By Haruzo Hida

ISBN-10: 1461466563

ISBN-13: 9781461466567

ISBN-10: 1461466571

ISBN-13: 9781461466574

This ebook incorporates a certain account of the results of the author's contemporary Annals paper and JAMS paper on mathematics invariant, together with *μ*-invariant, *L*-invariant, and comparable subject matters. This booklet may be considered as an introductory textual content to the author's prior ebook *p-Adic Automorphic types on Shimura Varieties*. Written as a down-to-earth creation to Shimura kinds, this article contains many examples and purposes of the speculation that offer motivation for the reader. because it is restricted to modular curves and the corresponding Shimura kinds, this publication is not just a superb source for specialists within the box, however it is usually available to complex graduate scholars learning quantity thought. Key subject matters contain non-triviality of mathematics invariants and targeted values of *L*-functions; elliptic curves over advanced and *p*-adic fields; Hecke algebras; scheme concept; elliptic and modular curves over earrings; and Shimura curves.

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**Extra resources for Elliptic Curves and Arithmetic Invariants**

**Sample text**

We have ζlbv m Φ|ζlm (z) Φj (z) = v mod lm ∞ ζlbv m = v λ(n)(ζlm z)n n=1 (b+n)v λ(n)z n = ζlm v mod lm n>0,n≡−b lm n≥1 = lm λ(n)z n , mod which is nonconstant, and Pj is also nonconstant. 8. 8 when λ has values in F× q for a general p-power q. 3 CM Periods and L-Values We continue our outline of the principal results of this book; later we will ﬁll in all the details. This section has a twofold purpose: 22 1 Nontriviality of Arithmetic Invariants 1. To give a concise proof of the rationality and integrality result (due to Shimura [Sh6] and Katz [K2]) of the special values of (Archimedean and p-adic) arithmetic modular forms at CM points (Sects.

Serre (see [Se2]). Because of this limit principle, V (B) is complete under p-adic topology (see Sect. 4). 4 CM Elliptic Curve An elliptic curve E/A is said to have complex multiplication if End(E/A ) contains an order O of the integer ring O of an imaginary quadratic ﬁeld M ⊂ C. An order O is a subring of ﬁnite index in O. If E(C) = C/Lw has complex multiplication, we have O ·Lw ⊂ Lw , and hence O = {α ∈ M |α·Lw ⊂ Lw }. In this way, we have a representation ρ : M × → GL2 (Q) such that αw = ρ(α)w a b 0 +b for α ∈ M × .

Since γ changes the level structure φN , the deﬁnition is more involved. For that, we actually need to extend the level structure to the prime-to-p Tate module η (p) : (A(p∞) )2 ∼ E[m] ⊗Z Q so that we can let = V (p) (E) = lim ←−p m (p) (p) γ act by η → η ◦ γ. This is tantamount to introducing the structure of Shimura variety, and this will be done in Chap. 7. Deﬁne (dr f )|ρ(α)(E, φp , η (p) ) := (dr f )(E, φp ◦ αp , η (p) ◦ ρ(p) (α)) for the image αp ∈ Op of α. 33 Recall the elliptic curve E(a) with complex multiplication by an imaginary quadratic ﬁeld M in which p splits into (p) = pp.

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