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Geometry of numbers --- Convex bodies --- Lattice theory

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Geometry of numbers --- Geometrie des nombres

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This thesis is devoted to the study of certain cases of a conjecture of Greenberg and Benois on derivative of p-adic L-functions using the method of Greenberg and Stevens. We first prove this conjecture in the case of the symmetric square of a parallel weight 2 Hilbert modular form over a totally real field where p is inert and whose associated automorphic representation is Steinberg in p, assuming certain hypotheses on the conductor. This is a direct generalization of (unpublished) results of Greenberg and Tilouine. Subsequently, we deal with the symmetric square of a finite slope, elliptic, modular form which is Steinberg at p. To construct the two-variable p-adic L-function, necessary to apply the method of Greenberg and Stevens, we have to appeal to the recently developed theory of nearly overconvergent forms of Urban. We further strengthen the above result, removing the assumption that the conductor of the form is even, using the construction of the p-adic L-function by Bocherer and Schmidt. In the final chapter we recall the definition and the calculation of the algebraic L-invariant a la Greenberg-Benois, and explain how some of the above-mentioned results could be generalized to higher genus Siegel modular forms.

511.9 --- Geometry of numbers --- Theses --- Academic collection --- 511.9 Geometry of numbers

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Reihentext + Geometry of Numbers From the reviews: "The work is carefully written. It is well motivated, and interesting to read, even if it is not always easy... historical material is included... the author has written an excellent account of an interesting subject." (Mathematical Gazette) "A well-written, very thorough account ... Among the topics are lattices, reduction, Minkowski's Theorem, distance functions, packings, and automorphs; some applications to number theory; excellent bibliographical references." (The American Mathematical Monthly).

511.9 --- 511.9 Geometry of numbers --- Geometry of numbers --- Number theory. --- Geometry. --- Number Theory. --- Mathematics --- Euclid's Elements --- Number study --- Numbers, Theory of --- Algebra --- Geometry of numbers. --- Numbers, Geometry of --- Number theory

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The Geometry of Numbers presents a self-contained introduction to the geometry of numbers, beginning with easily understood questions about lattice-points on lines, circles, and inside simple polygons in the plane. Little mathematical expertise is required beyond an acquaintance with those objects and with some basic results in geometry. The reader moves gradually to theorems of Minkowski and others who succeeded him. On the way, he or she will see how this powerful approach gives improved approximations to irrational numbers by rationals, simplifies arguments on ways of representing integers as sums of squares, and provides a natural tool for attacking problems involving dense packings of spheres. An appendix by Peter Lax gives a lovely geometric proof of the fact that the Gaussian integers form a Euclidean domain, characterizing the Gaussian primes, and proving that unique factorization holds there. In the process, he provides yet another glimpse into the power of a geometric approach to number theoretic problems.

Geometry of numbers. --- Number theory. --- Number study --- Numbers, Theory of --- Algebra --- Numbers, Geometry of --- Number theory