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23
The Chebotarev invariant of a finite group
"... Abstract. We consider invariants of a finite group G related to the number of random (independent, uniformly distributed) conjugacy classes which are required to generate it. These invariants are intuitively related to problems of Galois theory. We find grouptheoretic expressions for them and inves ..."
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Abstract. We consider invariants of a finite group G related to the number of random (independent, uniformly distributed) conjugacy classes which are required to generate it. These invariants are intuitively related to problems of Galois theory. We find grouptheoretic expressions for them and investigate their values both theoretically and numerically. 1.
The large sieve and Galois representations
, 2008
"... Abstract. We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action o ..."
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Abstract. We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis. 1.
ON SOME QUESTIONS RELATED TO THE GAUSS CONJECTURE FOR FUNCTION FIELDS
"... Abstract. We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infi ..."
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Abstract. We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the socalled Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial. 1.
CM Jacobians
"... 0. Introduction. In this talk we discuss properties of the Torelli locus inside the moduli space of polarized abelian varieties over C. We would like to compare “canonical coordinates” on Ag ⊗ C on the one hand and see how they relate to Tg ⊂ Ag on the other hand. In order to make this more precise ..."
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0. Introduction. In this talk we discuss properties of the Torelli locus inside the moduli space of polarized abelian varieties over C. We would like to compare “canonical coordinates” on Ag ⊗ C on the one hand and see how they relate to Tg ⊂ Ag on the other hand. In order to make this more precise we study a special case: consider algebraic curves C over C
In a paper in progress (“The algebraic principle of the large sieve”), the author has developed
"... a general abstract form of the large sieve inequality that covers both classical instances and the ..."
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a general abstract form of the large sieve inequality that covers both classical instances and the
ON THE RANK OF QUADRATIC TWISTS OF ELLIPTIC CURVERS OVER FUNCTION FIELDS
, 2005
"... Abstract. We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/Fq(C) over a function field over a finite field that have rank � 2, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev dens ..."
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Abstract. We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/Fq(C) over a function field over a finite field that have rank � 2, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded. Let first E/Q be an elliptic curve over Q, and for fundamental quadratic discriminants d, let Ed denote the curve E twisted by the associated Kronecker character χd. Goldfeld conjectured that Ed is most of the time of minimal rank compatible with the root number of Ed, which in this case means 1 ∑ lim rankEd(Q) =
EXCEPTIONAL COVERS OF SURFACES
, 707
"... ABSTRACT. Consider a finite morphism f: X → Y of smooth, projective varieties over a finite field F. Suppose X is the vanishing locus in P N of r forms of degree at most d. We show that there is a constant C depending only on (N, r, d) and deg ( f) such that if F > C, then f(F) : X(F) → Y(F) i ..."
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ABSTRACT. Consider a finite morphism f: X → Y of smooth, projective varieties over a finite field F. Suppose X is the vanishing locus in P N of r forms of degree at most d. We show that there is a constant C depending only on (N, r, d) and deg ( f) such that if F > C, then f(F) : X(F) → Y(F) is injective if and only if it is surjective. 1.
WEIL NUMBERS GENERATED BY OTHER WEIL NUMBERS AND TORSION FIELDS OF ABELIAN VARIETIES
, 2005
"... Abstract. Using properties of the Frobenius eigenvalues, we show that, in a precise sense, “most ” isomorphism classes of (principally polarized) simple abelian varieties over a finite field are characterized, up to isogeny, by the sequence of their division fields, and a similar result for “most ” ..."
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Abstract. Using properties of the Frobenius eigenvalues, we show that, in a precise sense, “most ” isomorphism classes of (principally polarized) simple abelian varieties over a finite field are characterized, up to isogeny, by the sequence of their division fields, and a similar result for “most ” isogeny classes. Some global cases are also treated. 1.
DIVISIBILITY OF FUNCTION FIELD CLASS NUMBERS
, 2006
"... ABSTRACT. Let X → S be an abelian scheme over a finite field. We show that for a set of rational primes ℓ of density one, if q ≡ 1 mod ℓ is sufficiently large, then the proportion of s ∈ S(Fq) for which ℓ  ∣ ∣ Xs(Fq) ∣ ∣ is at least 1/ℓ − O(1/ℓ 2). We apply this to study class numbers of cyclic ..."
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ABSTRACT. Let X → S be an abelian scheme over a finite field. We show that for a set of rational primes ℓ of density one, if q ≡ 1 mod ℓ is sufficiently large, then the proportion of s ∈ S(Fq) for which ℓ  ∣ ∣ Xs(Fq) ∣ ∣ is at least 1/ℓ − O(1/ℓ 2). We apply this to study class numbers of cyclic covers of the projective line. 1.