Skip to main content
SpringerLink
Log in

The Riemann–Roch strategy

Complex lift of the Scaling Site

  • Chapter
  • First Online:
Advances in Noncommutative Geometry

  • 732 Accesses

Abstract

We describe the Riemann–Roch strategy which consists of adapting in characteristic zero Weil’s proof, of RH in positive characteristic, following the ideas of Mattuck–Tate and Grothendieck. As a new step in this strategy we implement the technique of tropical descent that allows one to deduce existence results in characteristic one from the Riemann–Roch result over \({\mathbb C}\). In order to deal with arbitrary distribution functions this technique involves the results of Bohr, Jessen, and Tornehave on almost periodic functions.

Our main result is the construction, at the adelic level, of a complex lift of the adèle class space of the rationals. We interpret this lift as a moduli space of elliptic curves endowed with a triangular structure. The equivalence relation yielding the noncommutative structure is generated by isogenies. We describe the tight relation of this complex lift with the GL(2)-system. We construct the lift of the Frobenius correspondences using the Witt construction in characteristic 1.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The product is the convolution in the variable q and the ordinary product in the variable y.

  2. 2.

    The global Tate module TE is best described at the conceptual level as the pro-etale fundamental group \({\pi _1^{\mathrm {alg}}}(E,0)\), where E is viewed as a curve over \({\mathbb C}\). Given \(\rho \in \mathrm {Hom} ({\mathbb Q}/{\mathbb Z}, E_{\mathrm {tor}})\) the corresponding element of \({\pi _1^{\mathrm {alg}}}(E,0)\) is given by the \((\rho (\frac 1 n))_{n\in {\mathbb N}}\).

References

  1. T. Baier, C. Florentino, J. Mourao, J. Nunes, Toric Kähler metrics seen from infinity, quantization and compact tropical amoebas. J. Differential Geom. 89 (2011), no. 3, 411–454.

    Article  MathSciNet  Google Scholar 

  2. M. Baker, S. Norine, Riemann-Roch and Abel-Jacobi theory on a finite graph, Advances in Mathematics 215 (2007), 766–788.

    Article  MathSciNet  Google Scholar 

  3. M. Baker, F. Shokrieh, Chip-firing games, potential theory on graphs, and spanning trees. J. Combin. Theory Ser. A 120 (2013), no. 1, 164–182.

    Article  MathSciNet  Google Scholar 

  4. A.S. Besicovitch, Almost Periodic Functions, Cambridge University Press, 1932.

    MATH  Google Scholar 

  5. A. Bjorner, L. Lovasz, P. W. Shor, Chip-firing games on graphs, European J. Combin., 12(4), (1991), 283–291.

    Article  MathSciNet  Google Scholar 

  6. H. Bohr, Almost Periodic Functions, Chelsea Pub. Co., New York, 1947.

    MATH  Google Scholar 

  7. S. Bochner, Beitrage zur Theorie der fastperiodischen Funktionen, Math. Annalen, 96: 119–147.

    Google Scholar 

  8. E. Bombieri, J. Lagarias Complements to Li’s criterion for the Riemann hypothesis. J. Number Theory 77 (1999), no. 2, 274–287.

    Article  MathSciNet  Google Scholar 

  9. V. Borchsenius, B. Jessen, Mean motion and values of the Riemann zeta function. Acta Math. 80 (1948), 97–166.

    Article  MathSciNet  Google Scholar 

  10. J.B. Bost, A. Connes, Hecke algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (New Series) Vol.1 (1995) N.3, 411–457.

    Article  MathSciNet  Google Scholar 

  11. A. Connes, Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Selecta Math. (N.S.) 5 (1999), no. 1, 29–106.

    Article  MathSciNet  Google Scholar 

  12. A. Connes, H. Moscovici, The local index formula in noncommutative geometry, GAFA, Vol. 5 (1995), 174–243.

    MathSciNet  MATH  Google Scholar 

  13. A. Connes, H. Moscovici, Modular Hecke Algebras and their Hopf Symmetry, Mosc. Math. J. 4 (2004), no. 1, 67–109, 310.

    Article  MathSciNet  Google Scholar 

  14. A. Connes, H. Moscovici, Rankin-Cohen Brackets and the Hopf Algebra of Transverse Geometry, Mosc. Math. J. 4 (2004), no. 1, 111–130, 311.

    Article  MathSciNet  Google Scholar 

  15. A. Connes, H. Moscovici, Transgressions of the Godbillon-Vey class and Rademacher functions, in “Noncommutative Geometry and Number Theory”, pp.79–107. Vieweg Verlag, 2006.

    Google Scholar 

  16. A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives American Mathematical Society Colloquium Publications, 55. American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2008.

    Google Scholar 

  17. A. Connes, C. Consani, Characteristic one, entropy and the absolute point, “ Noncommutative Geometry, Arithmetic, and Related Topics”, the Twenty-First Meeting of the Japan-U.S. Mathematics Institute, Baltimore 2009, JHUP (2012), 75–139.

    Google Scholar 

  18. A. Connes, The Witt construction in characteristic one and Quantization. Noncommutative geometry and global analysis, 83–113, Contemp. Math., 546, Amer. Math. Soc., Providence, RI, 2011.

    Google Scholar 

  19. A. Connes, C. Consani, Universal thickening of the field of real numbers. Advances in the theory of numbers, 11–74, Fields Inst. Commun., 77, Fields Inst. Res. Math. Sci., Toronto, ON, 2015.

    Google Scholar 

  20. A. Connes, C. Consani, Schemes over \({{\mathbb F}}_1\)and zeta functions, Compositio Mathematica 146 (6), (2010) 1383–1415.

    Article  MathSciNet  Google Scholar 

  21. A. Connes, C. Consani, From monoids to hyperstructures: in search of an absolute arith- metic, in Casimir Force, Casimir Operators and the Riemann Hypothesis, de Gruyter (2010), 147–198.

    Google Scholar 

  22. A. Connes, C. Consani, The Arithmetic Site, Comptes Rendus Mathematique Ser. I 352 (2014), 971–975.

    Article  MathSciNet  Google Scholar 

  23. A. Connes, C. Consani, Geometry of the Arithmetic Site. Advances in Mathematics 291 (2016) 274–329.

    Article  MathSciNet  Google Scholar 

  24. A. Connes, C. Consani, The Scaling Site, C.R. Mathematique, Ser. I 354 (2016) 1–6.

    Article  MathSciNet  Google Scholar 

  25. A. Connes, C. Consani, Geometry of the Scaling Site, Selecta Math. New Ser. 23 no. 3 (2017), 1803–1850.

    Article  MathSciNet  Google Scholar 

  26. A. Connes, C. Consani, Homological algebra in characteristic one. Higher Structures Journal 3 (2019), no. 1, 155–247.

    MathSciNet  MATH  Google Scholar 

  27. A. Connes, An essay on the Riemann Hypothesis. In “Open problems in mathematics”, Springer (2016), volume edited by Michael Rassias and John Nash.

    Google Scholar 

  28. A. Gathmann and M. Kerber. A Riemann-Roch theorem in tropical geometry. Math. Z., 259(1):217–230, 2008.

    Article  MathSciNet  Google Scholar 

  29. J. Golan, Semi-rings and their applications, Updated and expanded version of The theory of semi-rings, with applications to mathematics and theoretical computer science [Longman Sci. Tech., Harlow, 1992. Kluwer Academic Publishers, Dordrecht, 1999.

    Google Scholar 

  30. A. Grothendieck Sur une note de Mattuck-Tate J. reine angew. Math. 200, 208–215 (1958).

    MathSciNet  MATH  Google Scholar 

  31. B. Jessen, Börge; Über die Nullstellen einer analytischen fastperiodischen Funktion. Eine Verallgemeinerung der Jensenschen Formel. (German) Math. Ann. 108 (1933), no. 1, 485–516.

    Article  Google Scholar 

  32. Jessen, Tornehave Mean motions and zeros of almost periodic functions. Acta math. 77, 137–279 (1945).

    Article  MathSciNet  Google Scholar 

  33. M. Laca, N. Larsen and S. Neshveyev, Phase transition in the Connes-MarcolliGL 2system, J. Noncommut. Geom. 1 (2007), no. 4, 397–430.

    Article  MathSciNet  Google Scholar 

  34. V. Kolokoltsov, V. P. Maslov, Idempotent analysis and its applications. Mathematics and its Applications, 401. Kluwer Academic Publishers Group, Dordrecht, 1997.

    Book  Google Scholar 

  35. G. Litvinov, Tropical Mathematics, Idempotent Analysis, Classical Mechanics and Geometry. Spectral theory and geometric analysis, 159–186, Contemp. Math., 535, Amer. Math. Soc., Providence, RI, 2011.

    Google Scholar 

  36. R. Meyer, On a representation of the idele class group related to primes and zeros ofL-functions. Duke Math. J. Vol.127 (2005), N.3, 519–595.

    Article  MathSciNet  Google Scholar 

  37. G. Mikhalkin and I. Zharkov, Tropical curves, their Jacobians and theta functions. In Curves and abelian varieties, volume 465 of Contemp. Math., p 203–230. Amer. Math. Soc., Providence, RI, 2008.

    Google Scholar 

  38. J.S. Milne, Canonical models of Shimura curves, manuscript, 2003 (www.jmilne.org).

  39. M. Laca, S. Neshveyev, M. Trifkovic Bost-Connes systems, Hecke algebras, and induction. J. Noncommut. Geom. 7 (2013), no. 2, 525–546.

    Article  MathSciNet  Google Scholar 

  40. A. Robert, A course in p-adic analysis. Graduate Texts in Mathematics, 198. Springer-Verlag, New York, 2000.

    Google Scholar 

  41. W. Rudin, Real and complex analysis. McGraw-Hill, New York, 1966.

    MATH  Google Scholar 

  42. J. von Neumann, Almost Periodic Functions in a Group I, Trans. Amer. Math. Soc., 36 no. 3 (1934) pp. 445–492

    Article  MathSciNet  Google Scholar 

  43. A. Weil, Basic Number Theory, Reprint of the second (1973) edition. Classics in Mathematics. Springer-Verlag, 1995.

    Google Scholar 

  44. S. Yoshitomi, Generators of modules in tropical geometry. ArXiv Math.AG, 10010448.

    Google Scholar 

Download references

Acknowledgements

The second named author would like to thank Alain Connes for introducing her to the noncommutative geometric vision on the Riemann Hypothesis and for sharing with her many mathematical ideas and insights.

Caterina Consani is partially supported by the Simons Foundation collaboration Grant no. 353677. She would like to thank Collège de France for some financial support.

Author information

Authors and Affiliations

  1. IHES, Bures sur Yvette, France

    Alain Connes

  2. Collège de France, Ohio State University, Columbus, OH, USA

    Alain Connes

  3. Johns Hopkins University, Baltimore, MD, USA

    Caterina Consani

Authors
  1. Alain Connes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Caterina Consani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Caterina Consani .

Editor information

Editors and Affiliations

  1. Department of Physics Institute for Mathematics, Astrophysics and Particle Physics, Nijmegen, The Netherlands

    Ali Chamseddine

  2. Department of Mathematics, Johns Hopkins University, Baltimore, MD, USA

    Caterina Consani

  3. Department of Mathematics, Pennsylvania State University, University Park, PA, USA

    Nigel Higson

  4. Department of Mathematics, The University of Western Ontario, London, ON, Canada

    Masoud Khalkhali

  5. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Henri Moscovici

  6. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Guoliang Yu

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Connes, A., Consani, C. (2019). The Riemann–Roch strategy. In: Chamseddine, A., Consani, C., Higson, N., Khalkhali, M., Moscovici, H., Yu, G. (eds) Advances in Noncommutative Geometry. Springer, Cham. https://doi.org/10.1007/978-3-030-29597-4_2

Download citation

Publish with us

Policies and ethics

Search

Navigation