Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves
نویسندگان
چکیده
منابع مشابه
HYPERGEOMETRIC FUNCTIONS OVER Fq AND TRACES OF FROBENIUS FOR ELLIPTIC CURVES
We present explicit relations between the traces of Frobenius endomorphisms of certain families of elliptic curves and special values of 2F1hypergeometric functions over Fq for q ≡ 1(mod 6) and q ≡ 1(mod 4).
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Let E/Q be an elliptic curve over the field of rational numbers, with EndQ̄(E) = Z. Let K be a fixed imaginary quadratic field over Q, and x a positive real number. For each prime p of good reduction for E, let πp(E) be a root of the characteristic polynomial of the Frobenius endomorphism of E over the finite field Fp. Let ΠE(K;x) be the number of primes p ≤ x such that the field extension Q(πp(...
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The Sato-Tate conjecture asserts that given an elliptic curve without complex multiplication, the primes whose Frobenius elements have their trace in a given interval (2α √ p, 2β √ p) have density given by 2 π R β α √ 1− t2 dt. We prove that this conjecture is true on average in a more general setting.
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In [7], Greene introduced the notion of general hypergeometric series over finite fields or Gaussian hypergeometric series, which are analogous to classical hypergeometric series. The motivation for his work was to develop the area of character sums and their evaluations through parallels with the theory of hypergeometric functions. The basis for this parallel was the analogy between Gauss sums...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2010
ISSN: 0002-9939,1088-6826
DOI: 10.1090/s0002-9939-2010-10609-3