scholarly journals An arithmetic intersection formula for denominators of Igusa class polynomials

2015 ◽  
Vol 137 (2) ◽  
pp. 497-533 ◽  
Author(s):  
Kristin Lauter ◽  
Bianca Viray
Keyword(s):  
Class Polynomials ◽  
Intersection Formula

scholarly journals FINE DELIGNE–LUSZTIG VARIETIES AND ARITHMETIC FUNDAMENTAL LEMMAS

2019 ◽  
Vol 7 ◽  
Author(s):  
XUHUA HE ◽  
CHAO LI ◽  
YIHANG ZHU
Keyword(s):  
Fixed Points ◽  
Character Formula ◽  
Shimura Varieties ◽  
Fundamental Lemma ◽  
Residual Characteristic ◽  
Intersection Formula

We prove a character formula for some closed fine Deligne–Lusztig varieties. We apply it to compute fixed points for fine Deligne–Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport–Zink spaces arising from the arithmetic Gan–Gross–Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.

scholarly journals Transversal intersection formula for compacta

1998 ◽  
Vol 85 (1-3) ◽  
pp. 93-117 ◽  
Author(s):  
A.N. Dranishnikov ◽  
D. Repovš ◽  
E.V. Ščepin
Keyword(s):  
Transversal Intersection ◽  
Intersection Formula

scholarly journals Constructing supersingular elliptic curves with a given endomorphism ring

2014 ◽  
Vol 17 (A) ◽  
pp. 71-91 ◽  
Author(s):  
Ilya Chevyrev ◽  
Steven D. Galbraith
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Endomorphism Ring ◽  
Quaternion Algebra ◽  
Maximal Order ◽  
Running Time ◽  
Successive Minima ◽  
Supersingular Elliptic Curves ◽  
Class Polynomials ◽  
Theoretical Results

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{O}$ be a maximal order in the quaternion algebra $B_p$ over $\mathbb{Q}$ ramified at $p$ and $\infty $. The paper is about the computational problem: construct a supersingular elliptic curve $E$ over $\mathbb{F}_p$ such that ${\rm End}(E) \cong \mathcal{O}$. We present an algorithm that solves this problem by taking gcds of the reductions modulo $p$ of Hilbert class polynomials.New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice $\mathcal{O}^T$ of $\mathcal{O}$, the order $\mathcal{O}$ is effectively characterized by the three successive minima and two other short vectors of $\mathcal{O}^T\! .$ The desired conditions turn out to hold whenever the $j$-invariant $j(E)$, of the elliptic curve with ${\rm End}(E) \cong \mathcal{O}$, lies in $\mathbb{F}_p$. We can then prove that our algorithm terminates with running time $O(p^{1+\varepsilon })$ under the aforementioned conditions.As a further application we present an algorithm to simultaneously match all maximal order types with their associated $j$-invariants. Our algorithm has running time $O(p^{2.5 + \varepsilon })$ operations and is more efficient than Cerviño’s algorithm for the same problem.

scholarly journals Computing Class Polynomials for Abelian Surfaces

2014 ◽  
Vol 23 (2) ◽  
pp. 129-145 ◽  
Author(s):  
Andreas Enge ◽  
Emmanuel Thomé
Keyword(s):  
Abelian Surfaces ◽  
Class Polynomials

scholarly journals Denominators of Igusa class polynomials

10.5802/pmb.6 ◽  
2015 ◽  
pp. 5-29 ◽  
Author(s):  
Kristin Lauter ◽  
Bianca Viray
Keyword(s):  
Class Polynomials

scholarly journals Twists of Shimura Curves

2014 ◽  
Vol 66 (4) ◽  
pp. 924-960 ◽  
Author(s):  
James Stankewicz
Keyword(s):  
Local Field ◽  
Rational Numbers ◽  
Shimura Curves ◽  
Shimura Curve ◽  
Modulo P ◽  
First Time ◽  
Class Polynomials

AbstractConsider a Shimura curve XD0(N) over the rational numbers. We determine criteria for the twist by an Atkin–Lenher involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan and Livné on Qp points when p |D and for the first time give criteria for Qp points when p |N. We also give congruence conditions for roots modulo p of Hilbert class polynomials.

Sign in / Sign up

Export Citation Format

Share Document