Geodesic
Finiteness of the number of classes of vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ with jumps of height $2$
Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 5, Tome 511 (2022), pp. 137-160 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider vector bundles of rank $2$ with jumps of heights $1$ and $2$ and a trivial generic fiber on the arithmetic surface $\mathbb{P}^1_{\mathbb{Z}}$. The finiteness of the number of isomorphism classes of such vector bundles with a fixed discriminant and, as a consequence, with a fixed genus is obtained. 
@article{ZNSL_2022_511_a4,
     author = {V. M. Polyakov},
     title = {Finiteness of the number of classes of vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ with jumps of height $2$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {137--160},
     year = {2022},
     volume = {511},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2022_511_a4/}
}
TY  - JOUR
AU  - V. M. Polyakov
TI  - Finiteness of the number of classes of vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ with jumps of height $2$
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2022
SP  - 137
EP  - 160
VL  - 511
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2022_511_a4/
LA  - ru
ID  - ZNSL_2022_511_a4
ER  - 
%0 Journal Article
%A V. M. Polyakov
%T Finiteness of the number of classes of vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ with jumps of height $2$
%J Zapiski Nauchnykh Seminarov POMI
%D 2022
%P 137-160
%V 511
%U http://geodesic.mathdoc.fr/item/ZNSL_2022_511_a4/
%G ru
%F ZNSL_2022_511_a4
V. M. Polyakov. Finiteness of the number of classes of vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ with jumps of height $2$. Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 5, Tome 511 (2022), pp. 137-160. http://geodesic.mathdoc.fr/item/ZNSL_2022_511_a4/

[1] A. L. Smirnov, “On filtrations of vector bundles over ${\textbf P}^1_{\mathbb Z}$”, Arithmetic and Geometry, London Math. Soc. Lect. Note Series, 420, Cambridge Univ. Press, 2015, 436–457 | MR

[2] L. N. Vasershtein, “O gruppe $SL_2$ nad dedekindovymi koltsami arifmeticheskogo tipa”, Matem. sb., 89(131):2(10) (1972), 313–322

[3] D. Quillen, “Projetive modules over polynomial rings”, Invent. Math., 36 (1976), 167–171 | DOI | MR

[4] A. A. Suslin, “Proektivnye moduli nad koltsom mnogochlenov svobodny”, Dokl. AN SSSR, 229:5 (1976), 1063–1066 | MR

[5] A. Grothendieck, “Sur la classification des fibrés holomorphes sur la sphére de Riemann”, Amer. Journal of Math., 79:1 (1957) | DOI | MR

[6] A. Grothendieck, Éléments de géométrie algébrique, v. II, Publications Mathématiques de l'IHÉS, 8, Étude globale élémentaire de quelques classes de morphismes, 1961 | MR

[7] Ch. C. Hanna, “Subbundles of vector bundles on the projective line”, J. Algebra, 52:2 (1978), 322–327 | DOI | MR

[8] G. Horrocks, “Projective modules over an extension of a local ring”, Proc. London Math. Soc. (3), 14 (1964), 714–718 | DOI | MR

[9] R. Khartskhorn, Algebraicheskaya geometriya, Mir, M., 1981

[10] J. W. S. Cassels, Rational Quadratic Forms, Academic press, 1978 | MR

[11] A. L. Smirnov, “Vektornye rassloeniya na ${\textbf P}^1_{\mathbb Z}$ s prostymi podskokami”, Zap. nauchn. semin. POMI, 452, 2016, 202–217

[12] S. S. Yakovenko, “Vektornye rassloeniya na ${\textbf P}^1_{\mathbb Z}$ s obschim sloem $\mathcal{O}+\mathcal{O}(1)$ i prostymi podskokami”, Zap. nauchn. sem. POMI, 452, 2016, 218–237

[13] A. A. Beilinson, “Kogerentnye puchki na $P^n$ i problemy lineinoi algebry”, Funkts. analiz i ego prilozheniya, 12:3 (1978), 68–69 | MR

[14] K. Okonek, M. Shnaider, Kh. Shpindler, Vektornye rassloeniya na kompleksnykh proektivnykh prostranstvakh, Mir, M., 1984

[15] D. A. Cox, Primes of the Form $x^2 + ny^2$: Fermat, Class Field Theory, and Complex Multiplication, Monographs and textbooks in pure and applied math., Wiley, 1989