Arakelov computations in genus 3 curves

Jordi Guàrdia

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 157-165
  • ISSN: 1246-7405

Abstract

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Arakelov invariants of arithmetic surfaces are well known for genus 1 and 2 ([4], [2]). In this note, we study the modular height and the Arakelov self-intersection for a family of curves of genus 3 with many automorphisms: C n : Y 4 = X 4 - ( 4 n - 2 ) X 2 Z 2 + Z 4 . Arakelov calculus involves both analytic and arithmetic computations. We express the periods of the curve C n in terms of elliptic integrals. The substitutions used in these integrals provide a splitting of the jacobian of C n as a product of three elliptic curves. Using the corresponding isogeny, we determine the stable model of the arithmetic surface given by C n . Once we have the periods and the stable model of C n , we can study the modular height and the self-intersection of the canonical sheaf. We can give a good estimate for the modular height, which reflects its logarithmic behaviour. We provide a lower bound for the self-intersection, which shows that it can be arbitrarily large. We present here all our calculations on the curves C n , almost without proofs. Details can be found in [5].

How to cite

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Guàrdia, Jordi. "Arakelov computations in genus $3$ curves." Journal de théorie des nombres de Bordeaux 13.1 (2001): 157-165. <http://eudml.org/doc/248720>.

@article{Guàrdia2001,
abstract = {Arakelov invariants of arithmetic surfaces are well known for genus 1 and 2 ([4], [2]). In this note, we study the modular height and the Arakelov self-intersection for a family of curves of genus 3 with many automorphisms:\begin\{equation*\} C\_n: Y^4 = X^4 - (4n - 2)X^2 Z^2 + Z^4.\end\{equation*\}Arakelov calculus involves both analytic and arithmetic computations. We express the periods of the curve $C_n$ in terms of elliptic integrals. The substitutions used in these integrals provide a splitting of the jacobian of $C_n$ as a product of three elliptic curves. Using the corresponding isogeny, we determine the stable model of the arithmetic surface given by $C_n$. Once we have the periods and the stable model of $C_n$, we can study the modular height and the self-intersection of the canonical sheaf. We can give a good estimate for the modular height, which reflects its logarithmic behaviour. We provide a lower bound for the self-intersection, which shows that it can be arbitrarily large. We present here all our calculations on the curves $C_n$, almost without proofs. Details can be found in [5].},
author = {Guàrdia, Jordi},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Arakelov theory; genus 3 curves; modular height; self-intersection of the dualizing sheaf},
language = {eng},
number = {1},
pages = {157-165},
publisher = {Université Bordeaux I},
title = {Arakelov computations in genus $3$ curves},
url = {http://eudml.org/doc/248720},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Guàrdia, Jordi
TI - Arakelov computations in genus $3$ curves
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 157
EP - 165
AB - Arakelov invariants of arithmetic surfaces are well known for genus 1 and 2 ([4], [2]). In this note, we study the modular height and the Arakelov self-intersection for a family of curves of genus 3 with many automorphisms:\begin{equation*} C_n: Y^4 = X^4 - (4n - 2)X^2 Z^2 + Z^4.\end{equation*}Arakelov calculus involves both analytic and arithmetic computations. We express the periods of the curve $C_n$ in terms of elliptic integrals. The substitutions used in these integrals provide a splitting of the jacobian of $C_n$ as a product of three elliptic curves. Using the corresponding isogeny, we determine the stable model of the arithmetic surface given by $C_n$. Once we have the periods and the stable model of $C_n$, we can study the modular height and the self-intersection of the canonical sheaf. We can give a good estimate for the modular height, which reflects its logarithmic behaviour. We provide a lower bound for the self-intersection, which shows that it can be arbitrarily large. We present here all our calculations on the curves $C_n$, almost without proofs. Details can be found in [5].
LA - eng
KW - Arakelov theory; genus 3 curves; modular height; self-intersection of the dualizing sheaf
UR - http://eudml.org/doc/248720
ER -

References

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  1. [1] A. Abbes, E. Ullmo, Auto-intersection du dualisant relatif des courbes modulaires X0(N). J. reine angew. Math.484 (1997), 1-70. Zbl0934.14016MR1437298
  2. [2] J.-B. Bost, Fonctions de Green-Anakelov, fonctions théta et courbes de genre 2. C.R. Acad. Sci. Paris Série I305 (1987), 643-646. Zbl0638.14016MR917587
  3. [3] Bost J.-B., J.-F. Mestre, L. Moret-Bailly, Sur le calcul explicite des "classes de Chern" des surfaces arithmétiques de genre 2. Astérisque183 (1990), 69-105. Zbl0731.14017MR1065156
  4. [4] G. Faltings, Calculus on arithmetic surfaces. Ann. of Math.119 (1984), 387-424. Zbl0559.14005MR740897
  5. [5] J. Guàrdia, Geometria aritmética en una famlia de corbes de genere 3. Thesis, Universitat de Barcelona, 1998. 
  6. [6] A. Moriwaki, Lower bound of self-intersection of dualizing sheaves on arithmetic surfaces with reducible fibres. Compositio Mathematica105 (1997), 125-140. Zbl0917.14012MR1386111
  7. [7] M. Raynaud, Hauteurs et isogénies. Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, exp. VII. Astérisque127 (1985), 199-234. MR801923
  8. [8] E. Ullmo, Positivité et discrétion des points algébriques des courbes. Annals of Math.147 (1998), 167-179. Zbl0934.14013MR1609514

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