<p><b>A groundbreaking contribution to number theory that unifies classical and modern results</b><br><br>This book develops a new theory of <i>p</i>-adic modular forms on modular curves, extending Katz''s classical theory to the supersingular locus. The main novelty is to move to infinite level and extend coefficients to period sheaves coming from relative <i>p</i>-adic Hodge theory. This makes it possible to trivialize the Hodge bundle on the infinite-level modular curve by a "canonical differential" that restricts to the Katz canonical differential on the ordinary Igusa tower. Daniel Kriz defines generalized <i>p</i>-adic modular forms as sections of relative period sheaves transforming under the Galois group of the modular curve by weight characters. He introduces the fundamental de Rham period, measuring the position of the Hodge filtration in relative de Rham cohomology. This period can be viewed as a counterpart to Scholze''s Hodge-Tate period, and the two periods satisfy a Lege
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The first edition of this work has become the standard introduction to the theory of p-adic numbers at both the advanced undergraduate and beginning graduate level. This second edition includes a deeper treatment of p-adic functions in Ch. 4 to include...
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P-adic Analytic Functions describes the definition and properties of p-adic analytic and meromorphic functions in a complete algebraically closed ultrametric field.Various properties of p-adic exponential-polynomials are examined, such as the Hermite-Lindemann...
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