More Like this

1. Base Change and Iwasawa Main Conjectures for GL2

   https://doi.org/10.1093/imrn/rnaf082  

   Burungale, Ashay; Castella, Francesc; Skinner, Christopher (April 2025, International Mathematics Research Notices)

   Abstract Let $$E$$ be an elliptic curve defined over $${\mathbb{Q}}$$ of conductor $$N$$, $$p$$ an odd prime of good ordinary reduction such that $E[p]$ is an irreducible Galois module, and $$K$$ an imaginary quadratic field with all primes dividing $Np$ split. We prove Iwasawa main conjectures for the $${\mathbb{Z}}_{p}$$-cyclotomic and $${\mathbb{Z}}_{p}$$-anticyclotomic deformations of $$E$$ over $${\mathbb{Q}}$$ and $K,$ respectively, dispensing with any of the ramification hypotheses on $E[p]$ in previous works. The strategy employs base change and the two-variable zeta element associated to $$E$$ over $$K$$, via which the sought after main conjectures are deduced from Wan’s divisibility towards a three-variable main conjecture for $$E$$ over a quartic CM field containing $$K$$ and certain Euler system divisibilities. As an application, we prove cases of the two-variable main conjecture for $$E$$ over $$K$$. The aforementioned one-variable main conjectures imply the $$p$$-part of the conjectural Birch and Swinnerton-Dyer formula for $$E$$ if $$\operatorname{ord}_{s=1}L(E,s)\leq 1$$. They are also an ingredient in the proof of Kolyvagin’s conjecture and its cyclotomic variant in our joint work with Grossi [1]. 

   more » « less
2. On Weissler’s Conjecture on the Hamming Cube I

   https://doi.org/10.1093/imrn/rnaa363  

   Ivanisvili, P; Nazarov, F (December 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract Let $$1\leq p \leq q <\infty $$ and let $$w \in \mathbb{C}$$. Weissler conjectured that the Hermite operator $$e^{w\Delta }$$ is bounded as an operator from $$L^{p}$$ to $$L^{q}$$ on the Hamming cube $$\{-1,1\}^{n}$$ with the norm bound independent of $$n$$ if and only if $$\begin{align*} |p-2-e^{2w}(q-2)|\leq p-|e^{2w}|q. \end{align*}$$It was proved in [ 1], [ 2], and [ 17] in all cases except $$2<p\leq q <3$$ and $$3/2<p\leq q <2$$, which stood open until now. The goal of this paper is to give a full proof of Weissler’s conjecture in the case $p=q$. Several applications will be presented. 

   more » « less
3. Sporadic Cubic Torsion

   https://doi.org/10.2140/ant.2021.15.1837  

   Derickx, Maarten; Etropolski, Anastassia; van Hoeij, Mark; Morrow, Jackson S.; Zureick-Brown, David. (November 2021, Algebra and number theory)

   Let K be a number field, and let E/K be an elliptic curve over K. The Mordell–Weil theorem asserts that the K-rational points E(K) of E form a finitely generated abelian group. In this work, we complete the classification of the finite groups which appear as the torsion subgroup of E ( K ) for K a cubic number field. To do so, we determine the cubic points on the modular curves X1(N) for N = 21,22,24,25,26,28,30,32,33,35,36,39,45,65,121. As part of our analysis, we determine the complete lists of N for which J0(N), J1(N), and J1(2,2N) have rank 0. We also provide evidence to a generalized version of a conjecture of Conrad, Edixhoven, and Stein by proving that the torsion on J1(N)(Q) is generated by Galois-orbits of cusps of X1(N) for N ≤55, N ̸=54. 

   more » « less
4. On the nonvanishing of generalised Kato classes for elliptic curves of rank 2

   https://doi.org/10.1017/fms.2021.85  

   Castella, Francesc; Hsieh, Ming-Lun (January 2022, Forum of Mathematics, Sigma)

   Abstract Let $$E/\mathbf {Q}$$ be an elliptic curve and $p>3$ be a good ordinary prime for E and assume that $L(E,1)=0$ with root number $+1$ (so $$\text {ord}_{s=1}L(E,s)\geqslant 2$$ ). A construction of Darmon–Rotger attaches to E and an auxiliary weight 1 cuspidal eigenform g such that $$L(E,\text {ad}^{0}(g),1)\neq 0$$ , a Selmer class $$\kappa _{p}\in \text {Sel}(\mathbf {Q},V_{p}E)$$ , and they conjectured the equivalence $$ \begin{align*} \kappa_{p}\neq 0\quad\Longleftrightarrow\quad{\textrm{dim}}_{{\mathbf{Q}}_{p}}\textrm{Sel}(\mathbf{Q},V_{p}E)=2. \end{align*} $$ In this article, we prove the first cases on Darmon–Rotger’s conjecture when the auxiliary eigenform g has complex multiplication. In particular, this provides a new construction of nontrivial Selmer classes for elliptic curves of rank 2. 

   more » « less
5. A Note on Visible Islands

   https://doi.org/10.1556/012.2022.01524  

   Leuchtner, Sophie; Nicolás, Carlos M.; Suk, Andrew (June 2022, Studia Scientiarum Mathematicarum Hungarica)

   Given a finite point set P in the plane, a subset S⊆P is called an island in P if conv(S) ⋂ P = S . We say that S ⊂ P is a visible island if the points in S are pairwise visible and S is an island in P. The famous Big-line Big-clique Conjecture states that for any k ≥ 3 and l ≥ 4, there is an integer n = n(k, l ), such that every finite set of at least n points in the plane contains l collinear points or k pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by replacing each point in a Horton set by a triple of collinear points. Hence, there are arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size 13. 

   more » « less