- March 2017 (5)
- February 2017 (2)
- January 2017 (2)
- December 2016 (3)
- November 2016 (2)
- October 2016 (3)
- August 2016 (1)
- June 2016 (1)
- May 2016 (3)
- April 2016 (1)
- March 2016 (4)
- October 2015 (1)
- September 2015 (1)
- August 2015 (1)
- July 2015 (1)
- June 2015 (3)
- May 2015 (3)
- April 2015 (2)
- March 2015 (3)
- February 2015 (1)
- January 2015 (5)
- December 2014 (2)
- November 2014 (2)
- October 2014 (2)
- September 2014 (6)
- August 2014 (7)
- July 2014 (5)
- June 2014 (3)
- May 2014 (5)
- April 2014 (3)
- March 2014 (3)
- February 2014 (2)
- January 2014 (2)
- December 2013 (1)
- November 2013 (2)
- October 2013 (5)
- September 2013 (3)
- August 2013 (2)
- July 2013 (3)
- June 2013 (7)
- May 2013 (9)
- April 2013 (5)
- March 2013 (3)
- February 2013 (2)
- January 2013 (6)
- December 2012 (6)
- November 2012 (4)
- October 2012 (11)
Meta
I doubt it. The constructions carried out to get a cover with positive betti number are likely highly non-congruence. Maybe one could hope to promote virtual positive b_1 in a non-congruence cover to a congruence cover? If one has a tower of congruence covers, then there is a finite-sheeted cover of bounded index of each of these (in a compatible tower) which has positive b_1 (by intersecting with the non-congruence cover with positive b_1). Maybe the rank of b_1 of these covers grows unbounded? One could hope to show that this implies that the b_1 is positive downstairs eventually. But the 1-forms may be highly non-automorphic, in some sense.
Any such cohomology classes will generate (in a congruence tower at some other prime $p$) an admissible $latex \mathrm{G}(\mathbf{Q}_p)$-representation. Such representations are either infinite dimensional or trivial, and in the latter case the cohomology had to come from $latex \mathrm{G}(\mathbf{R})$-invariant forms, and so already come from a congruence subgroup. So certainly any new cohomology in a non-congruence cover generates more such cohomology. But yes, I agree, it’s hard to see this having any relation to the cohomology downstairs.