On the Davenport–Heilbronn theorems and second order terms

Bhargava, Manjul; Shankar, Arul; Tsimerman, Jacob

doi:10.1007/s00222-012-0433-0

Cited by 102 publications

(223 citation statements)

References 31 publications

(70 reference statements)

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“…Moreover, by Proposition , the binary cubic forms

f \in V (Z)

having discriminant in the complement

T_{m} - S_{i}

have discriminant divisible by

p_{j}^{2}

for some

j ⩾ i

. By the uniformity estimate in [, Proposition 29], we have

\sum_{k \in T_{m} ∖ S; false| k false| < X} {normalSel}_{ϕ} false(E_{k} false)

\sum_{j ⩾ i} O false(X / p_{j}^{2} false) = O false(X / p_{i} false)

, where the implied constant is independent of

i

. That is, the total number of Selmer elements of

E_{k}

over all

k \in T_{m} ∖ S_{i}

| k | < X

, is

O (X / p_{i})

, while the total number of Selmer elements of

E_{k}

over all

| k | < X

is of course

≫ X

.…”

Section: The Average Size Of Normalselϕfalse(ekfalse)mentioning

confidence: 97%

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The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k

Bhargava

Elkies

Ari

2019

J. London Math. Soc.

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The elliptic curve Ek:y2=x3+k admits a natural 3‐isogeny ϕk:Ek→E−27k. We compute the average size of the ϕk‐Selmer group as k varies over the integers. Unlike previous results of Bhargava and Shankar on n‐Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on k; this sensitivity can be precisely controlled by the Tamagawa numbers of Ek and E−27k. As a consequence, we prove that the average rank of the curves Ek, k∈Z, is less than 1.21 and over 23% (respectively, 41%) of the curves in this family have rank 0 (respectively, 3‐Selmer rank 1).

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“…Moreover, by Proposition , the binary cubic forms

f \in V (Z)

having discriminant in the complement

T_{m} - S_{i}

have discriminant divisible by

p_{j}^{2}

for some

j ⩾ i

. By the uniformity estimate in [, Proposition 29], we have

\sum_{k \in T_{m} ∖ S; false| k false| < X} {normalSel}_{ϕ} false(E_{k} false)

\sum_{j ⩾ i} O false(X / p_{j}^{2} false) = O false(X / p_{i} false)

, where the implied constant is independent of

i

. That is, the total number of Selmer elements of

E_{k}

over all

k \in T_{m} ∖ S_{i}

| k | < X

, is

O (X / p_{i})

, while the total number of Selmer elements of

E_{k}

over all

| k | < X

is of course

≫ X

.…”

Section: The Average Size Of Normalselϕfalse(ekfalse)mentioning

confidence: 97%

“…Proof The proof is exactly as in [, Theorem 2.21], though we use the uniformity estimate

N false(Z_{p}; X false) = O false(X / p^{2} false)

of [, Proposition 29] in place of the uniformity estimate [, Theorem 2.13]. Here,

Z_{p}

is the set of integral cubic forms of non‐fundamental discriminant at

p

□

…”

Section: The Average Size Of Normalselϕfalse(ekfalse)mentioning

confidence: 99%

The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k

Bhargava

Elkies

Ari

2019

J. London Math. Soc.

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“…Similar observations have been made when testing the Cohen-Lenstra heuristicsin words, there are unusually few quadratic fields with p dividing the order of the class group compared to the number predicted by the heuristics, even in cases where the heuristics have been proven correct. In the Cohen-Lenstra case with p D 3, this discrepancy between asymptotic and observed behavior is explained by a secondary main term that is negative [1,15,18]. It is interesting to wonder whether the same is the case in the present context.…”

Section: Datamentioning

confidence: 64%

“…1 The largest -invariant we found in our computations is 14, which is the 3-adic -invariant of Q. p 956238/. 2 Computations with p-adic random matrices that we carry out in Section 4 suggest the following conjecture: Here the .K/ denotes the -invariant of the cyclotomic Z p -extension.…”

Section: The -Invariantmentioning

confidence: 89%