This open access textbook presents a comprehensive treatment of the arithmetic theory of
quaternion algebras and orders, a subject with applications in diverse areas of mathematics …

Abstract The supersingular Endomorphism Ring problem is the following: given a
supersingular elliptic curve, compute all of its endomorphisms. The presumed hardness of …

Modular Forms is a graduate student-level introduction to the classical theory of modular
forms and computations involving modular forms, including modular functions and the theory …

Let K be a totally real field. By the asymptotic Fermat's Last Theorem over K we mean the
statement that there is a constant BK such that for any prime exponent p> BK, the only …

Using modularity, level lowering, and explicit computations with Hilbert modular forms,
Galois representations, and ray class groups, we show that for 3≤ d≤ 2 3, where d≠ 5, 1 7 …

We obtain additional Diophantine applications of the methods surrounding Darmon's
program for the generalized Fermat equation developed in the first part of this series of …

Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^ 3+ x^
3+(x+ 1)^ 3= z^ 2$ in integers $ x $, $ z $. The generalization $(x-1)^ k+ x^ k+ (x+ 1)^ k= z^ n …

We develop a variety of new techniques to treat Diophantine equations of the shape $ x^ 2+
D= y^ n $, based upon bounds for linear forms in $ p $-adic and complex logarithms, the …

Abstract The Langlands Programme, formulated by Robert Langlands in the 1960s and
since much developed and refined, is a web of interrelated theory and conjectures …

We exhibit a method to numerically compute power series expansions of modular forms on a
cocompact Fuchsian group, using the explicit computation of a fundamental domain and …