The topic of this book is the theory of degenerations of abelian varieties and its application to
the construction of compactifications of moduli spaces of abelian varieties. These …

Elliptic curves with small embedding degree and large prime-order subgroup are key
ingredients for implementing pairing-based cryptographic systems. Such “pairing-friendly” …

Fermat's problem, also ealled Fermat's last theorem, has attraeted the attention of
mathematieians far more than three eenturies. Many clever methods have been devised to …

Rigid (analytic) spaces were invented to describe degenerations, reductions, and moduli of
algebraic curves and abelian varieties. This work, a revised and greatly expanded new …

Let p be a prime number and let f be a modular form of level N with p6jN, weight k and
character", with coe cients in Fp. Suppose that f is an eigenform for all Hecke operators Tl, l …

In this paper, we describe an algorithm to compute chains of (2, 2)-isogenies between
products of elliptic curves in the theta model. The description of the algorithm is split into …

Many techniques have been used to decide whether a variety over a number field has a
rational point. Some generalize Fermat's method of infinite descent, some use quadratic …

In the early years of the 1980s, while I was visiting the Institute for Ad vanced Study (lAS) at
Princeton as a postdoctoral member, I got a fascinating view, studying congruence modulo a …

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We survey the history of the Tate conjecture on algebraic cycles. The conjecture is closely
related with other big problems in arithmetic and algebraic geometry, including the Hodge …