Access structures of elliptic secret sharing schemes by Chen, Hao, Ling, San, Xing, Chaoping

“…In this correspondence, we give the access structures explicitly for the elliptic secret sharing schemes from algebraic-geometric (AG) codes associated with elliptic curves. Based on higher degree rational points on elliptic curves, we also construct some nonideal secret sharing schemes with weighted threshold structures.…”
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SPECIAL POINT PROBLEMS WITH ELLIPTIC MODULAR SURFACES by Pila, J

“…We prove a special point result for products of elliptic modular surfaces, elliptic curves, multiplicative groups and complex lines, and deduce a result about vanishing linear combinations of singular moduli and roots of unity. © University College London.…”

Modular Calabi-Yau threefolds in string compactifications by Elmi, MAJ

“…<p>All elliptic curves defined over Q are modular. This is the statement of the modularity theorem that relates arithmetic properties of an elliptic curve to a modular form for a subgroup of SL(2,Z). …”

Symmetric power functoriality for holomorphic modular forms by Newton, J, Thorne, JA

“…We establish the same result for a more general class of cuspidal Hecke eigenforms, including all those associated to semistable elliptic curves over 𝐐.…”

On class groups of imaginary quadratic fields by Wiles, A

“…Such existence theorems are useful in the arithmetic of elliptic curves and, potentially, also in certain lifting problems for reducible two‐dimensional Galois representations. …”

Explicit Coleman integration for hyperelliptic curves by Balakrishnan, J, Bradshaw, R, Kedlaya, K

“…Coleman's theory of p-adic integration figures prominently in several number-theoretic applications, such as finding torsion and rational points on curves, and computing p-adic regulators in K-theory (including p-adic heights on elliptic curves). We describe an algorithm for computing Coleman integrals on hyperelliptic curves, and its implementation in Sage.…”

Adjoint Selmer groups of automorphic Galois representations of unitary type by Newton, JJM, Thorne, JA

“…We obtain definitive results for the adjoint Selmer groups associated to non-CM Hilbert modular forms and elliptic curves over totally real fields.…”

A generalized successive resultant algorithm by Petit, C, Davenport, J, Pring, B

“…In this paper, we abstract the core SRA algorithm to arbitrary finite fields and present three instantiations of our general algorithm, one of which is novel and makes use of a series of isogenies derived from elliptic curves with sufficiently smooth order.…”

On a canonical class of Green currents for the unit sections of abelian schemes by Maillot, V, Rössler, D

“…This current generalizes the Siegel functions defined on elliptic curves. We prove generalizations of classical properties of Siegel functions, like distribution relations, limit formulae and reciprocity laws.…”

Centres of skewfields and completely faithful Iwasawa modules by Ardakov, K

“…This has an application to the study of Selmer groups of elliptic curves.…”

A non-abelian conjecture of Birch and Swinnerton-Dyer type for hyperbolic curves by Balakrishnan, J, Dan-Cohen, I, Kim, M, Wewers, S

“…For $\P^1\setminus \{0,1,\infty\}$ and the complement of the origin in semi-stable elliptic curves of rank 0, we compute the local image of global Selmer schemes, which then allows us to numerically confirm our conjecture in a wide range of cases.…”

Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry by Szendroi, B

“…The conjecture generalizes a proposal of Kontsevich relating monodromy transformations and self-equivalences, Supporting evidence is given in the case of elliptic curves, lattice-polarized K3 surfaces and Calabi-Yau threefolds. …”

Potential automorphy over CM fields by Allen, PB, Calegari, F, Caraiani, A, Gee, T, Helm, D, Le Hung, BV, Newton, J, Scholze, P, Taylor, R, Thorne, JA

“…We deduce that all elliptic curves E over F are potentially modular, and furthermore satisfy the Sato–Tate conjecture. …”

An analog of the Edwards model for Jacobians of genus 2 curves by Flynn, EV, Khuri-Makdisi, K

“…We give the explicit equations for a <strong>P</strong>³ x <strong>P</strong>³ embedding of the Jacobian of a curve of genus 2, which gives a natural analog for abelian surfaces of the Edwards curve model of elliptic curves. This gives a much more succinct description of the Jacobian variety than the standard version in <strong>P</strong>¹⁵. …”

A fixed point formula of Lefschetz type in Arakelov geometry IV: The modular height of C.M. abelian varieties by Koehler, K, Roessler, D

“…This theorem gives a formula for the Faltings height of abelian varieties with complex multiplication by a C.M. field whose Galois group over $\bf Q$ is abelian; it reduces to the formula of Chowla and Selberg in the case of elliptic curves. We show that the formula can be deduced from the arithmetic fixed point formula proved in the first paper of the series. …”

Remark on fundamental groups and effective Diophantine methods for hyperbolic curves by Kim, M

“…In this paper, we point out instead the analogy between the section conjecture and the finiteness conjecture for the Tate-Shafarevich group of elliptic curves. That is, the section conjecture should provide a terminating algorithm for finding all rational points on a hyperbolic curve equipped with a rational point.…”

The Atkin operator on spaces of overconvergent modular forms and arithmetic applications by Vonk, J, Jan Vonk

“…As an application, we explicitly construct Heegner-type points on elliptic curves. We then make a geometric study of the Atkin operator, and prove a potential semi-stability theorem for correspondences. …”

Variance of arithmetic sums and L-functions in Fq[t] by Hall, C, Keating, J, Roditty-Gershon, E

“…Our calculations apply, for example, to elliptic curves defined over F q [t].…”

Explicit Chaubauty–Kim for the split Cartan modular curve of level 13 by Balakrishnan, J, Dogra, N, Muller, J, Tuitman, J, Vonk, J

“…This is then applied to determine the rational points of the modular curve Xs (13), completing the classification of non-CM elliptic curves over Q with split Cartan level structure due to Bilu–Parent and Bilu–Parent–Rebolledo.…”

Covering collections and a challenge problem of Serre by Flynn, E, Wetherell, J

“…This is the only value of c$\le$81 for which the Fermat quartic X$^4$ + Y$^4$ = c Z$^4$ cannot be solved trivially, either by local considerations or maps to elliptic curves of rank 0, and it seems likely that our approach should give a method of attack for other nontrivial values of c.…”