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Twisted Teichmüller Curves
Release Date. Showing results. Weiss :. Theorem 4 C. On the other hand, the genus case of is non-trivial because the polarization has signature , i. In any event, we decided to include below an unified proof of the genus and cases just to show the role of played by Lyapunov exponents. For a proof and the original pictures! A consequence of this theorem is the fact that, in the language of this article of G. Nguyen and draw the closed cycles below:.
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In this picture, the waist curves of horizontal cylinders are represented by the cycles , and. The subspace in absolute homology spanned by them has dimension maximal because is a canonical symplectic basis of homology as one can see from the picture. Therefore, by G. Actually, as it was mentioned in the introduction to this post, one can say more about these Lyapunov exponents due to the recent works of D.
Zorich relating sum of exponents with slopes of divisors in moduli spaces.
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Moreover, M. Zorich and V. Delecroix available at SAGE after installation of sage-combinat package. Recall that, by definition, is a Prym eigenform, i. Since by our standing hypothesis , we can take such that and for all where is, as usual, the Galois conjugate of. In this way, we can attach a number to by computing the orbifold degree of the line bundle over and normalizing it by the Euler characteristic of.
For a characterization of in terms of intersection theory of some divisors of with , see Sections 1 and 5 of M. As it turns out, it is possible to show that is one of the Lyapunov exponents of the Kontsevich-Zorich cocycle over. As we mentioned above, the quantity admits an interpretation in terms of intersecion theory inside the Hilbert modular surface , and, as a matter of fact, the definition of can be naturally extended to any special curve Kobayashi geodesic in.
Posted in expository , math. AG , math. DS , Mathematics Tags: C. McMullen , C. Weiss , D.
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Nguyen , E. Moeller , Prym eigenforms , Prym varieties , Shimura curves , slopes of divisors , twisted diagonals , twisted Teichmueller curves. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email. This site uses Akismet to reduce spam. Learn how your comment data is processed. The Prym variety is the Abelian variety where i.
Remark 1 Notice that , so Remark 2 By Riemann-Hurwitz formula , , so that our standing hypothesis and the previous remark imply. In this way, the endomorphism ring of can be thought as We say that is self-adjoint if for all.
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Definition 1 We say that a polarized Abelian variety of complex dimension admits a real multiplication by if there exists a representation such that a is self-adjoint for all ; b is a proper subring of , i. McMullen : Theorem 2 McMullen is a closed -invariant locus of. Weiss : Theorem 4 C. Nguyen and draw the closed cycles below: In this picture, the waist curves of horizontal cylinders are represented by the cycles , and.
In any case, we are ready to say a few words on the proof of C. Share this: Twitter Facebook. Like this: Like Loading Leave a Reply Cancel reply Enter your comment here Fill in your details below or click an icon to log in:. Email required Address never made public. Name required. About Conferences Mathematical Prizes Mathematics expository guest blog math. AG math. AP math. CO math.