On the rank of ideal class groups

Wong, Siman

doi:10.1090/crmp/019/33

Cited by 8 publications

(9 citation statements)

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“…In [13], Soundararajan notes that it is possible to show that h 3 (D) |D| 1/3+ for all but X values D in a range [X, 2X]. Assuming both the Birch-Swinnerton-Dyer conjecture and the Riemann hypothesis, Wong [15] has shown that h 3 (D) |D| 1/4+ . Finally, we remark that it might appear that one could apply the quite general estimates for N q (X, Y ) in Theorems 3 and 4 to bound the g-part h g (−d) for any g 3.…”

Section: Remarksmentioning

confidence: 99%

The 3-Part of Class Numbers of Quadratic Fields

Pierce

2005

J. London Math. Soc.

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It is proved that the 3-part of the class number of a quadratic field Q( √ D) is O(|D| 55/112+ ) in general and O(|D| 5/12+ ) if |D| has a divisor of size |D| 5/6 . These bounds follow as results of nontrivial estimates for the number of solutions to the congruence x a ≡ y b modulo q in the ranges x X and y Y , where a, b are nonzero integers and q is a square-free positive integer. Furthermore, it is shown that the number of elliptic curves over Q with conductor N is O(N 55/112+ ) in general and O(N 5/12+ ) if N has a divisor of size N 5/6 . These results are the first improvements to the trivial bound O(|D| 1/2+ ) and the resulting bound O(N 1/2+ ) for the 3-part and the number of elliptic curves, respectively.

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Section: Remarksmentioning

confidence: 99%

The 3-Part of Class Numbers of Quadratic Fields

Pierce

2005

J. London Math. Soc.

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“…Questions about asymptotic properties of N (d, G, X) naturally arise in connection with tabulation of number fields of a given degree and Galois type. This quantity is also closely related to questions about p-rank of class groups of number fields ( [12], [22]). Given any such pair (d, G), Malle [14] …”

Section: Introductionmentioning

confidence: 99%

Densities of quartic fields with even Galois groups

Wong

2005

Proc. Amer. Math. Soc.

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Abstract. Let N (d, G, X) be the number of degree d number fields K with Galois group G and whose discriminant D K satisfies |D K | ≤ X. Under standard conjectures in diophantine geometry, we show that N (4, A 4 , X) X 2/3+ , and that there are N 3+ monic, quartic polynomials with integral coefficients of height ≤ N whose Galois groups are smaller than S 4 , confirming a question of Gallagher. Unconditionally we have N (4, A 4 , X) X 5/6+ , and that the 2-class groups of almost all Abelian cubic fields k have size D 1/3+ k . The proofs depend on counting integral points on elliptic fibrations.

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“…This follows from the Brauer Siegel estimate of the size of quadratic class groups. The next Theorem, which renders unconditional a result in [10], plus the work of Brumer and Silverman now improves this exponent to 1Â4+=. Combined with [5,Theorem 1], this gives rise to improved upper bounds for the number of S-unit solutions to equations of the form y 3 =f (x).…”

Section: â4+=mentioning

confidence: 98%

Exponents of Class Groups and Elliptic Curves

Wong

2001

Journal of Number Theory

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We show that the number of elliptic curves over Q with conductor N is < < = N 1Â4+= , and for almost all positive integers N, this can be improved to < < = N = . The second estimate follows from a theorem of Davenpart and Heilbronn on the average size of the 3-class groups of quadratic fields. The first estimate follows from the fact that the 3-class group of a quadratic field Q(-D) has size < < = |D| 1Â4+= , a non-trivial improvement over the Brauer Siegel estimate. Academic Press

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On the rank of ideal class groups

Cited by 8 publications

References 0 publications

The 3-Part of Class Numbers of Quadratic Fields

The 3-Part of Class Numbers of Quadratic Fields

Densities of quartic fields with even Galois groups

Exponents of Class Groups and Elliptic Curves

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