Koymans, P;
Milovic, DZ;
(2019)
Spins of prime ideals and the negative Pell equation x(2)-2py(2) =-1.
Compositio Mathematica
, 155
(1)
pp. 100-125.
10.1112/S0010437X18007601.
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Abstract
Let p ≡ 1 mod 4 be a prime number. We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i) 16- rank of the class group Cl(−4p) of the imaginary quadratic number field Q( √ −4p); (ii) 8-rank of the ordinary class group Cl(8p) of the real quadratic field Q( √ 8p); (iii) the solvability of the negative Pell equation x 2 − 2py2 = −1 over the integers; (iv) 2-part of the Tate-Šafarevič group X(Ep) of the congruent number elliptic curve Ep : y 2 = x 3 − p 2x. Our results are conditional on a standard conjecture about short character sums.
| Type: | Article |
|---|---|
| Title: | Spins of prime ideals and the negative Pell equation x(2)-2py(2) =-1 |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1112/S0010437X18007601 |
| Publisher version: | https://doi.org/10.1112/S0010437X18007601 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | class groups, negative Pell equation, sieve theory |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10073321 |
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