The number of extensions of a number field with fixed degree and bounded discriminant

Ellenberg, Jordan S.; Venkatesh, Akshay

doi:10.4007/annals.2006.163.723

Cited by 72 publications

(84 citation statements)

References 18 publications

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“…The best general bounds known for n ≥ 6 are due to Ellenberg and Venkatesh [24], who prove a bound of O(X n ε ). Conjectures for the density of discriminants of degree n number fields having a specified Galois group G (yielding the expected orders of growth but not the constants) have been suggested by Malle [32].…”

Section: Related and Future Workmentioning

confidence: 99%

Higher composition laws and applications

Bhargava¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. In 1801 Gauss laid down a remarkable law of composition on integral binary quadratic forms. This discovery, known as Gauss composition, not only had a profound influence on elementary number theory but also laid the foundations for ideal theory and modern algebraic number theory. Even today, Gauss composition remains one of the best ways of understanding ideal class groups of quadratic fields.The question arises as to whether there might exist similar laws of composition on other spaces of forms that could shed light on the structure of other algebraic number rings and fields. In this article we present several such higher analogues of Gauss composition, and we describe how each of these composition laws can be interpreted in terms of ideal classes in appropriate rings of algebraic integers. We also discuss several applications of these composition laws, including the resolution of a critical case of the Cohen-Lenstra-Martinet heuristics, and a solution of the longstanding problem of counting the number of quartic and quintic fields of bounded discriminant. In addition, we describe the mysterious relationship between these various composition laws and the exceptional Lie groups. Finally, we discuss prospects for future work and conclude with several open questions. Mathematics Subject Classification (2000). Primary 11R29; Secondary 11R45.

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Section: Related and Future Workmentioning

confidence: 99%

Higher composition laws and applications

Bhargava¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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“…. This was improved upon by Ellenberg-Venkatesh [EV06] for large n, who proved that there exist constants An depending on n and an absolute constant C such that…”

Section: Introductionmentioning

confidence: 98%

The weak form of Malle’s conjecture and solvable groups

Alberts

2020

Res. number theory

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For a fixed finite solvable group G and number field K, we prove an upper bound for the number of G-extensions L{K with restricted local behavior (at infinitely many places) and invpL{Kq ă X for a general invariant "inv". When the invariant is given by the discriminant for a transitive embedding of a nilpotent group G Ă Sn, this realizes the upper bound given in the weak form of Malle's conjecture. For other solvable groups, the upper bound depends on the size of torsion of the class group of number fields with fixed degree. In particular, the bounds we prove realize the upper bound given in the weak form of Malle's conjecture for the transitive embedding of a solvable group G Ă Sn if we assume that for each finite abelian group A the average size of class group torsion |HompClpLq, Aq| is smaller than X ǫ as L{K varies over certain families of extensions with invpL{Kq ă X.

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“…Geometrically speaking, we construct a coarse moduli space for these quartic fields, and then exploit the geometry of this space to get non-trivial estimates. In their recent work, Ellenberg and Venkatesh [9] improve Schmidt's estimate for large d. Their construction can be thought of as putting a level structure on this moduli space, and their main result follows by carefully controlling the size of the fiber of the map from this fine moduli space to our coarse moduli space. The proofs in [9] turn out to make relatively little use of the geometry of this fine moduli space; it would be interesting to see if we can get better results by combining our explicit geometric analysis with this general construction.…”

Section: ∼ C(d G)x E(dg) Log V(dg) X As X→∞mentioning

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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The number of extensions of a number field with fixed degree and bounded discriminant

Abstract: We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant ≤ X; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the number of extensions and upper bounds for Galois extensions.

Cited by 72 publications

References 18 publications

Higher composition laws and applications

Higher composition laws and applications

The weak form of Malle’s conjecture and solvable groups

Densities of quartic fields with even Galois groups

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