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1. Torsion phenomena for zero-cycles on a product of curves over a number field

   https://doi.org/10.1007/s40993-024-00519-4  

   Gazaki, Evangelia; Love, Jonathan (March 2024, Research in Number Theory)

   Abstract For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis a torsion group. In this article we consider a product$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true forX. For a product$$X=C_1\times C_2$$ $X=C_1\times C_2$of two curves over$$\mathbb {Q} $$ $Q$with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ $J_1\left(Q\right)\otimes J_2\left(Q\right)\stackrel{\epsilon }{\to }{\text{CH}}_0(C_1\times C_2)$is finite, where$$J_i$$ $J_i$is the Jacobian variety of$$C_i$$ $C_i$. Our constructions include many new examples of non-isogenous pairs of elliptic curves$$E_1, E_2$$ $E_1,E_2$with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$X=C_1\times \cdots \times C_d$$ $X=C_1\times \cdots \times C_d$for which the analogous map$$\varepsilon $$ $\epsilon$has finite image. 

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2. Quantitative Rigidity of Differential Inclusions in Two Dimensions

   https://doi.org/10.1093/imrn/rnad108  

   Lamy, Xavier; Lorent, Andrew; Peng, Guanying (May 2023, International Mathematics Research Notices)

   Abstract For any compact connected one-dimensional submanifold $$K\subset \mathbb R^{2\times 2}$$ without boundary that has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate $$\begin{align*} \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\, \textrm{d}x \leq C \int_{B_1} \operatorname{dist}^2(Du, K)\, \textrm{d}x, \qquad\forall u\in H^1(B_1;\mathbb R^2). \end{align*}$$This is an optimal generalization, for compact connected submanifolds of $$\mathbb R^{2\times 2}$$ without boundary, of the celebrated quantitative rigidity estimate of Friesecke, James, and Müller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $$K\subset{{\mathbb{R}}}^{2\times 2}$$ with respect to conformal–anticonformal decomposition, which provide a quasilinear elliptic partial differential equation satisfied by solutions of the exact differential inclusion $$Du\in K$$. We also give an example showing that no analogous result can hold true in $$\mathbb R^{n\times n}$$ for $$n\geq 3$$. 

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3. On the decisional Diffie–Hellman problem for class group actions on oriented elliptic curves

   https://doi.org/10.1007/s40993-022-00399-6  

   Castryck, Wouter; Houben, Marc; Vercauteren, Frederik; Wesolowski, Benjamin (December 2022, Research in Number Theory)

   Abstract We show how the Weil pairing can be used to evaluate the assigned characters of an imaginary quadratic order $${\mathcal {O}}$$ O in an unknown ideal class $$[{\mathfrak {a}}] \in {{\,\textrm{cl}\,}}({\mathcal {O}})$$ [ a ] ∈ cl ( O ) that connects two given $${\mathcal {O}}$$ O -oriented elliptic curves $$(E, \iota )$$ ( E , ι ) and $$(E', \iota ') = [{\mathfrak {a}}](E, \iota )$$ ( E ′ , ι ′ ) = [ a ] ( E , ι ) . When specialized to ordinary elliptic curves over finite fields, our method is conceptually simpler and often somewhat faster than a recent approach due to Castryck, Sotáková and Vercauteren, who rely on the Tate pairing instead. The main implication of our work is that it breaks the decisional Diffie–Hellman problem for practically all oriented elliptic curves that are acted upon by an even-order class group. It can also be used to better handle the worst cases in Wesolowski’s recent reduction from the vectorization problem for oriented elliptic curves to the endomorphism ring problem, leading to a method that always works in sub-exponential time. 

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4. EdSIDH: Supersingular Isogeny Die-Hellman Key Exchange on Edwards Curves

   https://doi.org/10.1007/978-3-030-05072-6_8  

   Azarderakhsh, Reza; Lang, B Elena; Jao, David; Koziel, Brian (January 2018, International Conference on Security, Privacy, and Applied Cryptography Engineering SPACE 2018: Security, Privacy, and Applied Cryptography Engineering)

   Problems relating to the computation of isogenies between elliptic curves defined over finite fields have been studied for a long time. Isogenies on supersingular elliptic curves are a candidate for quantum-safe key exchange protocols because the best known classical and quantum algorithms for solving well-formed instances of the isogeny problem are exponential. We propose an implementation of supersingular isogeny Diffie-Hellman (SIDH) key exchange for complete Edwards curves. Our work is motivated by the use of Edwards curves to speed up many cryptographic protocols and improve security. Our work does not actually provide a faster implementation of SIDH, but the use of complete Edwards curves and their complete addition formulae provides security benefits against side-channel attacks. We provide run time complexity analysis and operation counts for the proposed key exchange based on Edwards curves along with comparisons to the Montgomery form. 

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5. Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

   https://doi.org/10.1017/fmp.2022.14  

   Shankar, Ananth N.; Shankar, Arul; Tang, Yunqing; Tayou, Salim (January 2022, Forum of Mathematics, Pi)

   Abstract Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $$X_{\overline {K}}$$ has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K . 

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