We give a new construction of the moonshine module vertex operator algebra V , which was originally constructed in [FLM2]. We construct it as a framed VOA over the real number ﬁeld R. We also oﬀer ways to transform a structure of framed VOA into another framed VOA. As applications, we study the ﬁve framed VOA structures on VE8 and construct many framed VOAs including V from a small VOA. One of the advantages of our construction is that we are able to construct V as a framed VOA with a positive deﬁnite invariant bilinear form and we...
Elliptic units, which are obtained by evaluating modular units at quadratic
imaginary arguments of the Poincar´e upper half-plane, provide us with a rich
source of arithmetic questions and insights. They allow the analytic construction
of abelian extensions of imaginary quadratic fields, encode special values of zeta functions through the Kronecker limit formula, and are a prototype for
Stark’s conjectural construction of units in abelian extensions of number fields.
Our main empirical ndings are as follows. We nd that a range of nancial and macroeconomic
variables prove helpful in selecting funds that are capable of generating future alphas. In particular,
we nd evidence that a number of investment strategies (that use macroeconomic variables to
predict fund returns) generate out-of-sample alphas from 7-9%/year (after fund-level trading costs
and fees), when measured with a single-factor model, and from 12-13%/year with a four-factor
model that controls for fund exposures to size, book-to-market, and momentum.
Let p be a prime number, F a totally real ﬁeld such that [F (µp ) : F ] = 2 and [F : Q] is odd. For δ ∈ F × , let [ δ ] denote its class in F × /F ×p . In this paper, we show Main Theorem. There are inﬁnitely many classes [ δ ] ∈ F × /F ×p such that the twisted aﬃne Fermat curves Wδ : have no F -rational points. Remark. It is clear that if [ δ ] = [ δ ], then Wδ is isomorphic to Wδ over F . For any δ ∈ F × , Wδ /F has rational points locally everywhere.
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