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By Tsvetkov V. M.

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Let x' = bx + d and define a cubic in x' with Zc2 J coefficients by a'x'3 + b'x'2 + e'x' + d' 1 = 4(ax 3 + bx2 + ex + d- (bx + d) 2) , then c' ;/= d' mod 2. (5) If E has stable reduction modulo p but does not have good reduction, then E is said to have multiplicative reduction mod­ ulo p. If E does not have stable reduction, E is said to have additive reduction modulo p. Suppose E has stable reduction modulo an odd prime p. Among the defining equations of E satisfying ( 1 . 8 ) , one of them satisfies ( l .

Define a natural right action of the group GL 2 (Z/NZ) on the functors M (N) and M (N) as follows. For

MODULAR FORMS Arithmetically, the solution to the same problem is the link be­ tween elliptic curves and modular forms. It is a coincidence, I suppose, that the same object, modular curves, appears in both aspects. As a first example of modular curves, we introduce the j-line. 1 . Let S be a scheme over Q, and let E be an ellip­ tic curve over S. The j -invariant JE E f(S, 0) is a regular function satisfying the following condition. 2) . 2. Let S be any scheme over Q . 2) . (2) If two elliptic curves E and E' over S are isomorphic, then JE = )E1 .

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A 2-extension of the field of rational numbersof rational numbers by Tsvetkov V. M.

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