Recovering algebraic curves from L-functions of Hilbert class fields
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RESEARCH
Recovering algebraic curves from L-functions of Hilbert class fields Jeremy Booher
and José Felipe Voloch
* Correspondence:
[email protected] School of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
∗
Abstract In this paper, we prove that a smooth hyperbolic projective curve over a finite field can be recovered from L-functions associated to the Hilbert class field of the curve and its constant field extensions. As a consequence, we give a new proof of a result of Mochizuki and Tamagawa that two such curves with isomorphic fundamental groups are themselves isomorphic. Keywords: L-functions, Curves over finite fields, Fundamental groups
1 Introduction It has long been known that the zeta function of a global field does not determine the field uniquely (e.g. for function fields, take isogenous, but non-isomorphic, elliptic curves). Recently, there has been work done to recover a global field from more refined invariants of a similar nature. For example, [3] proves that a global field can be recovered from the collection of all its abelian L-functions. Another example is the conjecture [10, Conjecture 2.2] which predicts that the zeta functions of the Hilbert class field H(C) and successive iterates H(H(C)), . . . determines an algebraic curve C over a finite field up to Frobenius twist. These two approaches are naturally related to the classical work of Neukirch and Uchida of recovering global fields from their absolute Galois groups [12], and the more recent work of Mochizuki and Tamagawa (see [5,11]) of recovering algebraic curves from their fundamental groups, respectively. The purpose of this paper is twofold. First, we prove that a smooth proper curve of genus at least two (i.e. a proper hyperbolic curve) over a finite field can be recovered from L-functions associated to the Hilbert class field of the curve and its constant field extensions. Secondly, we show that two such curves with isomorphic fundamental groups are isomorphic. This gives a new proof of the weak isomorphism version of the theorem of Mochizuki and Tamagawa mentioned earlier. Our approach crucially depends on the work of Zilber [13,14], resolving a conjecture of Bogomolov et al. [2]. 2 L-functions Let q = pa be a prime power, and C be a smooth, projective, irreducible curve over Fq of genus at least one. A divisor D1 of degree one on C gives an Abel–Jacobi embedding of C
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into JC via P → [P − D1 ] on geometric points. Throughout, we fix a choice of degree one divisor and Abel–Jacobi embedding for each curve; a degree-one divisor exists exists by [7]. Definition 2.1 Let JC be the Jacobian of C and : JC → JC denote the Fq -Frobenius map. A Hilbert class field of C, with respect to a fixed Abel–Jacobi embedding of C into JC , is defined to be H(C) := ( − id)∗ (C) ⊂ JC . The function field of H(C) is a Hilbert class field of the function field of C in th
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