By Spencer J. Bloch
This e-book is the long-awaited booklet of the well-known Irvine lectures. brought in 1978 on the collage of California at Irvine, those lectures became out to be an access element to a number of intimately-connected new branches of mathematics algebraic geometry, comparable to regulators and distinct values of L-functions of algebraic kinds, specific formulation for them when it comes to polylogarithms, the idea of algebraic cycles, and at last the overall thought of combined reasons which unifies and underlies all the above (and a lot more). within the twenty years on account that, the significance of Bloch's lectures has now not lowered. A fortunate workforce of individuals operating within the above parts had the great fortune to own a duplicate of previous typewritten notes of those lectures. Now all people may have their very own replica of this vintage paintings.
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Additional info for Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves (CRM Monograph Series)
In the last part of this section, we obtain some preliminary results and make some conjectures as to what should be the case. 1 Definition To define the U operator, we start with the Frobenius endomorphism Frob : V ~ V, which, as was shown in the last section, is locally free of rank p. 3. The U Operator 43 a trace h o m o m o r p h i s m TrFrob : V ~V, defined by (TrFrobf)(E/A, T, z) : ~ f(E1, ~1, zl), where the sum is taken over the triples (E1,~'1,*1) which m a p (by quotient by the f u n d a m e n t a l subgroup) to the given triple (E,T,z).
Note, however, that the difficulty disappears if we consider all weights together, since it is easy to see that any m o d u l a r form of weight 1 over k will always have the same q-expansion as the reduction of some m o d u l a r form of weight p = 1 + (p - 1) (just multiply by Ep-1 and note that the reduction m a p for m o d u l a r forms of weight greater t h a n one is onto). C h a p t e r II T h e H e c k e and U O p e r a t o r s In this chapter we define p-adic versions of the classical Hecke operators.
I=2 with ord(a) > 0 (because E is supersingular), ord(ci) _> 1 for i ~ l(modp), and ord(cv) = 0 (because the formal group is of height 2). Note that since a (rood p) for any nonvanishing differential w on E, we have that, if ord(a) < 1, then ord(a) = ord(Ev_l(E,w)). We want to determine the curves (if any) that are mapped to E by quotient by their fundamental group. T h e o r e m I I . 3 . 5 Let 0 < ord(a) < p/(1 + p), so that the canonical subgroup H0 C E is defined, and let H1, H2, . . , Hp be the other finite fiat subgroup schemes of rank p orE.
Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves (CRM Monograph Series) by Spencer J. Bloch