Read e-book online Moduli of Supersingular Abelian Varieties PDF

Algebraic Geometry

By Ke-Zheng Li, Frans Oort

Abelian forms might be labeled through their moduli. In confident attribute the constitution of the p-torsion-structure is an extra, great tool. For that constitution supersingular abelian kinds will be thought of the main unique ones. they supply a kick off point for the fantastic description of assorted buildings. For low dimensions the moduli of supersingular abelian kinds is by means of now good understood. during this publication we offer an outline of the supersingular locus in all dimensions, specifically we compute the size of it: it seems to be equivalent to Äg.g/4Ü, and we show the variety of elements as a category quantity, therefore finishing a protracted historic line the place targeted circumstances have been studied and common effects have been conjectured (Deuring, Hasse, Igusa, Oda-Oort, Katsura-Oort).

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Otherwise, STEPSIZE CONTROL FOR PATH TRACKING 27 7 Exit on branch b. When a step is successfully completed, the step success algorithm is executed. It tries to reduce the computational cost of the next step by decreasing precision or increasing the stepsize. However, the extent of these changes is limited by the necessity of conforming to Eqs. 4 and C. Moreover, to avoid acting too aggressively, which might waste computation by failing during the next correction cycle, the frequency of relaxation in precision or stepsize is limited.

156, Springer-Verlag, 1970. [HM] H. Hironaka and H. Matsumura, Formal functions and formal embeddings, J. Math. Soc. Japan 20 (1968), 52–82. R. Shafarevich, Basic Algebraic Geometry (Second, Revised and Expanded Edition), Part 1, Springer-Verlag, 1994. [S1] R. Speiser, Extending meromorphic functions on Pm ×Pn , J. Reine Angew. Math. 305 (1979), 126–133. [S2] R. Speiser, Projective varieties of low codimension in characteristic p > 0, Trans. Amer. Math. Soc. 240 (1978), 329–343. it Contemporary Mathematics Volume 496, 2009 Stepsize control for path tracking Daniel J.

Figure 2 is a graph of the log of the condition number, precision and stepsize in relation to tracking parameter t for a typical path having a near singularity and requiring the use of at least 96 bits of precision before returning to double precision. 01 0 Figure 2. Graph of the log of the condition number, precision and stepsize against the tracking parameter t 4. Conclusion Adaptive stepsize and adaptive multiprecision techniques may be employed to enhance the efficiency and especially the reliability of path-tracking methods, as with homotopy continuation.

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