In this note, we provide evidence for a certain twisted version of the parity conjecture for Jacobians, introduced in prior work of V. Dokchitser, Green, Konstantinou and the author. To do this, we use arithmetic duality theorems for abelian varieties to study the determinant of certain endomorphisms acting on p-infinity Selmer groups.
This paper is the same as ANT-0265, but with a few minor mistakes corrected. Let E be an elliptic curve over Q with good ordinary reduction at a prime p. We show that the parity of the (co)-rank of the p-Selmer group of E is as predicted by the conjecture of Birch and Swinnerton-Dyer.
Let $p$ be an odd prime, and let $K/K_0$ be a quadratic extension of number fields. Denote by $K_\pm$ the maximal $\mathbb{Z}_p$-power extensions of $K$ that are Galois over $K_0$, with $K_+$ abelian over $K_0$ and $K_-$ dihedral over $K_0$. In this paper we show that for a Galois representation over $K_0$ satisfying certain hypotheses, if it has odd Selmer rank over $K$ then for one of $K_\pm$ its Selmer rank over $L$ is bounded below by $[L:K]$ for $L$ ranging over the finite subextensions of $K$ in $K_\pm$. Our method or proof generalizes a method of Mazur--Rubin, building upon results of N…
In section 2, we show a $p$-parity result in a $D_{2p^{n}}$-extension of number fields $L/K$ ($p\geq 5$) for the twist $1\oplus η\oplus τ$: W(E/K,1\oplus η\oplus τ)=(-1)^{< 1\oplusη\oplus 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 abov…
We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a generalisation of a formula of Kramer and Tunnell relating local invariants of the curve, which may be of independent interest. A new feature of this generalisation is the appearance of terms which govern whether or not the Cassels--Tate pairing on the Jacobian is alternating, which first appeared in work of Poonen--Stoll. We establish the local formula in many ins…
Let $p$ be an odd prime number. Let $K$ be the $p$-th cyclotomic field and $F$ its maximal real subfield. We give general formulae of the root numbers of the Jacobian varieties of the Fermat curves $X^p+Y^p=δ$ where $δ$ is an integer. As an application of these general formulae, we derive the equidistribution of the root numbers for the families of Jacobian varieties of the Fermat curves. When $p\nmid δ$, we bound the Selmer groups of these Jacobian varieties. Moreover, if $p$ is regular and all prime ideals of $K$ dividing $δ$ are inert in $K/F$, the Selmer groups are explicitly determined an…
We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the ``independence'' of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.
Let $p$ be an odd prime. We attach appropriate signed Selmer groups to an elliptic curve $E$, where $E$ is assumed to have semistable reduction at all primes above $p$. We then compare the Iwasawa $λ$-invariants of these signed Selmer groups for two congruent elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension in the spirit of Greenberg-Vatsal and B. D. Kim. As an application of our comparsion formula, we show that if the $p$-parity conjecture is true for one of the congruent elliptic curves, then it is also true for the other elliptic curve. In the midst of proving this latter result…
We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we determine the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun--Mazur--Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich--Tate group (if finite) may have order twice a square. In particular, the statistics for parities of 2-Selmer ranks and 2-infinity Selmer ranks need no longer agree…
Let $K/F$ be a finite Galois extension of number fields and $σ$ be an absolutely irreducible, self-dual representation of $\mathrm{Gal}(K/F)$. Let $p$ be an odd prime and consider two elliptic curves $E_1, E_2$ with good, ordinary reduction at primes above $p$ and equivalent mod-$p$ Galois representations. In this article, we study the variation of the parity of the multiplicities of $σ$ in the representation space associated to the $p^\infty$-Selmer group of $E_i$ over $K$. We also compare the root numbers for the twist of $E_i/F$ by $σ$ and show that the $p$-parity conjecture holds for the t…
We study the $2$-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set $A_E$ of 2-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it contains all integers larger than $c$ and having the same parity as $c$. We also find sufficient conditions on $A_E$ such that $A_E$ is equal to $\Z_{\ge t_E}$ for some number $t_E$. When all points in $E[2]$ are rational, we give an upper bound for $t_E$.
This note shows how to use the framework of Euler characteristic formulae to study Selmer groups of abelian varieties in certain dihedral or anticyclotomic extensions of CM fields via Iwasawa main conjectures, and in particular how to verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this setting. When the Selmer group is cotorsion with respect to the associated Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with respect to the associated Iwasawa algebra, we give a con…
For a prime $\ell$ and an abelian variety $A$ over a global field $K$, the $\ell$-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the $\mathbb{Z}_{\ell}$-corank of the $\ell^{\infty}$-Selmer group and the analytic rank agree modulo $2$. Assuming that $\mathrm{char} K > 0$, we prove that the $\ell$-parity conjecture holds for the base change of $A$ to the constant quadratic extension if $\ell$ is odd, coprime to $\mathrm{char} K$, and does not divide the degree of every polarization of $A$. The techniques involved in the proof include the étale cohomo…
We study the parity of rank of $2$-${\rm Selmer}$ groups associated to $π/3$ and $2π/3$-congruent numbers. Our second result gives some positive densities about $π/3$ and $2π/3$ non-congruent numbers which can support the even part of Goldfeld's conjecture. We give some necessary conditions such that $n$ is non $π/3$-congruent number for elliptic curves $E_n$ whose Shafarevich-Tate group is non-trivial. In the last section, we show that for $n=pq\equiv 5(resp. \ 11)\pmod{24}$, the density of non $π/3$($resp.$ $2π/3$)-congruent numbers is at least 75\%, where $p,q$ are primes.
This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]), then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r_2, the number of complex places of K. We also obtain results abo…
We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the…
Given two elliptic curves $E_1$ and $E_2$ defined over the field of rational numbers, $\mathbb{Q}$, with good reduction at an odd prime $p$ and equivalent mod $p$ Galois representation, we compare the $p$-Selmer rank, global and local root numbers of $E_1$ and $E_2$ over number fields.
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 a…
Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p odd 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.
Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that the associated $p$-adic Galois representation has large image. We establish a new and refined "Birch and Swinnerton-Dyer type" formula for Bloch-Kato Selmer groups of the central critical twist of $f$ via Kolyvagin derivatives of $L$-values instead of complex analytic or $p$-adic variation of $L$-values only under the Iwasawa main conjecture localized at the augmentation ideal. Our formula determines the exact rank and module structure of the Selmer groups and is insensitive to weight, the local behavior of $f$ at $p$, and analytic…
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 $\mathrm{Sha}(E/K)[p^\infty]$ for a prime $p$ this is equivalent to the $p$-parity conjecture: the global root number matches the parity of the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group. We complete the proof of the $p$-parity conjecture for elliptic curves that have a $p$-isogeny for $p > 3$ (the cases $p \le 3$ were known). T. and V. Dokchitser have showed …
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.
The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity 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^\infty is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of Gal(K^\infty/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 b…
Assuming specific instances of two general conjectures in arithmetic algebraic geometry (bijectivity of $p$-adic regulator maps, injectivity of $p$-adic Abel-Jacobi maps), we prove several cases of the $p$-part of the Tamagawa number conjecture ($p$-TNC) of Bloch-Kato and Fontaine-Perrin-Riou for (homological) motives of modular forms of even weight $\geq4$ in analytic rank $1$. More precisely, we prove our results for a large class of newforms $f$ and prime numbers $p$ that are ordinary for $f$ and such that the weight of $f$ is congruent to $2$ modulo $2(p-1)$. Inspired by work of W. Zhang i…
