A Cohen-Lenstra phenomenon for elliptic curves

David, Chantal; Smith, Ethan

doi:10.1112/jlms/jdt036

Cited by 5 publications

(22 citation statements)

References 21 publications

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“…where the sum is taken over all isomorphism classes of elliptic curves over F p and | Aut p (E)| is the number of F p -automorphisms of E. It is worth noting here that | Aut p (E)| = 2 for all but a bounded number of isomorphism classes E over F p , and hence In [DS14b], the authors studied the weighted number of isomorphism classes of elliptic curves over any prime finite field with group of points isomorphic to G, i.e., they studied…”

Section: Introductionmentioning

confidence: 99%

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The Frequency of Elliptic Curve Groups over Prime Finite Fields

Chandee

David²,

Koukoulopoulos

et al. 2016

Can. j. math.

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Abstract. Letting p vary over all primes and E vary over all elliptic curves over the finite field F p , we study the frequency to which a given group G arises as a group of points E(F p ). It is well-known that the only permissible groups are of the form G m,k := Z/mZ × Z/mkZ. Given such a candidate group, we let M(G m,k ) be the frequency to which the group G m,k arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for M(G m,k ) assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for M(G m,k ), pointwise and on average. In particular, we show that M(G m,k ) is bounded above by a constant multiple of the expected quantity when m ≤ k A and that the conjectured asymptotic for M (G m,k ) holds for almost all groups G m,k when m ≤ k 1/4−ǫ . We also apply our methods to study the frequency to which a given integer N arises as the group order #E(F p ).

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Section: Introductionmentioning

confidence: 99%

“…Before stating the main theorem of [DS14b], we fix some more notation. Given a group G = G m,k , we let Aut(G) denote its automorphism group (as a group).…”

Section: Introductionmentioning

confidence: 99%

The Frequency of Elliptic Curve Groups over Prime Finite Fields

Chandee

David²,

Koukoulopoulos

et al. 2016

Can. j. math.

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“…Finally, we remark that the Cohen-Lenstra heuristics apply to a broader class of situations where finite abelian groups arise as co-kernels of random sub-lattices of Z n . For instance, [10] contains average results on the group of Z/pZ-rational points of an elliptic curve which are consistent with the Cohen-Lenstra heuristics (of course the rank can be at most two in this setting), and (in much the same spirit as our present consideration of missing class groups) [2] considers the question of which finite rank 2 abelian groups occur as the group of Z/pZ-rational points of some elliptic curve E over Z/pZ.…”

Section: Partitions Ofmentioning

confidence: 99%

“…n=1 d k (n) n e −n/Z log n ≤ e 1/Z ∞ a=1 e −a/Z a log(a) k ≤ (log(e 2 · Z)) 3k ,for Z large enough. Inserting this into (3.4) and taking Z = exp (log x)10 , we obtain…”

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confidence: 99%

Missing Class Groups and Class Number Statistics for Imaginary Quadratic Fields

Holmin

Jones

Kurlberg

et al. 2017

Experimental Mathematics

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Abstract. The number F (h) of imaginary quadratic fields with class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F (h). The unconditional computation of F (h) for h ≤ 100 was completed by M. Watkins, using ideas of Goldfeld and Gross-Zagier; Soundararajan has more recently made conjectures about the order of magnitude of F (h) as h → ∞ and determined its average order. In the present paper, we refine Soundararajan's conjecture to a conjectural asymptotic formula and also consider the subtler problem of determining the number F (G) of imaginary quadratic fields with class group isomorphic to a given finite abelian group G. Using Watkins' tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance (Z/3Z) 3 does not). This observation is explained in part by the Cohen-Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine heuristics of Cohen-Lenstra together with our refinement of Soundararajan's conjecture to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of "missing" class groups, for the case of p-groups as p tends to infinity. Furthermore, conditionally on the Generalized Riemann Hypothesis, we extend Watkins' data, tabulating F (h) for odd h ≤ 10 6 and F (G) for G a p-group of odd order with |G| ≤ 10 6 . The numerical evidence matches quite well with our conjectures.

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“…but we do not have a closed expression for its value. For the case d = 2, Parks [33], with contributions from Giri, obtained Theorem 1.6 with a different technique, without using Gekeler's theorem, but following similar steps as in the original proofs of Theorems 1.2, 1.4, 1.5 and 1.8 [16,11,4,13,14]. Theorem 1.4 of [33] (or, rather, its proof) implies that…”

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confidence: 99%

Sums of Euler products and statistics of elliptic curves

2016

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Abstract. We present several results related to statistics for elliptic curves over a finite field F p as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove known results such as the average LangTrotter conjecture, the average Koblitz conjecture, and the vertical Sato-Tate conjecture, even for very short intervals, not accessible by previous methods. We also compute statistics for new questions, such as the problem of amicable pairs and aliquot cycles, first introduced by Silverman and Stange. Our technique is rather flexible and should be easily applicable to a wide range of similar problems. The starting point of our results is a theorem of Gekeler which gives a reinterpretation of Deuring's theorem in terms of an Euler product involving random matrices, thus making a direct connection between the (conjectural) horizontal distributions and the vertical distributions. Our main technical result then shows that, under certain conditions, a weighted average of Euler products is asymptotic to the Euler product of the average factors.

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A Cohen-Lenstra phenomenon for elliptic curves

Cited by 5 publications

References 21 publications

The Frequency of Elliptic Curve Groups over Prime Finite Fields

The Frequency of Elliptic Curve Groups over Prime Finite Fields

Missing Class Groups and Class Number Statistics for Imaginary Quadratic Fields

Sums of Euler products and statistics of elliptic curves

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