TATE-SHAFAREVICH GROUPS AND FROBENIUS FIELDS OF REDUCTIONS OF ELLIPTIC CURVES
نویسندگان
چکیده
منابع مشابه
Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves
Let E/Q be a fixed elliptic curve over Q which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W. Duke have obtained an asymptotic formula for the number of primes p ≤ x such that the reduction of E modulo p has a trivial Tate-Shafarevich group. Recent results of A. C. Cojocaru and C. David lead to a better error term. We introduce a new arg...
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The Tate-Shafarevich groups of certain elliptic curves over Fq(t) are related, via étale cohomology, to the group of points of an elliptic curve with complex multiplication. The Cassels-Tate pairing is computed under this identification.
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Generalizing results of Stroeker and Top we show that the 2-ranks of the TateShafarevich groups of the elliptic curves y = (x + k)(x + k) can become arbitrarily large. We also present a conjecture on the rank of the Selmer groups attached to rational 2-isogenies of elliptic curves. 1991 Mathematics Subject Classification: 11 G 05
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Using elliptic modular functions, Kronecker proved a number of recurrence relations for suitable class numbers of positive binary quadratic forms. For instance if F (N) denotes the number of uneven classes of positive binary quadratic forms with determinant −N, then
متن کاملModeling the Distribution of Ranks, Selmer Groups, and Shafarevich Tate Groups of Elliptic Curves
Using only linear algebra over Zp, we de ne a discrete probability distribution on the set of isomorphism classes of short exact sequences of Zp-modules, and then conjecture that as E varies over elliptic curves over a xed global eld k, the distribution of 0→ E(k)⊗Qp/Zp → Selp∞ E →X[p∞]→ 0 is that one. This single conjecture would explain many of the known theorems and conjectures on ranks, Sel...
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ژورنال
عنوان ژورنال: The Quarterly Journal of Mathematics
سال: 2009
ISSN: 0033-5606,1464-3847
DOI: 10.1093/qmath/hap001