An elliptic curve analogue of pillai’s lower bound on primitive roots

AbstractLet k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.

Download Full-text

LOWER BOUNDS FOR THE NORMALIZED HEIGHT AND NON-DENSE SUBSETS OF SUBVARIETIES OF ABELIAN VARIETIES

International Journal of Number Theory ◽

10.1142/s1793042110003010 ◽

2010 ◽

Vol 06 (03) ◽

pp. 471-499 ◽

Cited By ~ 1

Author(s):

EVELINA VIADA

Keyword(s):

Lower Bound ◽

Elliptic Curve ◽

Abelian Variety ◽

Lower Bounds ◽

Finite Rank ◽

Height Function ◽

Algebraic Numbers ◽

Algebraic Subgroup ◽

Rank Zero ◽

The Third

This work is the third part of a series of papers. In the first two, we considered curves and varieties in a power of an elliptic curve. Here, we deal with subvarieties of an abelian variety in general. Let V be a proper irreducible subvariety of dimension d in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. Then, for V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d + 1 translated by the points close to a subgroup Γ of finite rank, are non-Zariski-dense in V. If Γ has rank zero, it is sufficient to assume that V is weak-transverse. The notion of closeness is defined using a height function.

Download Full-text

Twisted Hasse-Weil L-Functions and the Rank of Mordell-Weil Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1997-037-7 ◽

1997 ◽

Vol 49 (4) ◽

pp. 749-771 ◽

Cited By ~ 1

Author(s):

Lawrence Howe

Keyword(s):

Functional Equation ◽

Lower Bound ◽

Irreducible Representation ◽

Elliptic Curve ◽

Elliptic Curves ◽

Root Number

AbstractFollowing a method outlined by Greenberg, root number computations give a conjectural lower bound for the ranks of certain Mordell–Weil groups of elliptic curves. More specifically, for PQn a PGL2(Z/pnZ)–extension of Q and E an elliptic curve over Q, define the motive E ⊗ ρ, where ρ is any irreducible representation of Gal(PQn /Q). Under some restrictions, the root number in the conjectural functional equation for the L-function of E ⊗ ρ is easily computable, and a ‘–1’ implies, by the Birch and Swinnerton–Dyer conjecture, that ρ is found in E(PQn) ⊗ C. Summing the dimensions of such ρ gives a conjectural lower bound ofp2n–p2n–1–p–1for the rank of E(PQn).

Download Full-text

On thep-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004116000852 ◽

2016 ◽

Vol 164 (1) ◽

pp. 67-98 ◽

Cited By ~ 2

Author(s):

YUKAKO KEZUKA

Keyword(s):

Lower Bound ◽

Elliptic Curve ◽

Elliptic Curves ◽

Lower Bounds ◽

Complex Multiplication ◽

Ring Of Integers ◽

Quadratic Twists ◽

The Family ◽

Adic Valuation

AbstractWe study infinite families of quadratic and cubic twists of the elliptic curveE=X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complexL-series ats=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the sameL-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

Download Full-text

Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative reduction

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004112000564 ◽

2012 ◽

Vol 154 (2) ◽

pp. 303-324 ◽

Cited By ~ 2

Author(s):

CHERN–YANG LEE

Keyword(s):

Lower Bound ◽

Elliptic Curve ◽

Elliptic Curves ◽

Iwasawa Theory ◽

Curve Extension ◽

Tate Curve ◽

Iwasawa Module

AbstractThis paper studies the compact p∞-Selmer Iwasawa module X(E/F∞) of an elliptic curve E over a False Tate curve extension F∞, where E is defined over ℚ, having multiplicative reduction at the odd prime p. We investigate the p∞-Selmer rank of E over intermediate fields and give the best lower bound of its growth under certain parity assumption on X(E/F∞), assuming this Iwasawa module satisfies the H(G)-Conjecture proposed by Coates–Fukaya–Kato–Sujatha–Venjakob.

Download Full-text

SIMULTANEOUS ARITHMETIC PROGRESSIONS ON ALGEBRAIC CURVES

International Journal of Number Theory ◽

10.1142/s1793042111004198 ◽

2011 ◽

Vol 07 (04) ◽

pp. 921-931 ◽

Cited By ~ 1

Author(s):

RYAN SCHWARTZ ◽

JÓZSEF SOLYMOSI ◽

FRANK DE ZEEUW

Keyword(s):

Lower Bound ◽

Elliptic Curve ◽

Arithmetic Progression ◽

Upper Bound ◽

Algebraic Curve ◽

Algebraic Curves ◽

Arithmetic Progressions ◽

Real Algebraic Curve

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (xi, yσ(i)), where [Formula: see text] and [Formula: see text] are arithmetic progressions and σ is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over ℚ. We show that 4319 is such a bound for curves over ℝ. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the "crossing inequality" gives a lower bound. Together these bound the length of an s.a.p. on the curve. We also extend this method to bound the k for which a real algebraic curve can contain k points from a k × k grid. Lastly, these results are extended to complex algebraic curves.

Download Full-text

SELF-POINTS ON ELLIPTIC CURVES OF PRIME CONDUCTOR

International Journal of Number Theory ◽

10.1142/s1793042109002456 ◽

2009 ◽

Vol 05 (05) ◽

pp. 911-932

Author(s):

CHRISTOPHE DELAUNAY ◽

CHRISTIAN WUTHRICH

Keyword(s):

Lower Bound ◽

Elliptic Curve ◽

Elliptic Curves ◽

Cyclic Subgroup ◽

Infinite Order ◽

Field Of Definition ◽

Modular Parametrization ◽

Weil Group ◽

Definition Of

Let E be an elliptic curve of conductor p. Given a cyclic subgroup C of order p in E[p], we construct a modular point PC on E, called self-point, as the image of (E,C) on X0(p) under the modular parametrization X0(p) → E. We prove that the point is of infinite order in the Mordell–Weil group of E over the field of definition of C. One can deduce a lower bound on the growth of the rank of the Mordell–Weil group in its PGL 2(ℤp)-tower inside ℚ(E[p∞]).

Download Full-text