ARITHMETIC STATISTICS STUDY GROUP
1. Week 1: Statement of main results, brief overview of the methods
involved in the proof
Given any elliptic curve over Q, one can nd an equation for E of the form
E : y 2 = x3 + Ax + B
for some integers A and B . We may always assume that for each prime p, if p4 divides A then
p6 does not divide B , in which case A and B are unique. We dene the (naive) height h(E)
of E as
h(E) := max{4|A|3 , 27B 2 }.
The following theorem will be the focus of the study group.
(BhargavaShankar). When elliptic curves over Q are ordered by their naive
height, the average rank is bounded, and is in fact at most 1.5.
Theorem 1.1
Subsequent work of BhargavaShankar has now reduced the bound to 0.885. It is conjectured that the actual answer is 0.5, with 50 percent of elliptic curves having rank 0 and 50
percent having rank 1.
The above theorem is a corollary of the following more rened result.
Let n ∈ {2, 3, 4, 5}. Then when elliptic curves over Q are ordered by their
naive height, the average size of the n-Selmer group is equal to σ(n), the sum of divisors of n.
Theorem 1.2.
The case n = 2 gives the bound of 1.5 for the average rank stated above, whilst the smaller
bound of 0.885 needs larger values of n. Whilst the method of proof is similar in each case,
we'll focus on the n = 2 case which is the least technical to prove.
In each case, the proof follows the following strategy.
• `parameterise' Selmer elements - for example, elements of the 2-Selmer group of an
elliptic curve correspond to equivalence classes of (integral) binary quartic forms satisfying certain irreducibility conditions and local conditions, elements of the 3-Selmer
group similarly correspond to classes of ternary cubic forms satisfying some additional
conditions. For n = 4, 5 the situation is a bit more complicated, but similar correspondences exist.
• count the parameterising objects using geometry of numbers arguments.
Theorem 1.1 is just one of many results in `arithmetic statistics' that follow a similar strategy, and the problem of counting integral forms with bounded invariants is classical.
For example, letting h(D) denote the number of SL2 (Z)-equivalence classes of integral,
irreducible binary quadratic forms of discriminant D, a theorem of Mertens (1874) gives
X
h(D) ∼
−X<D<0
1
π 3
X2.
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ARITHMETIC STATISTICS STUDY GROUP
In light of the classical correspondence between binary quadratic forms and class groups,
this can be viewed as giving the average size of the class group of imaginary orders up to a
given discriminant.
A more recent theorem of Davenport (1951) is the following.
(Davenport). Let h(D) denote the number of GL2 (Z)-equivalence classes of
irreducible integral binary cubic forms having discriminant D. Then we have
Theorem 1.3
X
h(D) ∼
−X<D<0
and
X
0<D<X
h(D) ∼
π2
X
24
π2
X.
72
If one additionally imposes local conditions on the binary cubic forms, one can obtain the
DavenportHeilbronn theorem.
Let N+ (X) denote the number of cubic elds K with discriminant 0 <
Disc(K) < X . Then
Theorem 1.4.
1
X + o(X).
12ζ(3)
Similarly, if N− (X) denotes the number of cubic elds K with discriminant −X < Disc(K) <
0, then
1
N− (X) =
X + o(X).
4ζ(3)
N+ (X) =
One can also average the size of the 3-torsion subgroup of the ideal class group of quadratic
elds similarly.
Recently, in [8], BhargavaShankar and Tsimmerman employed the averaging techniques
underpinning the proof of Theorem 1.1 to give a new proof of Theorem 1.3 with much improved
error terms.
In this study group we will aim to cover the following topics:
• Heuristics and conjectures for the average size of the rank of elliptic curves, average
sizes of Selmer groups of elliptic curves, and number of number elds of given degree
and bounded discriminant
• The association between objects of arithmetic interest and orbits of integral forms,
specically the DeloneFadeev correspondence between binary cubic forms and cubic
rings and the correspondence between 2-Selmer elements and binary quartic forms
• Introduction to geometry of numbers methods and Davenport's theorem relating the
number of lattice points inside bounded regions of Rn dened by algebraic equations
and its volume
• The DavenportHeilbronn theorem via the methods of BhargavaShankarTsimmerman
(this acts as a good warmup to the more complicated arguments required for Theorem 1.1)
• Counting binary quartic forms with bounded invariants and the proof of Theorem 1.1
• Applications to other statistical questions related to the arithmetic of elliptic curves
and the proportion of elliptic curves know to satsify BSD
• Survey of more exotic situations where the counting methods work
ARITHMETIC STATISTICS STUDY GROUP
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2. Week 2: Introduction to Selmer groups and examples of 2-descent
As covered, for example in Chapter X of Silverman's book `The arithmetic of elliptic curves'.
3. Week 3: Heuristics for Selmer groups and ranks
• Numerology from [5, 4] including conjectural average size of n-Selmer groups .
• Probabilistic models for Selmer groups as in [10, 3, 11].
4. Week 4: Associating binary cubic forms to cubic rings and binary quartic
forms to elements of the 2-Selmer group of an elliptic curve
• DeloneFadeev correspondence [8, Section 2]
• The Kummer sequence for p = 2 geometrically (see Darmon's notes [9]).
• Existence of a degree 2 divisor on locally soluble 2-coverings and binary quartic forms
(reference as above).
• [6, Section 3.1]
5. Week 5: Geometry of numbers
Proof of Davenport's lemma, see pages 13-17 of Frank Thorne's lecture notes [13].
6. Week 6: The number of binary cubic forms having bounded invariants,
Davenport Heilbronn theorem
Following [8, Sections 3,4,5,8].
7. Week 7: Counting binary quartic forms I; setting up the problem
Following [6, Sections 2.1, 2.2 and 2.3].
8. Week 8: Counting binary quartic forms II; computing the volume, passing
to Selmer elements
Following [6, Sections 2.4,2.5, 3].
9. Week 9; Higher Selmer groups, applications to BSD
Overview of the 3-Selmer, 4-Selmer and 5-Selmer papers of Bhargava and Shankar [7, 4, 5],
positive proportions of E/Q satisfy BSD etc. (see also [1]). Discussion of why 5 is the limit of
the method to date.
10. Week 10; Extensions (time dependent)
Additional situations in which the averaging method applies and applications. Possible
topics to be covered include: Selmer groups for Jacobians of hyperelliptic curves [2], sparsity
of points on hyperelliptic curves over Q [12], Jack Thorne's work on average sizes of Selmer
groups for Jacobians of non-hyperelliptic genus 3 curves [14]. Additional topics of choice.
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ARITHMETIC STATISTICS STUDY GROUP
References
[1] Majul Bhargava, Chris Skinner, and Wei Zhang. A majority of elliptic curves over Q satisfy the birch and
swinnerton-dyer conjecture. Preprint, http://arxiv.org/abs/1407.1826, 2014.
[2] Manjul Bhargava and Benedict H. Gross. The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point. In Automorphic representations and L-functions,
volume 22 of Tata Inst. Fundam. Res. Stud. Math., pages 2391. Tata Inst. Fund. Res., Mumbai, 2013.
[3] Manjul Bhargava, Danie M.l Kane, Hendrik W. Lenstra jr., Bjorn Poonen, and Eric Rains. Modeling the distribution of ranks, selmer groups, and shafarevich-tate groups of elliptic curves. Preprint,
http://arXiv:1304.3971v2.
[4] Manjul Bhargava and Arul Shankar. The average number of elements in the 4-selmer groups of elliptic
curves is 7. Preprint, http://arxiv.org/pdf/1312.7333, 2013.
[5] Manjul Bhargava and Arul Shankar. The average size of the 5-selmer group of elliptic curves is 6, and the
average rank is less than 1. Preprint, http://arxiv.org/abs/1312.7859, 2013.
[6] Manjul Bhargava and Arul Shankar. Binary quartic forms having bounded invariants, and the boundedness
of the average rank of elliptic curves. Ann. of Math. (2), 181(1):191242, 2015.
[7] Manjul Bhargava and Arul Shankar. Ternary cubic forms having bounded invariants, and the existence of
a positive proportion of elliptic curves having rank 0. Ann. of Math. (2), 181(2):587621, 2015.
[8] Manjul Bhargava, Arul Shankar, and Jacob Tsimerman. On the Davenport-Heilbronn theorems and second
order terms. Invent. Math., 193(2):439499, 2013.
[9] Henri Darmon. Curves: Diophantine aspects. http://www.crm.umontreal.ca/sms/2014/pdf/darmon2.pdf.
[10] Bjorn Poonen. Selmer group heuristics and sieves. Arizona winter school notes,
http://swc.math.arizona.edu/aws/2014/2014PoonenNotes.pdf, 2014.
[11] Bjorn Poonen and Eric Rains. Random maximal isotropic subspaces and Selmer groups. J. Amer. Math.
Soc., 25(1):245269, 2012.
[12] Bjorn Poonen and Michael Stoll. Most odd degree hyperelliptic curves have only one rational point. Ann.
of Math. (2), 180(3):11371166, 2014.
[13] Frank
Thorne.
Davenport's
lemma.
Geometry
of
numbers
lecture
notes,
http://people.math.sc.edu/thornef/math788g/gon-11-16.pdf.
[14] Jack A. Thorne. E6 and the arithmetic of a family of non-hyperelliptic curves of genus 3. Forum Math.
Pi, 3:e1, 41, 2015.