The rationality of quaternionic Darmon points over genus fields of real quadratic fields

Darmon points on p-adic tori and Jacobians of Shimura curves over Q were introduced in previous joint works with Rotger as generalizations of Darmon's Stark-Heegner points. In this article we study the algebraicity over extensions of a real quadratic field K of the projections of Darmon points to elliptic curves. More precisely, we prove that linear combinations of Darmon points on elliptic curves weighted by certain genus characters of K are rational over the predicted genus fields of K. This extends to an arbitrary quaternionic setting the main theorem on the rationality of Stark-Heegner points obtained by Bertolini and Darmon, and at the same time gives evidence for the rationality conjectures formulated in a joint paper with Rotger and by M. Greenberg in his article on Stark-Heegner points. In light of this result, quaternionic Darmon points represent the first instance of a systematic supply of points of Stark-Heegner type other than Darmon's original ones for which explicit rationality results are known.Comment: 34 page

An irreducibility criterion for group representations, with arithmetic applications

We prove a criterion for the irreducibility of an integral group representation \rho over the fraction field of a noetherian domain R in terms of suitably defined reductions of \rho at prime ideals of R. As applications, we give irreducibility results for universal deformations of residual representations, with a special attention to universal deformations of residual Galois representations associated with modular forms of weight at least 2.Comment: 11 page

Quaternion algebras, Heegner points and the arithmetic of Hida families

Given a newform f, we extend Howard's results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekov\'a\v{r}'s extended Selmer groups attached to suitable big Galois representations and we formulate two-variable Iwasawa main conjectures both in the definite case and in the indefinite case. Moreover, in the definite case we propose refined conjectures \`a la Greenberg on the vanishing at the critical points of (twists of) the L-functions of the modular forms in the Hida family of f living on the same branch as f.Comment: Heavily revised and shortened version, to appear in Manuscripta Mathematic

How to add a boundary condition

Given a conformal QFT local net of von Neumann algebras B_2 on the two-dimensional Minkowski spacetime with irreducible subnet A\otimes\A, where A is a completely rational net on the left/right light-ray, we show how to consistently add a boundary to B_2: we provide a procedure to construct a Boundary CFT net B of von Neumann algebras on the half-plane x>0, associated with A, and locally isomorphic to B_2. All such locally isomorphic Boundary CFT nets arise in this way. There are only finitely many locally isomorphic Boundary CFT nets and we get them all together. In essence, we show how to directly redefine the C* representation of the restriction of B_2 to the half-plane by means of subfactors and local conformal nets of von Neumann algebras on S^1.Comment: 20 page

A kinematically decoupled component in NGC4778

We present a kinematical and photometrical study of a member, NGC4778, of the nearest (z=0.0137) compact group: Hickson 62. Our analysis reveals that Hickson 62a, also known as NGC4778, is an S0 galaxy with kinematical and morphological peculiarities, both in its central regions (r < 5'') and in the outer halo. In the central regions, the rotation curve shows the existence of a kinematically decoupled stellar component, offset with respect to the photometric center. In the outer halo we find an asymmetric rotation curve and a velocity dispersion profile showing a rise on the SW side, in direction of the galaxy NGC4776.Comment: Proceedings of the first workshop of astronomy and astrophysics for student

Checking the Staats: How Long Is Too Long to Give Adequate Public Notice in Broadening Reissue Patent Applications?

A classic property rights question looms large in the field of patent law: where do the rights of inventors end and the rights of the public begin? The right of inventors to modify the scope of their claimed inventions, even after the patent issues, is in direct tension with the concepts of public notice and the public domain. The Patent Act currently permits broadening of claims so long as a reissue application demonstrating intent to broaden is filed within two years of the original patent issue. Over the years, however, this relatively straightforward statutory provision has sparked numerous disputes over its meaning and application. On September 8, 2011, the Court of Appeals for the Federal Circuit heard oral arguments or In re Staats. In this case, Apple Computer, Inc. appeals the rejection of a continuation reissue patent application. The U.S. Patent & Trademark Office and the Board of Patent Appeals and Interferences rejected the application on the grounds that Apple attempted to broaden the scope of its patent claims in a manner not “foreseeable” more than eight years after the patent first issued. Apple contends that the language of the statute and prior case law permit its interpretation, and the application should be allowed in the interest of innovation. This issue is hardly a new one—this submission highlights nearly 140 years of case law, legislative history, and statutory shaping pertaining to broadening reissues. We analyze the issues raised in the briefs from Staats, as well as the oral arguments. Finally, we discuss from a practitioner’s perspective what the Federal Circuit could do—and should do—in the field of broadening reissues