Average root numbers

in families of elliptic curves

Author:
Ottavio G. Rizzo

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1597-1603

MSC (1991):
Primary 11G05; Secondary 11D25, 11C08, 28C10

DOI:
https://doi.org/10.1090/S0002-9939-99-05167-9

Published electronically:
February 18, 1999

MathSciNet review:
1641093

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a height measure on to count rational numbers. Through it, we prove a density result on the average value of the root numbers of families of twists of elliptic curves.

**1.**Tom M. Apostol,*Introduction to analytic number theory*, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1976. MR**55:7892****2.**Pierre Deligne,*Les constantes des equations fonctionnelles des fonctions*, Modular functions of one variable, II, Lecture Notes in Math. 349, Springer-Verlag, Berlin, 1973, pp. 501-597. MR**58:22020****3.**Nelson Dunford and Jacob T. Schwarz,*Linear operators, part I*, Wiley, New York, 1988. MR**90g:47001a****4.**Liem Mai,*The average analytic rank of a family of elliptic curves*, J. Number Theory**45**(1993), 45-60. MR**95d:11080****5.**Ottavio G. Rizzo,*On the variations of root numbers in families of elliptic curves*, Ph.D. thesis, Brown University, Providence, RI, 1997.**6.**David E. Rohrlich,*Variation of the root number in families of elliptic curves*, Compos. Math.**87**(1993), no. 2, 119-151. MR**94d:11045****7.**-,*Elliptic curves and the Weil-Deligne group*, Elliptic Curves and Related Topics (Hershy Kisilevsky and M. Ram Murty, eds.), CRM Proceedings & Lecture Notes, vol. 4, Centre de Recherches Mathématiques, Amer. Math. Soc., 1994, pp. 125-157. MR**95a:11054****8.**Walter Rudin,*Real and complex analysis*, second ed., McGraw-Hill, 1974. MR**49:8783****9.**Joseph H. Silverman,*Advanced topics in the arithmetic of elliptic curves*, GTM 151, Springer-Verlag, New York, 1994. MR**96b:11074****10.**John Tate,*Number theoretic background*, Automorphic Forms, Representations and -Functions, part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1977, pp. 3-26. MR**80m:12009****11.**Don Zagier and Gerhard Kramarz,*Numerical investigations related to the -series of certain elliptic curves*, J. Indian Math. Soc.**52**(1987), 51-69. MR**90d:11072**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
11G05,
11D25,
11C08,
28C10

Retrieve articles in all journals with MSC (1991): 11G05, 11D25, 11C08, 28C10

Additional Information

**Ottavio G. Rizzo**

Affiliation:
Department of Mathematics, Brown University, Box 1917, Providence, Rhode Island 02912

Address at time of publication:
Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6

Email:
otto@math.brown.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-05167-9

Received by editor(s):
September 15, 1997

Published electronically:
February 18, 1999

Additional Notes:
This research was partially written while the author was supported by a grant of the Istituto Nazionale di Alta Matematica of Rome.

Communicated by:
David E. Rohrlich

Article copyright:
© Copyright 1999
American Mathematical Society