## 43 Citations

Applications to cryptography of twisting commutative algebraic groups

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2008

Arithmetic local constants for abelian varieties with extra endomorphisms

- Mathematics
- 2016

This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than Z. We then study the growth of…

Class Numbers, Cyclic Simple Groups and Arithmetic

- Mathematics
- 2022

Here we initiate a program to study relationships between ﬁnite groups and arithmetic-geometric invariants in a systematic way. To do this we ﬁrst introduce a notion of optimal module for a ﬁnite…

Ranks of abelian varieties in cyclotomic twist families

- Mathematics
- 2021

Let A be an abelian variety over a number field F , and suppose that Z[ζn] embeds in EndF̄ A, for some root of unity ζn of order n = 3 . Assuming that the Galois action on the finite group A[1− ζn]…

Kummer Theories for Algebraic Tori and Normal Basis Problem

- Mathematics
- 2017

. We discuss the inverse Galois problem with normal basis, concerning Kummertheories for algebraic tori, in the framework of group schemes. The unit group scheme of a group algebra plays an important…

Twists of genus three curves and their Jacobians

- Mathematics
- 2008

An algebraic curve is a curve defined over by polynomial equations with coefficients in a given field. This thesis treats problems which arise from genus three curves over finite fields. An important…

Finding Large Selmer Rank via an Arithmetic Theory of Local Constants

- Mathematics
- 2005

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and…

Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion

- Mathematics
- 2011

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number eld K. We prove that the fraction of twists (of a given elliptic…

ON SELMER RANK PARITY OF TWISTS

- MathematicsJournal of the Australian Mathematical Society
- 2016

In this paper we study the variation of the $p$ -Selmer rank parities of $p$ -twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions,…

## References

SHOWING 1-10 OF 44 REFERENCES

On the arithmetic of abelian varieties

- Mathematics
- 1972

In w 1 we consider the situation: L/K is a finite separable field extension, A is an abelian variety over L, and A, is the abelian variety over K obtained from A by restriction of scalars. We study…

Isogeny Classes of Abelian Varieties with no Principal Polarizations

- Mathematics
- 2001

We provide a simple method of constructing isogeny classes of abelian varieties over certain fields k such that no variety in the isogeny class has a principal polarization. In particular, given a…

Algebraic Groups and Their Birational Invariants

- Mathematics
- 1998

Since the late 1960s, methods of birational geometry have been used successfully in the theory of linear algebraic groups, especially in arithmetic problems. This book--which can be viewed as a…

Shafarevich-Tate Groups of Nonsquare Order

- Mathematics
- 2004

Let A denote an abelian variety over ℚ. We give the first known examples in which #Ш(A/ℚ) is neither a square nor twice a square. For example, let E be the elliptic curve y2 + y = x3 — x of conductor…

Finding Large Selmer Rank via an Arithmetic Theory of Local Constants

- Mathematics
- 2005

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and…

Linear representations of finite groups

- MathematicsGraduate texts in mathematics
- 1977

Representations and characters: generalities on linear representations character theory subgroups, products, induced representation compact groups examples. Representations in characteristic zero:…

A G ] 7 A ug 2 00 5 On the Structure of Weil Restrictions of Abelian Varieties

- Mathematics
- 2008

We give a description of endomorphism rings of Weil restrictions of abelian varieties with respect to finite Galois extensions of fields. The results are applied to study the isogeny decompositions…

A note on the cyclotomic polynomial

- Mathematics
- 1964

The n -th roots of unity 1, ω, …, ω n - 1 , where ω = exp (2π i \ n ), are linearly dependent in the field Q of rationals since, for instance, their sum vanishes. We are here concerned with the…

Supersingular Abelian Varieties in Cryptology

- Mathematics, Computer ScienceCRYPTO
- 2002

This paper determines exactly which values can occur as the security parameters of supersingular abelian varieties (in terms of the dimension of theAbelian variety and the size of the finite field), and gives constructions of supersedication varieties that are optimal for use in cryptography.

Adeles and algebraic groups

- Mathematics
- 1982

I. Preliminaries on Adele-Geometry.- 1.1. Adeles.- 1.2. Adele-spaces attached to algebraic varieties.- 1.3. Restriction of the basic field.- II. Tamagawa Measures.- 2.1. Preliminaries.- 2.2. The case…