# L-functions, ranks of elliptic curves, and random matrix theory (07w5114)

Arriving in Banff, Alberta Sunday, July 8 and departing Friday July 13, 2007

## Organizers

Brian Conrey (American Institute of Mathematics)

Michael Rubinstein (University of Waterloo)

Nina Snaith (University of Bristol)

## Objectives

Random matrix theory has had an astounding impact on analytic number theory.

Thanks to the insight gained from random matix models, we now know the answers to essentially any question having to do with averages of L-functions or with the distribution of their zeros. Many of these we cannot prove; however the theoretical results combined with numerical evidence paint an absolutely convincing picture.

The statistics of moments and the distribution of zeros largely deal with values of L-functions in an intermediate range. Remarkably, the distribution of very small values as modeled by random matrix theory, together with theorems about special values of L-functions, give a glimpse of what is happening on sparse sets. One such example is as follows.

Let $E: y^2=x^3+Ax+B$ be an elliptic curve over $Q$. There is an L-function, $L_E(S)$ attached to $E$; this L-function is a Dirichlet series which can be expressed as an Euler product and which has a functional equation. According to the conjecture of Birch and Swinnerton-Dyer, the rank of the elliptic curve should equal the order of vanishing at the central point of the L-function associated to the elliptic curve. (The central point is the point of symmetry for the functional equation.) Thus, conjecturally at least, the distribution of

ranks of elliptic curves can be studied by examining the L-functions at the

central point.

After the work of Katz and Sarnak, Keating and Snaith, and Conrey, Farmer, Keating, Rubinstein, and Snaith, we know how to model the distribution of L-values using characeristic polynomials of orthogonal matrices. On the other hand, there are formulas, given in terms of the coefficients of certain ternary quadratic forms, for the special values of the L-functions of this family at the central point. Using these formulas a ``disretisation'' can be identified which can be used to predict how often the L-function actually vanishes.

Assuming the conjectures of Birch and Swinnerton-Dyer, one can predict how often rank two curves appear in this family.

The prediction is that the number of rank two curves within the family of quadratic twists $E_d: dy^2=x^3+Ax+B$ of $E$ for $dle X$ is $c_E X^{3/4} (log X)^{b_E}$. Interestingly, $b_E$ can take on four different values according to the 2-torsion of $E$ and whether the discriminant of $E$ is a square. The value of $c_E$ depends on a combination of random matrix theory, the arithmetical

constant $a_k$ in the moments of the L-functions of the twists, and probabilistic group theory in the form of Delaunay's version for the Tate-Shafarevich group of the Cohen-Lenstra heuristics for class groups. Even with all of this, there is still an aspect of $c_E$, a factor at a few bad primes, that we do not yet understand. The numerics of this example are extensive. Tornaria, Mao, Baruch, Rosson, Rodriguez-Villegas in various collaborations have developed the algorithms necessary to determine ternary quadratic forms whose theta series provide the arithmetical part of the special value of the L-function of the twisted elliptic curve. More than 2000 of these theta series

have been computed and for each of them several hundred million Fourier coefficients have been determined by Rubinstein, thus giving information about hundreds of millions of quadratic twists of 2000 elliptic curves.

The basic paradigm described above has been extended in various ways. If one begins with a rational new form $f$ of weight 4 and considers the vanishings of order two of its twisted L-series, the frequency is $c_f X^{1/4}log^{b_f}X$; this indicates the frequency with which the Chow group has rank two. Some of these forms are associated with Calabi-Yau threefolds (see for example the work of Helena Verrill) and so are of interest to a wide community of scientists.

A small amount of data has been gathered here; more would be extremely valuable. For weight 6 newforms, only a finite number of twists are expected to vanish. Investigating how this finite number varies with the level is of interest.

The work of David, Fearnley, and Kivilevsky involves cubic twists of a given elliptic curve $E$. Here the family of twists is modeled by a unitary family of matrices, whereas the underlying symmetry type of quadratic twists is orthogonal. Here the number of rank two occurrences when twisting by cubic characters with conductor smaller than $x$ is expected to be $sim c_3(E) x^{1/2} log^{b_3(E)} x$. Not much has been done to determine $b_3(E)$ and $c_3(E_)$.

Considering vanishing for twists of odd weight rational new forms would also be very interesting. No one knows any theta series to assist with numerical experiments here.

Another application for these ideas is in the context of Siegel modular forms which also have formulas for the special values of their twists.

A main problem which remains to be modeled is the frequency of ranks three and higher occuring among quadratic twists of an elliptic curve. This raises the key question of what is the correct choice of random matrix for predicting the values of a particular derivative of L-functions that have been selected from a family of quadratic twists for the property that all derivatives lower than the one in question vanish. There have been two random matrix models proposed, but due to the rarity of ranks three and higher, it has so far proved difficult to obtain convincing numerical evidence for either. This is a question that it would be very desirable to resolve so that random matrix theory could be used confidently to examine high ranks of elliptic curves.

From the number theory side, complications in the modeling of the frequency of rank three curves arise because of the appearance of the regulator in the conjectural formula of Birch and Swinnerton-Dyer for the value of the derivative of an elliptic curve L-function which vanishes to order one.

From attempting to model this rank 3 situation, an interesting development has occurred. It appears, based on numerical work, that for a rank one elliptic curve, the regulator and the size of the Tate-Shafarevich group are not independent. It may even be the case that the product of these two for the elliptic curve $E_d$ may always be larger than $d^{theta-epsilon}$ for some

positiv $theta$. If so, it would be very interesting to learn why. In this situation the data is very sparse, but Watkins has some ideas about how to compute the rquired L-values essentially as quickly as what we can do for quadratic twists; this involves computing Heegner points.

The goal of this workshop is to bring together experts in a variety of areas

to try to:

* Give a good statistical model for rank three quadratic twists.

* Find a fast algorithm to compute which curves from a quadratic twist family have rank 3.

* Understand why the size of the regulator times the size of Sha is so big for rank one curves.

* Produce theta series for thousands of weight 4 and 6 forms.

* Find an algorithm to give the theta series for odd weight forms.

* Refine our understanding of the value of c_E.

* Investigate the constants $b_3(E)$ and $c_3(E)$ which arise in the case of cubic twists.

Invitees to this workshop include people who have already worked on problems

related to these as well as people who potentially have the expertise to assist in the solution of these open problems.

Thanks to the insight gained from random matix models, we now know the answers to essentially any question having to do with averages of L-functions or with the distribution of their zeros. Many of these we cannot prove; however the theoretical results combined with numerical evidence paint an absolutely convincing picture.

The statistics of moments and the distribution of zeros largely deal with values of L-functions in an intermediate range. Remarkably, the distribution of very small values as modeled by random matrix theory, together with theorems about special values of L-functions, give a glimpse of what is happening on sparse sets. One such example is as follows.

Let $E: y^2=x^3+Ax+B$ be an elliptic curve over $Q$. There is an L-function, $L_E(S)$ attached to $E$; this L-function is a Dirichlet series which can be expressed as an Euler product and which has a functional equation. According to the conjecture of Birch and Swinnerton-Dyer, the rank of the elliptic curve should equal the order of vanishing at the central point of the L-function associated to the elliptic curve. (The central point is the point of symmetry for the functional equation.) Thus, conjecturally at least, the distribution of

ranks of elliptic curves can be studied by examining the L-functions at the

central point.

After the work of Katz and Sarnak, Keating and Snaith, and Conrey, Farmer, Keating, Rubinstein, and Snaith, we know how to model the distribution of L-values using characeristic polynomials of orthogonal matrices. On the other hand, there are formulas, given in terms of the coefficients of certain ternary quadratic forms, for the special values of the L-functions of this family at the central point. Using these formulas a ``disretisation'' can be identified which can be used to predict how often the L-function actually vanishes.

Assuming the conjectures of Birch and Swinnerton-Dyer, one can predict how often rank two curves appear in this family.

The prediction is that the number of rank two curves within the family of quadratic twists $E_d: dy^2=x^3+Ax+B$ of $E$ for $dle X$ is $c_E X^{3/4} (log X)^{b_E}$. Interestingly, $b_E$ can take on four different values according to the 2-torsion of $E$ and whether the discriminant of $E$ is a square. The value of $c_E$ depends on a combination of random matrix theory, the arithmetical

constant $a_k$ in the moments of the L-functions of the twists, and probabilistic group theory in the form of Delaunay's version for the Tate-Shafarevich group of the Cohen-Lenstra heuristics for class groups. Even with all of this, there is still an aspect of $c_E$, a factor at a few bad primes, that we do not yet understand. The numerics of this example are extensive. Tornaria, Mao, Baruch, Rosson, Rodriguez-Villegas in various collaborations have developed the algorithms necessary to determine ternary quadratic forms whose theta series provide the arithmetical part of the special value of the L-function of the twisted elliptic curve. More than 2000 of these theta series

have been computed and for each of them several hundred million Fourier coefficients have been determined by Rubinstein, thus giving information about hundreds of millions of quadratic twists of 2000 elliptic curves.

The basic paradigm described above has been extended in various ways. If one begins with a rational new form $f$ of weight 4 and considers the vanishings of order two of its twisted L-series, the frequency is $c_f X^{1/4}log^{b_f}X$; this indicates the frequency with which the Chow group has rank two. Some of these forms are associated with Calabi-Yau threefolds (see for example the work of Helena Verrill) and so are of interest to a wide community of scientists.

A small amount of data has been gathered here; more would be extremely valuable. For weight 6 newforms, only a finite number of twists are expected to vanish. Investigating how this finite number varies with the level is of interest.

The work of David, Fearnley, and Kivilevsky involves cubic twists of a given elliptic curve $E$. Here the family of twists is modeled by a unitary family of matrices, whereas the underlying symmetry type of quadratic twists is orthogonal. Here the number of rank two occurrences when twisting by cubic characters with conductor smaller than $x$ is expected to be $sim c_3(E) x^{1/2} log^{b_3(E)} x$. Not much has been done to determine $b_3(E)$ and $c_3(E_)$.

Considering vanishing for twists of odd weight rational new forms would also be very interesting. No one knows any theta series to assist with numerical experiments here.

Another application for these ideas is in the context of Siegel modular forms which also have formulas for the special values of their twists.

A main problem which remains to be modeled is the frequency of ranks three and higher occuring among quadratic twists of an elliptic curve. This raises the key question of what is the correct choice of random matrix for predicting the values of a particular derivative of L-functions that have been selected from a family of quadratic twists for the property that all derivatives lower than the one in question vanish. There have been two random matrix models proposed, but due to the rarity of ranks three and higher, it has so far proved difficult to obtain convincing numerical evidence for either. This is a question that it would be very desirable to resolve so that random matrix theory could be used confidently to examine high ranks of elliptic curves.

From the number theory side, complications in the modeling of the frequency of rank three curves arise because of the appearance of the regulator in the conjectural formula of Birch and Swinnerton-Dyer for the value of the derivative of an elliptic curve L-function which vanishes to order one.

From attempting to model this rank 3 situation, an interesting development has occurred. It appears, based on numerical work, that for a rank one elliptic curve, the regulator and the size of the Tate-Shafarevich group are not independent. It may even be the case that the product of these two for the elliptic curve $E_d$ may always be larger than $d^{theta-epsilon}$ for some

positiv $theta$. If so, it would be very interesting to learn why. In this situation the data is very sparse, but Watkins has some ideas about how to compute the rquired L-values essentially as quickly as what we can do for quadratic twists; this involves computing Heegner points.

The goal of this workshop is to bring together experts in a variety of areas

to try to:

* Give a good statistical model for rank three quadratic twists.

* Find a fast algorithm to compute which curves from a quadratic twist family have rank 3.

* Understand why the size of the regulator times the size of Sha is so big for rank one curves.

* Produce theta series for thousands of weight 4 and 6 forms.

* Find an algorithm to give the theta series for odd weight forms.

* Refine our understanding of the value of c_E.

* Investigate the constants $b_3(E)$ and $c_3(E)$ which arise in the case of cubic twists.

Invitees to this workshop include people who have already worked on problems

related to these as well as people who potentially have the expertise to assist in the solution of these open problems.