Parity Conjecture Research Articles

Compatibility of arithmetic and algebraic local constants (the case )

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).

The p-parity conjecture for elliptic curves with a p-isogeny

Abstract For an elliptic curve E over a number field K, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell–Weil rank. Assuming finiteness of Ш ⁢ ( E / K ) ⁢ [ p ∞ ] ${\Sha(E/K)[p^{\infty}]}$ for a prime p this is equivalent to the p-parity conjecture: the global root number matches the parity of the ℤ p ${\mathbb{Z}_{p}}$ -corank of the p ∞ ${p^{\infty}}$ -Selmer group. We complete the proof of the p-parity conjecture for elliptic curves that have a p-isogeny for p > 3 ${p>3}$ (the cases p ≤ 3 ${p\leq 3}$ were known). Tim and Vladimir Dokchitser have showed this in the case when E has semistable reduction at all places above p by establishing respective cases of a conjectural formula for the local root number. We remove the restrictions on reduction types by proving their formula in the remaining cases. We apply our result to show that the p-parity conjecture holds for every E with complex multiplication defined over K. Consequently, if for such an elliptic curve Ш ⁢ ( E / K ) ⁢ [ p ∞ ] ${\Sha(E/K)[p^{\infty}]}$ is infinite, it must contain ( ℚ p / ℤ p ) 2 ${(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{2}}$ .

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

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.

Elliptic curves over quadratic fields with fixed torsion subgroup and positive rank

For each of the torsion groups Z/2Z⊕Z/10Z,Z/2Z⊕Z/12Z,Z/15Z we find the quadratic field with the smallest absolute value of its discriminant such that there exists an elliptic curve with that torsion and positive rank. For the torsion groups Z/11Z, Z/14Z we solve the analogous problem after assuming the Parity conjecture.

On a problem of Diophantus for rationals

Let q be a nonzero rational number. We investigate for which q there are infinitely many sets consisting of five nonzero rational numbers such that the product of any two of them plus q is a square of a rational number. We show that there are infinitely many square-free such q and on assuming the Parity Conjecture for the twists of an explicitly given elliptic curve we derive that the density of such q is at least one half. For the proof we consider a related question for polynomials with integral coefficients. We prove that, up to certain admissible transformations, there is precisely one set of non-constant linear polynomials such that the product of any two of them except one combination, plus a given linear polynomial is a perfect square.

Invariance of the parity conjecture for $p$-Selmer groups of elliptic curves in a $D_{2p^{n}}$-extension

We show a p-parity result in a D 2p n -extension of number fields L/K (p≥5) for the twist 1⊕η⊕τ: W(E/K,1⊕η⊕τ)=(-1) 1⊕η⊕τ,X p (E/L) , where E is an elliptic curve over K, η and τ are respectively the quadratic character and an irreductible representation of degree 2 of Gal (L/K)=D 2p n , and X p (E/L) is the p-Selmer group. The main novelty is that we use a congruence result between ε 0 -factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the p-parity conjecture (using the machinery of the Dokchitser brothers).

Root numbers and parity of ranks of elliptic curves

The purpose of the paper is to complete several global and local results concerning parity of ranks of elliptic curves. Primarily, we show that the Shafarevich-Tate conjecture implies the parity conjecture for all elliptic curves over number fields, give a formula for local and global root numbers of elliptic curves and complete the proof of a conjecture of Kramer and Tunnell in characteristic 0. The method is to settle the outstanding local formulae by deforming from local fields to totally real number fields and then using global parity results.

On the Birch-Swinnerton-Dyer quotients modulo squares

Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of Sha, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of Sha[6^\infty] and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of E/K agrees with the parity of the analytic rank, as determined by the root number. We also prove the p-parity conjecture for all elliptic curves over Q and all primes p: the parities of the p^\infty-Selmer rank and the analytic rank agree.

Pairings and functional equations over the GL2 -extension†

We construct a pairing on the dual Selmer group over the GL 2-extension ℚ(E[p∞]) of an elliptic curve without complex multiplication and with good ordinary reduction at a prime p ⩾ 5 whenever it satisfies certain, conjectured, torsion properties. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of the p-adic L-function. As an application we reduce the parity conjecture for the p-Selmer rank and the analytic root number for the twists of elliptic curves with self-dual Artin representation to the case when the Artin representation factors through the quotient of Gal (ℚ(E[p∞])/ℚ) by its maximal pro-p normal subgroup. This gives a new proof of the parity conjecture whenever the elliptic curve E has a p-isogeny over the rationals.

On the parity of ranks of Selmer groups IV. With an appendix by Jean-Pierre Wintenberger

AbstractWe prove the parity conjecture for the ranks of p-power Selmer groups (p⁄=2) of a large class of elliptic curves defined over totally real number fields.

Regulator constants and the parity conjecture

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p ∞-Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/ℚ is semistable at 2 and 3, K/ℚ is abelian and K ∞ is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of $\mathop{\mathrm{Gal}}(K^{\infty}/{\mathbb{Q}})$ . We also give analogous results when K/ℚ is non-abelian, the base field is not ℚ and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their “regulator constants”, and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.

On the product of twists of rank two and a conjecture of Larsen

In this note we present examples of elliptic curves and infinite parametric families of pairs of integers (d,d′) such that, if we assume the parity conjecture, we can show that E d ,E d′ and E dd′ are all of positive even rank over ℚ. As an application, we show examples where a conjecture of M. Larsen holds.

The symmetric structure of the plus/minus Selmer groups of elliptic curves over totally real fields and the parity conjecture

By improving the techniques of [B.D. Kim, The parity conjecture for elliptic curves at supersingular reduction primes, Compos. Math. 143 (2007) 47–72] we prove some symmetric structure of the minus Selmer groups of elliptic curves for supersingular primes. This structure was already known for the Selmer groups for ordinary primes [J. Nekovar, On the parity of ranks of Selmer groups II, C. R. Math. Acad. Sci. Paris Ser. I 332 (2) (2001) 99–104; J. Nekovar, Selmer complexes, Astérisque 310 (2006)]. One consequence is the parity conjecture over a totally real field under some conditions.

On the Parity Conjecture for Multiple $L$-values of Conductor Four

On the Parity Conjecture for Multiple $L$-values of Conductor Four

Parity of ranks for elliptic curves with a cyclic isogeny

Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p ⩾ 3 and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E / K , in particular confirming the parity conjecture for such curves. We prove the analogous results for p = 2 under the additional assumption that E is not supersingular at primes above 2.

On the parity of ranks of Selmer groups. III.

We show that the parity conjecture for Selmer groups is invariant under deformation in p -adic families of self-dual pure Galois representations satisfying Panciskin's condition at all primes above p .

The parity conjecture for elliptic curves at supersingular reduction primes

In number theory, the Birch and Swinnerton-Dyer (BSD) conjecture for a Selmer group relates the corank of a Selmer group of an elliptic curve over a number field to the order of zero of the associated is a good supersingular reduction prime.

Root numbers and ranks in positive characteristic

For a global field K and an elliptic curve E η over K ( T ) , Silverman's specialization theorem implies rank ( E η ( K ( T ) ) ) ⩽ rank ( E t ( K ) ) for all but finitely many t ∈ P 1 ( K ) . If this inequality is strict for all but finitely many t, the elliptic curve E η is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q , and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q ( T ) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K = κ ( u ) over any finite field κ with characteristic ≠ 2 , we construct an explicit 2-parameter family E c , d of non-isotrivial elliptic curves over K ( T ) (depending on arbitrary c , d ∈ κ × ) such that, under the parity conjecture, each E c , d has elevated rank.

Ranks of elliptic curves

This paper gives a general survey of ranks of elliptic curves over the field of rational numbers. The rank is a measure of the size of the set of rational points. The paper includes discussions of the Birch and Swinnerton-Dyer Conjecture, the Parity Conjecture, ranks in families of quadratic twists, and ways to search for elliptic curves of large rank.

Some examples of 5 and 7 descent for elliptic curves over Q

We perform descent calculations for the families of elliptic curves over Q with a rational point of order n = 5 or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over Q may become arbitrarily large.