Explicit bounds for split reductions of simple abelian varieties

Achter, Jeffrey D.

doi:10.5802/jtnb.787

Cited by 7 publications

(2 citation statements)

References 22 publications

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“…The particular case of the forward direction of (1) when End A L ∼ = Z is an earlier result of Chavdarov [Cha97]. Achter [Ach12] gives explicit bounds on Π(A, geom. split)(X) in these cases (and one other).…”

Section: Geometrically Simplementioning

confidence: 79%

The square sieve and a Lang–Trotter question for generic abelian varieties

Bloom

2018

Journal of Number Theory

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Let A be a g-dimensional abelian variety over Q whose adelic Galois representation has open image in GSp 2g Z. We investigate the "Frobenius fields" Q(πp) = End(Ap)⊗ Q of the reduction of A modulo primes p at which this reduction is ordinary and simple. We obtain conditional and unconditional asymptotic upper bounds on the number of primes at which Q(πp) is a specified number field and, when A is two-dimensional, at which Q(πp) contains a specified real quadratic number field. These investigations continue the investigations of variants of the Lang-Trotter Conjectures on elliptic curves.

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Section: Geometrically Simplementioning

confidence: 79%

The square sieve and a Lang–Trotter question for generic abelian varieties

Bloom

2018

Journal of Number Theory

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“…We also remark that combining [Ach12] with [Lom16a] (for the case of real multiplication) and [Lom16b] (for the case of CM) one can give an explicit upper bound on the smallest place, as measured by its norm, for which A v is absolutely simple. In particular: Proposition 4.2.…”

Section: Auxiliary Results About Characteristic Polynomials Of Frobeniusmentioning

confidence: 99%

Computing the geometric endomorphism ring of a genus-2 Jacobian

Lombardo¹

2018

Math. Comp.

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We describe an algorithm, based on the properties of the characteristic polynomials of Frobenius, to compute End K (A) when A is the Jacobian of a nice genus-2 curve over a number field K. We use this algorithm to confirm that the description of the structure of the geometric endomorphism ring of Jac(C) given in the LMFDB (L-functions and modular forms database) is correct for all the genus 2 curves C currently listed in it. We also discuss the determination of the field of definition of the endomorphisms in some special cases.compute the structure of End Q (Jac(C)) for all the genus-2 curves admitting an odddegree model with small coefficients and for all the curves considered in the recent computational effort [BSS + 16]. In all cases our findings are in agreement with the data recorded in the [LMFDB], see section 8.

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Reductions of abelian varieties and K3 surfaces

Shankar,

Tang

2024

Journal of Number Theory

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Explicit bounds for split reductions of simple abelian varieties

Cited by 7 publications

References 22 publications

The square sieve and a Lang–Trotter question for generic abelian varieties

The square sieve and a Lang–Trotter question for generic abelian varieties

Computing the geometric endomorphism ring of a genus-2 Jacobian

Reductions of abelian varieties and K3 surfaces

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