Advanced Topics in the Arithmetic of Elliptic Curves by Joseph H. Silverman

By Joseph H. Silverman

In the advent to the 1st quantity of The mathematics of Elliptic Curves (Springer-Verlag, 1986), I saw that "the concept of elliptic curves is wealthy, diverse, and amazingly vast," and thus, "many vital issues needed to be omitted." I incorporated a short creation to 10 extra subject matters as an appendix to the 1st quantity, with the tacit knowing that at last there should be a moment quantity containing the main points. you're now preserving that moment quantity. it became out that even these ten subject matters wouldn't healthy regrettably, right into a unmarried publication, so i used to be pressured to make a few offerings. the next fabric is roofed during this ebook: I. Elliptic and modular services for the entire modular workforce. II. Elliptic curves with complicated multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron types, Kodaira-Neron type of certain fibers, Tate's set of rules, and Ogg's conductor-discriminant formulation. V. Tate's conception of q-curves over p-adic fields. VI. Neron's concept of canonical neighborhood peak functions.

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00 k=l C" A be a compact set, and let wi: z E C, wE A} and M = sup{lzl : z E C}. Since C is compact, we have E > 0 and M < 00. Let z E C, and let w E A satisfy Iwl > 2M. 1 w Z - W W 2 Z2 . _1_ ::; 21wM132 . w3 1_ ~ w 40 I. Elliptic and Modular Functions On the other hand, there are only finitely many terms in the sum with 0 < Iwl : : : 2M, and for z E C those terms all satisfy lIz I 1 Izl -1 M -1- + +1 -+ - 2. 1a] that the last series converges, which proves the series defining «(z; A) converges absolutely and uniformly on C.

Is a neighborhood of x, so {I(Tx)\Ux } xEX(l) is an open cover of X(l). Ix # 001 Let r = #I(Tx), and let gx be the holomorphic isomorphism gx : H ---+ {z E C : Izl < I}, Then the map is well defined and gives a local parameter at x. Ix = 001 We may take Tx = 00, so I(Tx) = {Tk}. Then 1/Jx (¢>( T)) = {oe27riT if ¢>( T) # 00, if ¢>(T) = 00 is well defined and gives a local parameter at x. 1. If I(Tx) = {I}, then the natural map is already a homeomorphism, so is a local parameter at x. Thus the only real complication occurs when x equals ¢>(i), ¢>(p), or ¢>(oo).

Z: A)} (d) We integrate ((z; A) around a fundamental parallelogram offset slightly so as not to contain points of A on its boundary. 5. 42 I. 5 The only pole of ( in D is a simple pole of residue 1 at z = O. ) Hence r «(z; A) dz laD = 27fi. On the other hand, using (b) we get some cancellation when computing the line integrals over opposite sides. Thus j «(z; A) dz = L1+L3 r ((a + tW2; A) W2dt + fO ((a + lo l WI + tW2; A) W2dt 1 = fo1 «(a + tW2; A) W2dt - fo1 ((a + tW2) + rl(wr)) W2dt = Similarly, Therefore -T)(Wr)W2' 43 §5.

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