The average analytic rank of elliptic curves

@article{HeathBrown2003TheAA,
  title={The average analytic rank of elliptic curves},
  author={D. R. Heath-Brown},
  journal={Duke Mathematical Journal},
  year={2003},
  volume={122},
  pages={591-623}
}
All the results in this paper are conditional on the Riemann Hypothesis for the L-functions of elliptic curves. Under this assumption, we show that the average analytic rank of all elliptic curves over Q is at most 2, thereby improving a result of Brumer [2]. We also show that the average within any family of quadratic twists is at most 3/2, improving a result of Goldfeld [3]. A third result concerns the density of curves with analytic rank at least R, and shows that the proportion of such… 
Average analytic ranks of elliptic curves over number fields
. We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are
A conditional determination of the average rank of elliptic curves
TLDR
It is said that nonreal zeros of elliptic curve $L$-functions in a family have a direct influence on the average rank in this family.
Variation in the number of points on elliptic curves and applications to excess rank
Michel proved that for a one-parameter family of elliptic curves over Q(T) with non-constant j(T) that the second moment of the number of solutions modulo p is p^2 + O(p^{3/2}). We show this bound is
On the distribution of analytic ranks of elliptic curves
TLDR
An upper bound for the probability for an elliptic curve with analytic rank $\leq a$ for $a \geq 11$ is given and an upper bound of n-th moments of analytic ranks of elliptic curves is given.
Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and
Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture
Abstract We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over $\mathbb{Q}$. We are primarily interested in the family of all twists
Note on the rank of quadratic twists of Mordell equations
Counting elliptic curves with local conditions and its applications
TLDR
An upper bound of $n$-th moments of analytic ranks of elliptic curves, and an upper bounds for the probability that an elliptic curve has analytic rank $\leq a$ for $a \geq 11$ under GRH for elliptic $L$-functions are given.
...
...

References

SHOWING 1-10 OF 18 REFERENCES
On the modularity of elliptic curves over Q
In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E/Q is an elliptic curve, then E is modular.
Ranks of elliptic curves
This paper gives a general survey of ranks of elliptic curves over the field of rational numbers. The rank is a measure of the size of the set of rational points. The paper includes discussions of
Low lying zeros of families of L-functions
In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at
Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?)
When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This
Zeroes of zeta functions and symmetry
Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence
Ring-Theoretic Properties of Certain Hecke Algebras
The purpose of this article is to provide a key ingredient of [W2] by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in [W2], a method
The Grothendieck Festschrift
The many diverse articles presented in these three volumes, collected on the occasion of Alexander Grothendieck’s sixtieth birthday and originally published in 1990, were offered as a tribute to one
Random Matrices
TLDR
This workshop was unusually diverse, even by MSRI standards; the attendees included analysts, physicists, number theorists, probabilists, combinatorialists, and more.
Formules explicites et minoration de conducteurs de vari'et'es alg'ebriques
© Foundation Compositio Mathematica, 1986, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions
Conjectures on elliptic curves over quadratic fields
...
...