Good Ordinary Reduction Research Articles

On the parity conjecture for abelian surfaces

AbstractAssuming finiteness of the Tate–Shafarevich group, we prove that the Birch–Swinnerton–Dyer conjecture correctly predicts the parity of the rank of semistable principally polarised abelian surfaces. If the surface in question is the Jacobian of a curve, we require that the curve has good ordinary reduction at 2‐adic places.

Weak approximation for 0-cycles on a product of elliptic curves

In the 1980’s Colliot-Thélène, Sansuc, Kato and S. Saito proposed conjectures related to local-to-global principles for 0-cycles on arbitrary smooth projective varieties over a number field. We give some evidence for these conjectures for a product \(X=E_1\times E_2\) of two elliptic curves. In the special case when \(X=E\times E\) is the self-product of an elliptic curve E over \(\mathbb {Q} \) with potential complex multiplication, we show that the places of good ordinary reduction are often involved in a Brauer–Manin obstruction for 0-cycles over a finite base change. We give many examples when these 0-cycles can be lifted to global ones.

On the soft p-converse to a theorem of Gross–Zagier and Kolyvagin

We give a proof of a soft version of the p-converse to a theorem of Gross–Zagier and Kolyvagin for non-CM elliptic curves with good ordinary reduction at \(p >3\) under the irreducibility assumption on the residual representation. In particular, no condition on the conductor is imposed. Combining with the known results, we obtain the equivalence for every elliptic curve E over \(\mathbb {Q}\).

Cyclic length in the Brauer group of a p-adic elliptic curve

Let K be a finite extension of , and E be an elliptic curve over K with good ordinary reduction. We study the cyclic length of the p-primary part of the Brauer group of E. In particular, for , we show that all elements in the p-torsion of are p-cyclic whenever p divides the number of -points of the reduction of E modulo p.

P∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^\\infty $$\\end{document}-Selmer groups and rational points on CM elliptic curves

Let E/Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E/{\\mathbb {Q}}$$\\end{document} be a CM elliptic curve and p a prime of good ordinary reduction for E. We show that if Selp∞(E/Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Sel}_{p^\\infty }(E/{\\mathbb {Q}})$$\\end{document} has Zp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_p$$\\end{document}-corank one, then E(Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E({\\mathbb {Q}})$$\\end{document} has a point of infinite order. The non-torsion point arises from a Heegner point, and thus ords=1L(E,s)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{ord}\\,}}_{s=1}L(E,s)=1$$\\end{document}, yielding a p-converse to a theorem of Gross–Zagier, Kolyvagin, and Rubin in the spirit of [49, 54]. For p>3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>3$$\\end{document}, this gives a new proof of the main result of [12], which our approach extends to all primes. The approach generalizes to CM elliptic curves over totally real fields [4].

Rational points on algebraic curves in infinite towers of number fields

We study a natural question in the Iwasawa theory of algebraic curves of genus \(>1\). Fix a prime number p. Let X be a smooth, projective, geometrically irreducible curve defined over a number field K of genus \(g>1\), such that the Jacobian of X has good ordinary reduction at the primes above p. Fix an odd prime p and for any integer \(n>1\), let \(K_n^{(p)}\) denote the degree-\(p^n\) extension of K contained in \(K(\mu _{p^{\infty }})\). We prove explicit results for the growth of \(\#X(K_n^{(p)})\) as \(n\rightarrow \infty \). When the Jacobian of X has rank zero and the associated adelic Galois representation has big image, we prove an explicit condition under which \(X(K_{n}^{(p)})=X(K)\) for all n. This condition is illustrated through examples. We also prove a generalization of Imai’s theorem that applies to abelian varieties over arbitrary pro-p extensions.

On order of vanishing of characteristic elements

AbstractLetpbe a fixed odd prime. LetEbe an elliptic curve defined over a number fieldFwith either good ordinary reduction or multiplicative reduction at each prime ofFabovep. We shall study the characteristic element of the Selmer group ofEover ap-adic Lie extension. In particular, we relate the order of vanishing of these characteristic elements evaluated at Artin representations to the Selmer coranks and their twists in the intermediate subextensions of thep-adic Lie extension.

Asymptotic growth of Mordell–Weil ranks of elliptic curves in noncommutative towers

AbstractLet E be an elliptic curve defined over a number field F with good ordinary reduction at all primes above p, and let $F_\infty $ be a finitely ramified uniform pro-p extension of F containing the cyclotomic $\mathbb {Z}_p$ -extension $F_{\operatorname {cyc}}$ . Set $F^{(n)}$ be the nth layer of the tower, and $F^{(n)}_{\operatorname {cyc}}$ the cyclotomic $\mathbb {Z}_p$ -extension of $F^{(n)}$ . We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $\lambda $ -invariant of the Selmer group over $F^{(n)}_{ \operatorname {cyc}}$ as $n\rightarrow \infty $ . This method has its origins in work of A. Cuoco, who studied $\mathbb {Z}_p^2$ -extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases.

Conjecture A andμ-invariant for Selmer groups of supersingular elliptic curves

Let p be an odd prime and let E be an elliptic curve defined over a number field F with good reduction at the primes above p. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over ℤ p 2 -extensions of an imaginary quadratic field where p splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the μ-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups, when E has good supersingular reduction, which are completely analogous to the classical Selmer group, when E has good ordinary reduction. In this survey paper we do not present any proofs, however, we have tried to give references of the discussed results for the interested reader.

On anticyclotomic variants of the p-adic Birch and Swinnerton-Dyer conjecture

We formulate analogues of the Birch and Swinnerton-Dyer conjecture for the p-adic L-functions of Bertolini, Darmon, and Prasanna attached to elliptic curves E/Q at primes p of good ordinary reduction. Using Iwasawa theory, we then prove, under mild hypotheses, one of the inequalities predicted by the “rank part” of our conjectures, as well as the predicted leading coefficient formula, up to a p-adic unit.

Euler characteristics and their congruences for multisigned Selmer groups

AbstractThe notion of the truncated Euler characteristic for Iwasawa modules is a generalization of the the usual Euler characteristic to the case when the Selmer groups are not finite. Let p be an odd prime, $E_{1}$ and $E_{2}$ be elliptic curves over a number field F with semistable reduction at all primes $v|p$ such that the $\operatorname {Gal}(\overline {F}/F)$ -modules $E_{1}[p]$ and $E_{2}[p]$ are irreducible and isomorphic. We compare the Iwasawa invariants of certain imprimitive multisigned Selmer groups of $E_{1}$ and $E_{2}$ . Leveraging these results, congruence relations for the truncated Euler characteristics associated to these Selmer groups over certain $\mathbb {Z}_{p}^{m}$ -extensions of F are studied. Our results extend earlier congruence relations for elliptic curves over $\mathbb {Q}$ with good ordinary reduction at p.

A proof of Perrin-Riou’s Heegner point main conjecture

Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $p>3$ be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate-Shafarevich group of $E$ over the anticyclotomic $\mathbf{Z}_p$-extension of $K$ in terms of Heegner points. In this paper, we give a proof of Perrin-Riou's conjecture under mild hypotheses. Our proof builds on Howard's theory of bipartite Euler systems and Wei Zhang's work on Kolyvagin's conjecture. In the case when $p$ splits in $K$, we also obtain a proof of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna.

The epsilon constant conjecture for higher dimensional unramified twists of (1)

AbstractLet $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$ , which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ . In this paper, we consider the $\mathrm {Gal}(N/K)$ -equivariant local $\varepsilon $ -conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$ . For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$ . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

Zero Cycles on a Product of Elliptic Curves Over a p-adic Field

Abstract We consider a product $X=E_1\times \cdots \times E_d$ of elliptic curves over a finite extension $K$ of ${\mathbb{Q}}_p$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has supersingular reduction. Under these assumptions, we prove that the Albanese kernel of $X$ is the direct sum of a finite group and a divisible group, extending work by Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map $CH_0(X)/p^n\rightarrow H^{2d}_{\acute{\textrm{e}}\textrm{t}}(X, \mu _{p^n}^{\otimes d})$. We give specific criteria that guarantee this map is injective for every $n\geq 1$. When all curves have good ordinary reduction, we show that it suffices to extend to a specific finite extension $L$ of $K$ for these criteria to be satisfied. This extends previous work by Yamazaki and Hiranouchi.

Galois symbol maps for abelian varieties over a $p$-adic field

We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a $p$-adic field. As a byproduct, one can calculate the “class group” in the sense of the class field theory for curves ove

Bounding the Iwasawa invariants of Selmer groups

AbstractWe study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

On the Refined Conjectures on Fitting Ideals of Selmer Groups of Elliptic Curves with Supersingular Reduction

AbstractIn this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the “weak main conjecture” à la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime $p$. We also prove the “strong main conjecture” suggested by the second named author under the validity of the $\pm $-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among “finite layer objects”, “$\pm $-objects”, and “fine objects” in Iwasawa theory. The case of good ordinary reduction is also treated.

Computing L-Invariants for the Symmetric Square of an Elliptic Curve

Let E be an elliptic curve over , and a prime of good ordinary reduction. The p-adic L-function for always vanishes at s = 1, even though the complex L-function does not have a zero there. The -invariant itself appears on the right-hand side of the formulawhere with . We first devise a method to calculate effectively, then show it is non-trivial for all elliptic curves E of conductor with , and almost all ordinary primes p < 17. Hence, in these cases at least, the order of the zero in at s = 1 is exactly one.

Some results about zero‐cycles on abelian and semi‐abelian varieties

AbstractIn this short note we extend some results obtained in [7]. First, we prove that for an abelian variety A with good ordinary reduction over a finite extension of with p an odd prime, the Albanese kernel of A is the direct sum of its maximal divisible subgroup and a torsion group. Second, for a semi‐abelian variety G over a perfect field k, we construct a decreasing integral filtration of Suslin's singular homology group, , such that the successive quotients are isomorphic to a certain Somekawa K‐group.

ON THE SELMER GROUP OF A CERTAIN -ADIC LIE EXTENSION

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication. Let $p\geq 5$ be a prime in $\mathbb{Q}$ and suppose that $E$ has good ordinary reduction at $p$. We study the dual Selmer group of $E$ over the compositum of the $\text{GL}_{2}$ extension and the anticyclotomic $\mathbb{Z}_{p}$-extension of an imaginary quadratic extension as an Iwasawa module.