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Rational Points on Modular Elliptic Curves
"... Based on an NSFCBMS lecture series given by the author at the University of Central Florida in Orlando from August 8 to 12, 2001, this monograph surveys some recent developments in the arithmetic of modular elliptic curves, with special emphasis on the Birch and SwinnertonDyer conjecture, the ..."
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Based on an NSFCBMS lecture series given by the author at the University of Central Florida in Orlando from August 8 to 12, 2001, this monograph surveys some recent developments in the arithmetic of modular elliptic curves, with special emphasis on the Birch and SwinnertonDyer conjecture, the construction of rational points on modular elliptic curves, and the crucial role played by modularity in shedding light on these questions.
Ordinary elliptic curves of high rank over ¯ Fp(x) with constant jinvariant
"... constant jinvariant ..."
The Rank of Elliptic Surfaces in Unramified Abelian Towers
, 2002
"... Let E > C be an elliptic surface defined over a number field K. For a finite covering C' > C defined over K, let = E C C be the corresponding elliptic surface over C . In this paper we give a strong upper bound for the rank of E in the case of unramified abelien coverings C ! C and under the ass ..."
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Let E > C be an elliptic surface defined over a number field K. For a finite covering C' > C defined over K, let = E C C be the corresponding elliptic surface over C . In this paper we give a strong upper bound for the rank of E in the case of unramified abelien coverings C ! C and under the assumption that the Tate conjecture is true for E =K. In the case that C is an elliptic curve and the map C = C ! C is the multiplicationbyn map, the bound for rank(E =K)) takes the form O n , which may be compared with the elementary bound of O(n²).
Elliptic curves of large rank and small conductor
"... Abstract. 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, t ..."
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Cited by 4 (2 self)
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Abstract. 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. 1
Elliptic curves, rank in families and random matrices
, 2004
"... This survey paper contains two parts. The first one is a written version of a lecture given at the “Random Matrix Theory and Lfunctions ” workshop organized at the Newton Institute in July 2004. This was meant as a very concrete and down to earth introduction to elliptic curves with some descriptio ..."
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Cited by 4 (0 self)
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This survey paper contains two parts. The first one is a written version of a lecture given at the “Random Matrix Theory and Lfunctions ” workshop organized at the Newton Institute in July 2004. This was meant as a very concrete and down to earth introduction to elliptic curves with some description of how random matrices become a tool for the (conjectural) understanding of the rank of MordellWeil groups by means of the Birch and SwinnertonDyer Conjecture; the reader already acquainted with the basics of the theory of elliptic curves can certainly skip it. The second part was originally the writeup of a lecture given for a workshop on the Birch and SwinnertonDyer Conjecture itself, in November 2003 at Princeton University, dealing with what is known and expected about the variation of the rank in families of elliptic curves. Thus it is also a natural continuation of the first part. In comparison with the original text and in accordance with the focus of the first part, more details about the input and confirmations of Random Matrix Theory have been added. Acknowledgments. I would like to thank the organizers of both workshops for
THE MAXIMUM SIZE OF LFUNCTIONS
, 2005
"... Abstract. We conjecture the true rate of growth of the maximum size of the Riemann zetafunction and other Lfunctions. We support our conjecture using arguments from random matrix theory, conjectures for moments of Lfunctions, and also by assuming a random model for the primes. ..."
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Abstract. We conjecture the true rate of growth of the maximum size of the Riemann zetafunction and other Lfunctions. We support our conjecture using arguments from random matrix theory, conjectures for moments of Lfunctions, and also by assuming a random model for the primes.
Lfunctions with large analytic rank and abelian varieties with large algebraic rank over function fields
"... The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of Lfunctions allows one to produce many examples of Lfunctions over function fields vanishing to high order at the center point of their functional equation. ..."
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The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of Lfunctions allows one to produce many examples of Lfunctions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch