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Mark Watkins

Publications and source records attributed to Mark Watkins.

10 recordsLinked to original sources
arXiv2006

Some heuristics about elliptic curves

We give some heuristics for counting elliptic curves with certain properties. In particular, we re-derive the Brumer-McGuinness heuristic for the number of curves with positive/negative discriminant up to $X$, which is an application of lattice-point counting. We then introduce heuristics (with refinements from random matrix theory) that allow us to predict how often we expect an elliptic curve $E$ with even parity to have $L(E,1)=0$. We find that we expect there to be about $c_1X^{19/24}(\log X)^{3/8}$ curves with $|Δ|<X$ with even parity and positive (analytic) rank; since Brumer and McGuinness predict $cX^{5/6}$ total curves, this implies that asymptotically almost all even parity curves have rank 0. We then derive similar estimates for ordering by conductor, and conclude by giving various data regarding our heuristics and related questions.

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arXiv2006

Symmetric powers of elliptic curve L-functions

The conjectures of Deligne, Be\uılinson, and Bloch-Kato assert that there should be relations between the arithmetic of algebro-geometric objects and the special values of their $L$-functions. We make a numerical study for symmetric power $L$-functions of elliptic curves, obtaining data about the validity of their functional equations, frequency of vanishing of central values, and divisibility of Bloch-Kato quotients.

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arXiv2006

Some remarks on Heegner point computations

We explain how to find a rational point on a rational elliptic curve of rank 1 using Heegner points. We give some examples, and list new algorithms that are due to Cremona and Delaunay. These are notes from a short course given at the Institut Henri Poincare in December 2004.

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arXiv2006

A note on integral points on elliptic curves

We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of~\Macsyma, with there being a unique rational nondegenerate solution. For the second case we found that resultants and/or Gröbner bases were not very efficacious. Instead, at the suggestion of Elkies, we used multidimensional $p$-adic Newton iteration, and were able to find a nondegenerate solution, albeit over a quartic number field. Due to our methodology, we do not have much hope of proving that there are no other solutions. For the third case we found a solution in a nonic number field, but we were unable to make much progress with the fourth case. We make a few concluding comments and include an appendix from Elkies regarding his calculations and correspondence with Zagier.

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arXiv2005

Discretisation for odd quadratic twists

The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point enters the picture, which does not necessarily take integral values (as does the order of the Shafarevich-Tate group). We discuss a couple of models and present data on this question.

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arXiv2005

Rank distribution in a family of cubic twists

In 1987, Zagier and Kramarz published a paper in which they presented evidence that a positive proportion of the even-signed cubic twists of the elliptic curve $x^3+y^3=1$ should have positive rank. We extend their data, showing that it is more likely that the proportion goes to zero.

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arXiv2004

Explicit lower bounds on the modular degree of an elliptic curve

We derive an explicit zero-free region for symmetric square L-functions of elliptic curves, and use this to derive an explicit lower bound for the modular degree of rational elliptic curves. The techniques are similar to those used in the classical derivation of zero-free regions for Dirichlet L-functions, but here, due to the work of Goldfield-Hoffstein-Lieman, we know that there are no Siegel zeros, which leads to a strengthened result.

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arXiv2004

Elliptic curves of large rank and small conductor

For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search methods, and tabulate, for each r=5,6,...,11, the five curves of lowest conductor, and (except for r=11) also the five of lowest absolute discriminant, that we found.

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arXiv2004

Modular Parametrizations of Neumann-Setzer Elliptic Curves

Suppose $p$ is a prime of the form $u^2+64$ for some integer $u$, which we take to be 3 mod 4. Then there are two Neumann--Setzer elliptic curves $E_0$ and $E_1$ of prime conductor $p$, and both have Mordell--Weil group $\Z/2\Z$. There is a surjective map $X_0(p)\xrightarrowπ E_0$ that does not factor through any other elliptic curve (i.e., $π$ is optimal), where $X_0(p)$ is the modular curve of level $p$. Our main result is that the degree of $π$ is odd if and only if $u \con 3\pmod{8}$. We also prove the prime-conductor case of a conjecture of Glenn Stevens, namely that that if $E$ is an elliptic curve of prime conductor $p$ then the optimal quotient of $X_1(p)$ in the isogeny class of $E$ is the curve with minimal Faltings height. Finally we discuss some conjectures and data about modular degrees and orders of Shafarevich--Tate groups of Neumann--Setzer curves.

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