Weighted Distribution of the 4-rank of Class Groups and Applications

Fouvry, Étienne; Klüners, Jürgen

doi:10.1093/imrn/rnq223

Cited by 4 publications

(4 citation statements)

References 19 publications

(53 reference statements)

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“…The above result is a generalization of work of Fouvry-Klüners [15] that is derived from their own previous results [14] completely verifying Gerth's extension [16] of the Cohen-Lenstra heuristics to the 4-rank of the narrow class group of quadratic fields. In [14], Fouvry-Klüners compute all moments for the 4-ranks of narrow class groups of quadratic fields ordered by discriminant.…”

Section: Introductionsupporting

confidence: 78%

The number of quartic $D_4$-fields ordered by conductor

Altug,

Shankar,

Varma

et al. 2017

Preprint

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We consider families of number fields of degree 4 whose normal closures over Q have Galois group isomorphic to D4, the symmetries of a square. To any such field L, one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image D4. We determine the asymptotic number of such quartic D4-fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order 4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant.Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with L-function methods. Both of these strategies fall short in the case of D4-fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of D4 combined with both approaches mentioned above to obtain exact asymptotics.

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Section: Introductionsupporting

confidence: 78%

The number of quartic $D_4$-fields ordered by conductor

Altug,

Shankar,

Varma

et al. 2017

Preprint

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“…The first significant achievement for families with arbitrary discriminants was made by Fouvry and Klüners [FK07], who translated Rédei’s theory on -ranks of class groups to sums of characters conducive to analytic techniques and then successfully dealt with these sums, basing some of their work on the techniques developed by Heath-Brown in [Hea93, Hea94]. Fouvry and Klüners subsequently developed their methods in various settings [FK10a, FK10b, FK10c, FK11], most notably obtaining impressive upper and lower bounds for the solvability of the negative Pell equation for general squarefree integers . When specialized to the one-prime-parameter family with prime, their results are as strong as the bounds in (2.1), so Theorem 4 can be viewed as the next natural step in the line of work initiated by Fouvry and Klüners.…”

Section: Discussion Of Resultsmentioning

confidence: 99%

Spins of prime ideals and the negative Pell equation

Koymans

Milovic

2018

Compositio Math.

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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 − 2py 2 = −1 over the integers; (iv) 2-part of the Tate-Šafarevič group X(E p ) of the congruent number elliptic curve E p : y 2 = x 3 − p 2 x. Our results are conditional on a standard conjecture about short character sums.

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“…The main tool behind the above theorem is the recent work of Fouvry and Klüners. For details see [FoKl], and more specifically [FoKl2,Theorem 1.1].…”

Section: Proof Of Theorem 310mentioning

confidence: 99%