scholarly journals Notes on the Parity Conjecture

2013 ◽  
pp. 201-249 ◽  
Author(s):  
Tim Dokchitser
Keyword(s):  
Parity Conjecture

scholarly journals A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM

2011 ◽  
Vol 07 (08) ◽  
pp. 2237-2247 ◽  
Author(s):  
LARRY ROLEN
Keyword(s):  
Elliptic Curve ◽  
Related Problem ◽  
Fixed Angle ◽  
Root Numbers ◽  
Tate Conjecture ◽  
Congruent Number ◽  
Number Problem ◽  
Parity Conjecture ◽  
Congruent Number Problem

We study a certain generalization of the classical Congruent Number Problem. Specifically, we study integer areas of rational triangles with an arbitrary fixed angle θ. These numbers are called θ-congruent. We give an elliptic curve criterion for determining whether a given integer n is θ-congruent. We then consider the "density" of integers n which are θ-congruent, as well as the related problem giving the "density" of angles θ for which a fixed n is congruent. Assuming the Shafarevich–Tate conjecture, we prove that both proportions are at least 50% in the limit. To obtain our result we use the recently proven p-parity conjecture due to Monsky and the Dokchitsers as well as a theorem of Helfgott on average root numbers in algebraic families.

Compatibility of arithmetic and algebraic local constants (the case )

2015 ◽  
Vol 151 (9) ◽  
pp. 1626-1646 ◽  
Author(s):  
Jan Nekovář
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Selmer Groups ◽  
Real Multiplication ◽  
Special Cases ◽  
Congruent Modulo ◽  
Multiplication Theorem ◽  
Parity Conjecture

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$ are compatible with the usual local constants at all primes not dividing $p$ and in two special cases also at primes dividing $p$. We deduce new cases of the $p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

scholarly journals The -parity conjecture for abelian varieties over function fields of characteristic

2014 ◽  
Vol 150 (4) ◽  
pp. 507-522 ◽  
Author(s):  
Fabien Trihan ◽  
Seidai Yasuda
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Prime Number ◽  
Abelian Variety ◽  
Function Field ◽  
Function Fields ◽  
Selmer Group ◽  
Smooth Variety ◽  
Tate Conjecture ◽  
Parity Conjecture

AbstractLet $A/K$ be an abelian variety over a function field of characteristic $p>0$ and let $\ell $ be a prime number ($\ell =p$ allowed). We prove the following: the parity of the corank $r_\ell $ of the $\ell $-discrete Selmer group of $A/K$ coincides with the parity of the order at $s=1$ of the Hasse–Weil $L$-function of $A/K$. We also prove the analogous parity result for pure $\ell $-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin–Tate conjecture.

scholarly journals PARALLEL WEIGHT 2 POINTS ON HILBERT MODULAR EIGENVARIETIES AND THE PARITY CONJECTURE

2019 ◽  
Vol 7 ◽  
Author(s):  
CHRISTIAN JOHANSSON ◽  
JAMES NEWTON
Keyword(s):  
Modular Forms ◽  
Weight Space ◽  
Small Radius ◽  
Real Field ◽  
Central Character ◽  
Hilbert Modular Forms ◽  
Irreducible Components ◽  
Hilbert Modular ◽  
Parity Conjecture ◽  
Totally Real

Let $F$ be a totally real field and let $p$ be an odd prime which is totally split in $F$ . We define and study one-dimensional ‘partial’ eigenvarieties interpolating Hilbert modular forms over $F$ with weight varying only at a single place $v$ above $p$ . For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch–Kato conjecture for finite slope Hilbert modular forms with trivial central character (with a technical assumption if $[F:\mathbb{Q}]$ is odd), by reducing to the case of parallel weight $2$ . As another consequence of our results on partial eigenvarieties, we show, still under the assumption that $p$ is totally split in $F$ , that the ‘full’ (dimension $1+[F:\mathbb{Q}]$ ) cuspidal Hilbert modular eigenvariety has the property that many (all, if $[F:\mathbb{Q}]$ is even) irreducible components contain a classical point with noncritical slopes and parallel weight $2$ (with some character at $p$ whose conductor can be explicitly bounded), or any other algebraic weight.

scholarly journals Regulator constants and the parity conjecture

2009 ◽  
Vol 178 (1) ◽  
pp. 23-71 ◽  
Author(s):  
Tim Dokchitser ◽  
Vladimir Dokchitser
Keyword(s):  
Parity Conjecture

scholarly journals Rank Growth of Elliptic Curves in Non-Abelian Extensions

2019 ◽  
Author(s):  
Robert J Lemke Oliver ◽  
Frank Thorne
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Positive Constant ◽  
Rank Zero ◽  
Abelian Extensions ◽  
Quadratic Twists ◽  
Rank One ◽  
Higher Rank ◽  
Parity Conjecture

Abstract Given an elliptic curve $E/\mathbb{Q}$, it is a conjecture of Goldfeld that asymptotically half of its quadratic twists will have rank zero and half will have rank one. Nevertheless, higher rank twists do occur: subject to the parity conjecture, Gouvêa and Mazur constructed $X^{1/2-\epsilon }$ twists by discriminants up to $X$ with rank at least two. For any $d\geq 3$, we build on their work to consider the rank growth of $E$ in degree $d$$S_d$-extensions of $\mathbb{Q}$ with discriminant up to $X$. We prove that there are at least $X^{c_d-\epsilon }$ such fields where the rank grows, where $c_d$ is a positive constant that tends to $1/4$ as $d\to \infty $. Moreover, subject to a suitable parity conjecture, we obtain the same result for fields for which the rank grows by at least two.

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