On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions

@article{Lei2020OnTM,
  title={On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions},
  author={Antonio Lei and Gautier Ponsinet},
  journal={Proceedings of the American Mathematical Society, Series B},
  year={2020}
}
<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a number field unramified at an odd prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml… 
View PDF on arXiv
8 Citations
Highly Influential Citations
2
Background Citations
2

8 Citations

Mordell–Weil ranks and Tate–Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions
Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text] where [Formula: see text] splits completely. Suppose that [Formula: see text] has good reduction at all…
On the control theorem for fine Selmer groups and the growth of fine Tate-Shafarevich groups in $\mathbb{Z}_p$-extensions
Let $A$ be an abelian variety defined over a number field $F$. We prove a control theorem for the fine Selmer group of the abelian variety $A$ which essentially says that the kernel and cokernel of…
On the Mordell-Weil Ranks of supersingular abelian varieties over $\mathbb{Z}_p^2$-extensions
Let p be a fixed odd prime and let K be an imaginary quadratic field in which p splits. Let A be an abelian variety defined over K with supersingular reduction at both primes above p in K. Under…
Bounding the Iwasawa invariants of Selmer groups
Abstract We 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…
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…
Chromatic Selmer groups and arithmetic invariants of elliptic curves
Chromatic Selmer groups are modified Selmer groups with local information for supersingular primes p. We sketch their role in establishing the p-primary part of the Birch–Swinnerton-Dyer formula in…
Functional equations for supersingular abelian varieties over $\mathbf{Z}_p^2$-extensions
Let K be an imaginary quadratic field and K8 be the Z 2 p-extension of K. Answering a question of Ahmed and Lim, we show that the Pontryagin dual of the Selmer group associated to a supersingular…
Machine Learning and Data Visualization to Evaluate a Robotics and Programming Project Targeted for Women

References

SHOWING 1-10 OF 27 REFERENCES
Iwasawa theory for elliptic curves at supersingular primes: A pair of main conjectures
OnL-functions of elliptic curves and cyclotomic towers
In this paper we shall strengthen a recent theorem of Ralph Greenberg [1] concerning L-functions of elliptic curves and anticyclotomic towers. Let K be an imaginary quadratic field and M an abelian…
Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions
We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose…
Ranks of rational points of the Jacobian varieties of hyperelliptic curves
Ranks of the rational points of abelian varieties over ramified fields, and Iwasawa theory for primes with non-ordinary reduction
Limites de représentations cristallines
  • L. Berger
  • Mathematics
    Compositio Mathematica
  • 2004
Let F be the fraction field of the ring of Witt vectors over a perfect field of characteristic p (for example $F=\mathbb{Q}_p$), and let GF be the absolute Galois group of F. The main result of this…
Wach modules and Iwasawa theory for modular forms
We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f be a normalized new eigenform and p an…
Galois Theory for the Selmer Group of an Abelian Variety
This paper concerns the Galois theoretic behavior of the p-primary subgroup SelA(F)p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the…
Iwasawa Theory for $p$-adic Representations
Several years ago Mazur and Wiles proved a fundamental conjecture of Iwasawa which gives a precise link between the critical values of the Riemann zeta function (and, more generally, Dirichlet…
On the p-adic L-function of a modular form at a supersingular prime
In this paper we study the two p-adic L-functions attached to a modular form f = ∑ anq at a supersingular prime p. When ap = 0, we are able to decompose both the sum and the difference of the two…
...
...