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Arithmetic moduli of elliptic curves
This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova"
P-ADIC Properties of Modular Schemes and Modular Forms
This expose represents an attempt to understand some of the recent work of Atkin, Swinnerton-Dyer, and Serre on the congruence properties of the q-expansion coefficients of modular forms from the
Zeroes of zeta functions and symmetry
Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence
Random matrices, Frobenius eigenvalues, and monodromy
Statements of the main results Reformulation of the main results Reduction steps in proving the main theorems Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants
Rigid Local Systems
Riemann introduced the concept of a "local system" on P1-{a finite set of points} nearly 140 years ago. His idea was to study "n"th order linear differential equations by studying the rank "n" local
Gauss Sums, Kloosterman Sums, And Monodromy Groups
The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's
Nilpotent connections and the monodromy theorem: Applications of a result of turrittin
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Exponential sums and di?erential equations
This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. These areas are the theory of linear differential
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