Counting elliptic curves of bounded Faltings height

Hortsch, Ruthi

doi:10.4064/aa8204-2-2016

Cited by 6 publications

(7 citation statements)

References 8 publications

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“…See Watkins [Wat08a, Section 4] for further discussion. Hortsch [Hor15] recently succeeded in counting elliptic curves of bounded Faltings height, however.…”

Section: Heuristics For Shafarevich-tate Groupsmentioning

confidence: 99%

A heuristic for boundedness of ranks of elliptic curves

et al. 2019

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We present a heuristic that suggests that ranks of elliptic curves E over Q are bounded. In fact, it suggests that there are only finitely many E of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.Our model is inspired in part by the Cohen-Lenstra heuristics for class groups [CL84], as reinterpreted by Friedman and Washington [FW89]. These heuristics predict that for a fixed odd prime p, the distribution of the p-primary part of the class group of a varying imaginary quadratic field is equal to the limit as n → ∞ of the distribution of the cokernel of the homomorphism Z n p A → Z n p given by a random matrix A ∈ M n (Z p ); see Section 4 for the precise conjecture. In analogy, and in agreement with conjectures of Delaunay [Del01, Del07, DJ14], Bhargava, Kane, Lenstra, Poonen, and Rains [BKLPR15] predicted that for a fixed prime p and r ∈ Z ≥0 , the distribution of the p-primary part of the Shafarevich-Tate group X(E) as E varies over rank r elliptic curves over Q ordered by height equals the limit as n → ∞ (through integers of the same parity as r) of the distribution of coker A for a random alternating matrix A ∈ M n (Z p ) subject to the condition rk Zp (ker A) = r; see Section 5 for the precise conjecture and the evidence for it.If imposing the condition rk Zp (ker A) = r yields a distribution conjecturally associated to the curves of rank r, then naturally we guess that if we choose A at random from the space M n (Z p ) alt of all alternating matrices without imposing such a condition, then the distribution of rk Zp (ker A) tends as n → ∞ to the distribution of the rank of an elliptic curve. This cannot be quite right, however: since an alternating matrix always has even rank, the parity of n dictates the parity of rk Zp (ker A). But if we choose n uniformly at random from {⌈η⌉, ⌈η⌉ + 1} (with η → ∞), then we find that rk Zp (ker A) equals 0 or 1 with probability 50% each, and rk Zp (ker A) ≥ 2 with probability 0%; for example, when n is even, we have rk Zp (ker A) = 0 unless det A = 0, and det A = 0 holds only when A lies on a (measure 0) hypersurface in the space M n (Z p ) alt of all alternating matrices. This 50%-50%-0% conclusion matches the elliptic curve rank behavior conjectured for quadratic twist families by Goldfeld [Gol79, Conjecture B] and Katz and Sarnak [KS99a, KS99b].So far, however, this model does not predict anything about the number of curves of each rank ≥ 2 except to say that asymptotically they should amount to 0% of curves. Instead of sampling from M n (Z p ), we could sample from the set M n (Z) alt,≤X of alternating integer matrices whose entries have absolute values bounded by X, and studybut this again would be 0 for each r ≥ 2. To obtain finer information, instead of taking the limit as X → ∞, we let X depend on the height H of the elliptic curve bein...

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“…See Watkins [Wat08a, Section 4] for further discussion. Hortsch [Hor15] recently succeeded in counting elliptic curves of bounded Faltings height, however.…”

Section: Heuristics For Shafarevich-tate Groupsmentioning

confidence: 99%

A heuristic for boundedness of ranks of elliptic curves

et al. 2019

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“…Moreover, we prove that the average size of the 2-Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is 3. This implies that the average rank of these elliptic curves is finite, and bounded by 1.5.1 See, however, work of Hortsch [26] obtaining asymptotics for the number of elliptic curves with bounded Faltings height.…”

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confidence: 99%

“…1 See, however, work of Hortsch [26] obtaining asymptotics for the number of elliptic curves with bounded Faltings height.…”

mentioning

confidence: 99%

Families of elliptic curves ordered by conductor

Shankar,

Wang

2019

Preprint

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In this article, we study the family of elliptic curves E/Q, having good reduction at 2 and 3, and whose j-invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves E such that the ratio ∆(E)/C(E) is squarefree; and second, the set of elliptic curves E such that ∆(E)/C(E) is bounded by a small power (< 3/4) of C(E). Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor.Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the 2-Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is 3. This implies that the average rank of these elliptic curves is finite, and bounded by 1.5.1 See, however, work of Hortsch [26] obtaining asymptotics for the number of elliptic curves with bounded Faltings height.

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“…See, however, work of Hortsch[Hor16] obtaining asymptotics for the number of elliptic curves with bounded Faltings height.…”

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confidence: 99%

Large families of elliptic curves ordered by conductor

Shankar¹,

Shankar²,

Wang³

2021

Compositio Math.

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In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$ , having good reduction at $2$ and $3$ , and whose $j$ -invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient $\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$ . Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$ -Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$ . The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

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Counting elliptic curves of bounded Faltings height

Cited by 6 publications

References 8 publications

A heuristic for boundedness of ranks of elliptic curves

A heuristic for boundedness of ranks of elliptic curves

Families of elliptic curves ordered by conductor

Large families of elliptic curves ordered by conductor

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