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17
The large sieve, monodromy and zeta functions of curves
, 2005
"... We prove a large sieve statement for the average distribution of Frobenius conjugacy classes in arithmetic monodromy groups over finite fields. As a first application we prove a stronger version of a result of Chavdarov on the “generic” irreducibility of the numerator of the zeta functions in a fami ..."
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Cited by 19 (6 self)
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We prove a large sieve statement for the average distribution of Frobenius conjugacy classes in arithmetic monodromy groups over finite fields. As a first application we prove a stronger version of a result of Chavdarov on the “generic” irreducibility of the numerator of the zeta functions in a family of curves with large monodromy.
ANALYTIC PROBLEMS FOR ELLIPTIC CURVES
, 2005
"... Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the dist ..."
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Cited by 6 (0 self)
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Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an interesting dichotomy for the occurence of the possible groups. (This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv). Contents
Reductions of an elliptic curve with almost prime orders
"... 1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exis ..."
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Cited by 3 (0 self)
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1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1
AN OPEN IMAGE THEOREM FOR A GENERAL CLASS OF ABELIAN VARIETIES
"... Abstract. Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Néron model of A/K has at least one fiber with potential toric dimension one, then for almost all rational primes ℓ, the Galois group of the splitting field of the ..."
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Cited by 3 (1 self)
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Abstract. Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Néron model of A/K has at least one fiber with potential toric dimension one, then for almost all rational primes ℓ, the Galois group of the splitting field of the ℓtorsion of A is GSp 2g (Z/ℓ). 1.
AN ANALOGUE OF THE SIEGELWALFISZ THEOREM FOR THE CYCLICITY OF CM ELLIPTIC CURVES MOD p
"... Abstract. Let E be a CM elliptic curve defined over Q and of conductor N. We establish an asymptotic formula, uniform in N and with improved error term, for the counting function of primes p for which the reduction mod p of E is cyclic. Our result resembles the classical SiegelWalfisz theorem regar ..."
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Cited by 2 (1 self)
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Abstract. Let E be a CM elliptic curve defined over Q and of conductor N. We establish an asymptotic formula, uniform in N and with improved error term, for the counting function of primes p for which the reduction mod p of E is cyclic. Our result resembles the classical SiegelWalfisz theorem regarding the distribution of primes in arithmetic progressions. 1.
Reduction mod p of subgroups of the MordellWeil group of an elliptic curve
 INT J. OF NUMBER THEORY
"... Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, Γp  ≥ p r r+2 +ɛ(p), wher ..."
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Cited by 2 (1 self)
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Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, Γp  ≥ p r r+2 +ɛ(p), where ɛ(p) is any function of p such that ɛ(p) → 0 as p → ∞. This is a consequence of two other results. Denote by Np the cardinality of Ep(Fp), where Fp is a finite field of p elements. Then for any δ> 0, the set of primes p for which Np has a divisor in the range (pδ−ɛ(p), pδ+ɛ(p) ) has density zero. Moreover, the set of primes p for which Γp  < p r r+2 −ɛ(p) has density zero.
A Titchmarsh divisor problem for elliptic curves, submitted
, 2014
"... Abstract. Let E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that∑ p≤x p≡1 (mod 4) τ(#E(Fp)) ∼ ..."
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Abstract. Let E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that∑ p≤x p≡1 (mod 4) τ(#E(Fp)) ∼