Algebra & Number Theory

Blow-ups and class field theory for curves

Daichi Takeuchi

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We propose another proof of geometric class field theory for curves by considering blow-ups of symmetric products of curves.

Article information

Algebra Number Theory, Volume 13, Number 6 (2019), 1327-1351.

Received: 20 May 2018
Revised: 19 February 2019
Accepted: 25 March 2019
First available in Project Euclid: 21 August 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G45: Geometric class field theory [See also 11R37, 14C35, 19F05]
Secondary: 14H30: Coverings, fundamental group [See also 14E20, 14F35]

geometric class field theory ramification theory generalized jacobian


Takeuchi, Daichi. Blow-ups and class field theory for curves. Algebra Number Theory 13 (2019), no. 6, 1327--1351. doi:10.2140/ant.2019.13.1327.

Export citation


  • S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergebnisse der Mathematik $($3$)$ 21, Springer, 1990.
  • J.-L. Brylinski, “Théorie du corps de classes de Kato et revêtements abéliens de surfaces”, Ann. Inst. Fourier $($Grenoble$)$ 33:3 (1983), 23–38.
  • Q. Guignard, “On the ramified class field theory of relative curves”, Algebra Number Theory 13:6 (2019), 1299–1326.
  • K. Kato, “Swan conductors for characters of degree one in the imperfect residue field case”, pp. 101–131 in Algebraic $K$-theory and algebraic number theory (Honolulu, 1987), edited by M. R. Stein and R. K. Dennis, Contemp. Math. 83, Amer. Math. Soc., Providence, RI, 1989.
  • S. Lang, “Unramified class field theory over function fields in several variables”, Ann. of Math. $(2)$ 64 (1956), 285–325.
  • G. Laumon, “Faisceaux automorphes liés aux séries d'Eisenstein”, pp. 227–281 in Automorphic forms, Shimura varieties, and $L$-functions, I (Ann Arbor, MI, 1988), edited by L. Clozel and J. S. Milne, Perspect. Math. 10, Academic Press, Boston, 1990.
  • I. Leal, “On ramification in transcendental extensions of local fields”, J. Algebra 495 (2018), 15–50.
  • A. Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge Tracts in Math. 153, Cambridge Univ. Press, 2003.
  • M. Rosenlicht, “Generalized Jacobian varieties”, Ann. of Math. $(2)$ 59 (1954), 505–530.
  • J.-P. Serre, Algebraic groups and class fields, Graduate Texts in Math. 117, Springer, 1988.
  • A. Grothendieck, Revêtements étales et groupe fondamental (Séminaire de Géométrie Algébrique du Bois Marie 1960–1961), Lecture Notes in Math. 224, Springer, 1971.
  • M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas, Tome 3: Exposés IX–XIX (Séminaire de Géométrie Algébrique du Bois Marie 1963–1964), Lecture Notes in Math. 305, Springer, 1973.