Certain hypotheses concerning<i>L</i>-functions

Friedlander, John B.

doi:10.2140/pjm.1977.69.37

Cited by 2 publications

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“…that are split in O K,f tends to infinity as d K,f tends to infinity. To prove this, he uses a result of Linnik and Vinogradov in [6], see also [4]. The central point in [6] is an upper bound for short character sums by Burgess, in which the exponent 1/4 + ε appears.…”

Section: Propositionmentioning

confidence: 99%

Special points on products of modular curves

Edixhoven¹

2005

Duke Math. J.

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We prove the André-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the Generalized Riemann Hypothesis. More explicitly, this means the following. Let n ≥ 0, and let Σ be a subset of C n consisting of points all of whose coordinates are j-invariants of elliptic curves with complex multiplications. Then we prove (under GRH) that the irreducible components of the Zariski closure of Σ are special sub-varieties, i.e., determined by isogeny conditions on coordinates and pairs of coordinates. A weaker variant (Thm. 1.3) is proved unconditionally.

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Section: Propositionmentioning

confidence: 99%

Special points on products of modular curves

Edixhoven¹

2005

Duke Math. J.

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“…For K an imaginary quadratic field, f a positive integer and x 2 a real number, let K;f x be the number of primes p 6 x that are split in O K;f , let [6], see also [4]. The central point in [6] is an upper bound for short character sums by Burgess, in which the exponent 1 4 + " appears.…”

Section: Existence Of Small Split Primesmentioning

confidence: 99%

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Edixhoven¹

1998

Compositio Mathematica

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Abstract. We prove, assuming the generalized Riemann hypothesis for imaginary quadratic fields, the following special case of a conjecture of Oort, concerning Zarsiski closures of sets of CM points in Shimura varieties. Let X be an irreducible algebraic curve in C 2 , containing infinitely many points of which both coordinates are j-invariants of CM elliptic curves. Suppose that both projections from X to C are not constant. Then there is an integer m 1 such that X is the image, under the usual map, of the modular curve Y0m. The proof uses some number theory and some topological arguments.Mathematics Subject Classifications (1991): 11F32, 11G15, 14K22, 14G35.

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Certain hypotheses concerningL-functions

Cited by 2 publications

References 4 publications

Special points on products of modular curves

Special points on products of modular curves

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