# Level Raising mod 2 and Obstruction to Rank Lowering

@article{Li2019LevelRM, title={Level Raising mod 2 and Obstruction to Rank Lowering}, author={Chao Li}, journal={International Mathematics Research Notices}, year={2019}, volume={2019}, pages={2332-2355} }

Given an elliptic curve E defined over Q, we are motivated by the 2-part of the Birch and Swinnerton-Dyer formula to study the relation between the 2-Selmer rank of E and the 2Selmer rank of an abelian variety A obtained by Ribet’s level raising theorem. For certain imaginary quadratic fields K satisfying the Heegner hypothesis, we prove that the 2-Selmer ranks of E and A over K have different parity, as predicted by the BSD conjecture. When the 2-Selmer rank of E is one, we further prove that…

## 2 Citations

On the parity of the index of ramified Heegner divisors

- Mathematics
- 2019

Let E be an elliptic curve defined over the rationals and let N be its conductor. Assume N is prime. In this paper, we prove that the index on E of the Heegner divisor of discriminant $$D=-~4N$$D=-4N…

Quadratic Twists of Abelian Varieties With Real Multiplication

- MathematicsInternational Mathematics Research Notices
- 2019

Let $F$ be a totally real number field and $A/F$ an abelian variety with real multiplication (RM) by the ring of integers $\mathcal {O}$ of a totally real field. Assuming $A$ admits an $\mathcal…

## References

SHOWING 1-10 OF 67 REFERENCES

Level raising mod 2 and arbitrary 2-Selmer ranks

- MathematicsCompositio Mathematica
- 2016

We prove a level raising mod $\ell =2$ theorem for elliptic curves over $\mathbb{Q}$ . It generalizes theorems of Ribet and Diamond–Taylor and also explains different sign phenomena compared to odd…

The ℓ-parity conjecture over the constant quadratic extension

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2017

Abstract For a prime ℓ and an abelian variety A over a global field K, the ℓ-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton–Dyer, the ℤℓ-corank of the ℓ∞-Selmer…

Ranks of twists of elliptic curves and Hilbert’s tenth problem

- Mathematics
- 2010

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has…

A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists

- Mathematics
- 2011

For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim \mathbb{F}_2 E^F(K)[2] + r_2$ for every…

Congruences between Heegner points and quadratic twists of elliptic curves

- Mathematics
- 2016

We establish a congruence formula between $p$-adic logarithms of Heegner points for two elliptic curves with the same mod $p$ Galois representation. As a first application, we use the congruence…

The Period-Index Obstruction for Elliptic Curves

- Mathematics
- 2002

Let K be a field and let E be an elliptic curve over K . Let GK be the absolute Galois group of K . The elements of the group HðGK ;EÞ are in one-to-one correspondence with isomorphism classes of…

Selmer groups and the indivisibility of Heegner points

- Mathematics
- 2014

For elliptic curves over Q, we prove the p-indivisibility of derived Heegner points for certain prime numbers p, as conjectured by Kolyvagin in 1991. Applications include the rened…

On the $p$-part of the Birch-Swinnerton-Dyer formula for multiplicative primes

- Mathematics
- 2017

Let $E/\mathbf{Q}$ be a semistable elliptic curve of analytic rank one, and let $p>3$ be a prime for which $E[p]$ is irreducible. In this note, following a slight modification of the methods of…

Quadratic twists of elliptic curves

- Mathematics
- 2013

The paper generalizes, for a wide class of elliptic curves defined over Q , the celebrated classical lemma of Birch and Heegner about quadratic twists with prime discriminants, to quadratic twists by…

Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves

- Mathematics
- 2013

Using maximal isotropic submodules in a quadratic module over Z_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of…