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The special L-value of the winding quotient of level a product of two distinct primes
 

Summary: The special L-value of the winding quotient of level
a product of two distinct primes
Amod Agashe
Abstract
Let p and q be two distinct primes and let Je denote the winding
quotient at level pq. We give an explicit formula that expresses the
special L-value of Je as a rational number, and interpret it in terms of
the Birch-Swinnerton-Dyer conjecture.
1 Introduction and results
Let N be a positive integer and let X0(N) denote the usual modular curve
of level N and J0(N) its Jacobian. Let {0, i} denote the projection of
the path from 0 to i in H P1
(Q) to X0(N)(C), where H is the com-
plex upper half plane. We have an isomorphism H1(X0(N), Z) R -
HomC(H0(X0(N), 1), C), obtained by integrating differentials along cy-
cles. Let e H1(X0(N), Z) R correspond to the map - {0,i}
under this isomorphism. It is called the winding element. Let T denote the
Hecke algebra, i.e., the sub-ring of endomorphisms of J0(N) generated by
the Hecke-operators Tl for primes l|N and by Up for primes p |N. We have
an action of T on H1(X0(N), Z) R. Let Ie be the annihilator of e with

  

Source: Agashe, Amod - Department of Mathematics, Florida State University

 

Collections: Mathematics