Contemporary Mathematics Volume 463, 2008 Results of Cohen-Lenstra type for quadratic function fields Jeffrey D. Achter ABSTRACT. Consider hyperelliptic curves C of fixed genus over a finite field lF. Let L be a finite abelian group of exponent dividng N. We give an asymptotic formula in IJFI, with explicit error term, for the proportion of C for which Jac(C)[NJ(lF) ~ L. 1. Introduction Let C be a smooth, proper curve of positive genus g over a finite field lF. Its Jacobian Jac(C) is a g-dimensional abelian variety. On one hand, if an explicit model of C is chosen, then there are efficient methods for computing in the finite abelian group Jac(C)(JF). Varying the coefficients of C yields a family of groups. On the other hand, since Jac(C)(JF) is isomorphic to the class group of the function field of C, studying these groups is tantamount to analyzing the class groups of certain families of global fields, an endeavor with a rich history of its own. The groups Jac(C)(JF) are extremely useful in public-key cryptography and computational number theory. For instance, the security of ElGamal's encryption scheme relies on the difficulty of the discrete logarithm problem in Z/p: given a, bE Z/px, find e such that ae = b mod p. There is an an analogous problem in the abelian group Jac(C)(JF): given A, B E Jac(C)(JF), find e such that eA = B. One can use to this to create an encryption scheme based on Jacobian varieties. Understanding the security of such a system relies on understanding the expected structure of the group Jac(C)(JF). Such groups also arise in primality testing [3] and integer factorization [19, 20] again, results on expected divisibility properties of IJac(C)(JF)I as C varies are crucial to estimates of efficiency. The Cohen-Lenstra heuristics conjecturally describe the frequency with which a given abelian group occurs as the class group of a quadratic imaginary number field [9]. Although these heuristics remain unproven, they have inspired detailed studies of class groups of function fields. Friedman and Washington conjecturally [10] describe the probability with which a given abelian £-group occurs as the £-Sylow part of the class group of a function field drawn randomly from all, or even all hyperelliptic, function fields. (Since the function field of a hyperelliptic curve admits a presentation as JF(x)[y]/(y2 - f(x)) for some polynomial f(x) E lF[x], such fields are a good analogue for quadratic number fields.) A variety of 2000 Mathematics Subject Classification. 11G20 11R58, 52B30. @2008 American Mathematical Society
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