## Equidistribution in Number Theory, An IntroductionAndrew Granville, Zeév Rudnick From July 11th to July 22nd, 2005, a NATO advanced study institute, as part of the series “Seminaire ́ de mathematiques ́ superieures”, ́ was held at the U- versite ́ de Montreal, ́ on the subject Equidistribution in the theory of numbers. There were about one hundred participants from sixteen countries around the world. This volume presents details of the lecture series that were given at the school. Across the broad panorama of topics that constitute modern number t- ory one nds shifts of attention and focus as more is understood and better questions are formulated. Over the last decade or so we have noticed incre- ing interest being paid to distribution problems, whether of rational points, of zeros of zeta functions, of eigenvalues, etc. Although these problems have been motivated from very di?erent perspectives, one nds that there is much in common, and presumably it is healthy to try to view such questions as part of a bigger subject. It is for this reason we decided to hold a school on “Equidistribution in number theory” to introduce junior researchers to these beautiful questions, and to determine whether di?erent approaches can in uence one another. There are far more good problems than we had time for in our schedule. We thus decided to focus on topics that are clearly inter-related or do not requirealotofbackgroundtounderstand. |

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### Contents

XIII | 34 |

XIV | 35 |

XV | 37 |

XVI | 41 |

XVII | 45 |

XVIII | 49 |

XIX | 51 |

XX | 59 |

XXI | 67 |

XXII | 74 |

XXIII | 84 |

XXIV | 86 |

XXV | 88 |

XXVI | 93 |

XXVII | 96 |

XXVIII | 98 |

XXIX | 100 |

XXXI | 103 |

XXXII | 104 |

XXXIII | 114 |

XXXIV | 126 |

XXXV | 139 |

XXXVI | 146 |

XXXVII | 154 |

XXXVIII | 161 |

XXXIX | 169 |

XL | 176 |

XLI | 178 |

XLII | 179 |

XLIII | 183 |

XLIV | 188 |

XLV | 189 |

XLVI | 197 |

### Common terms and phrases

Abelian varieties algebraic algebraic variety asymptotic automorphic Bi`evre bound circle method classical CM points coeﬃcients compact conjecture constant converges coprimality deduce deﬁned deﬁnition Del Pezzo surfaces denote diﬀerent divisors dynamical eigenfunctions Einsiedler elliptic curves entropy equations ergodic theory example exponential sums ﬁnd ﬁnite ﬁrst ﬁxed ﬂow Fourier function geodesic given Granville H-invariant Hamiltonian Heath-Brown Hecke operators inﬁnite integer intervals invariant Iwaniec lattice lectures Lemma Lindenstrauss linear Margulis Math measure classiﬁcation modular modular forms modulo non-trivial number ﬁeld number theory obtain orbit Pezzo surfaces polynomial prime number prime number theorem probability measure problem Proposition prove quadratic forms quantum random rational points Ratner’s theorem result Rudnick Sarnak sequence Shimura varieties singular space special subvarieties subgroup subset symplectic torsor torus Ullmo uniformly distributed mod unipotent unique ergodicity universal torsor variables vector Venkatesh Weyl Zariski