# Arakelov computations in genus $3$ curves

Journal de théorie des nombres de Bordeaux (2001)

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

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topGuàrdia, 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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- [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] 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] G. Faltings, Calculus on arithmetic surfaces. Ann. of Math.119 (1984), 387-424. Zbl0559.14005MR740897
- [5] J. Guàrdia, Geometria aritmética en una famlia de corbes de genere 3. Thesis, Universitat de Barcelona, 1998.
- [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] M. Raynaud, Hauteurs et isogénies. Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, exp. VII. Astérisque127 (1985), 199-234. MR801923
- [8] E. Ullmo, Positivité et discrétion des points algébriques des courbes. Annals of Math.147 (1998), 167-179. Zbl0934.14013MR1609514

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