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Statistics of moduli space of vector bundlesPublished in Elsevier Masson SAS

2019

Volume: 151

Pages: 13 - 33

Let X be a smooth projective curve of genus g≥1 over a finite field Fq with q elements, such that the function field Fq(X) is a geometric Galois extension of the rational function field Fq(x) with N=#Gal(Fq(X)/Fq(x)). Let ML(2,1) be the moduli space of rank 2 stable vector bundles over X‾=X×FqFq‾ with fixed determinant L, where L is a line bundle on X‾ of degree 1. Let Nq(ML(2,1)) be the cardinality of the set of Fq-rational points of ML(2,1). We give an estimate of Nq(ML(2,1)) in terms of N,q and g. Further we study the fluctuations of the quantity logNq(ML(2,1))−3(g−1)logq as the curve X as well as L varies over a large family of hyperelliptic curves (N=2) of genus g≥2. We find the limiting distribution of Nq(ML(2,1))−3(g−1)logq as g grows and q is fixed, in terms of its characteristic function. When both g and q grow we see that q3/2(logNq(ML(2,1))−3(g−1)logq) has a standard Gaussian distribution. © 2018 Elsevier Masson SAS

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About the journal

Journal | Data powered by TypesetBulletin des Sciences Mathematiques |
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Publisher | Data powered by TypesetElsevier Masson SAS |

ISSN | 00074497 |

Open Access | No |