Frobenius action on Jacobians of curves over finite fields

Author / Creator

Li, Wanlin, 1991-, author

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Online

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Summary

This thesis focuses on studying the eigenvalues of the Frobenius action on the l-adic Tate modules of Jacobians of curves over finite fields. Some of the results have applications to answering ques...

This thesis focuses on studying the eigenvalues of the Frobenius action on the l-adic Tate modules of Jacobians of curves over finite fields. Some of the results have applications to answering questions in analytic number theory over function fields. The study of zeros of L-functions associated to Dirichlet characters has been a topic of interest in analytic number theory. Questions and conjectures arising there could also be studied in the function field setting. With the field of rational numbers replaced by the field of rational functions over a finite field, those questions are closely related to the study of the Frobenius action on the l-adic Tate modules of Jacobians of curves over finite fields. Chowla conjectured that the L-function of any quadratic Dirichlet character does not vanish at the central point s=1/2. Soundararajan showed that Chowla's conjecture holds for a positive proportion of quadratic characters ordered by conductor. Over the function field F_q(t), the analogous statement can be phrased but the situation can be very different. Quadratic characters correspond to hyperelliptic curves over F_q and their L-functions are closely related to the Hasse-Weil zeta functions of the curves. To construct quadratic characters whose L-functions vanish at the central point s=1/2 is equivalent to constructing hyperelliptic curves whose Jacobians admit sqrt(q) as an eigenvalue of the Frobenius action on its l-adic Tate module. Over any given finite field F_q, I use the Honda-Tate theory and other previous results to show the existence of such hyperelliptic curves which then give quadratic characters over the function field F_q(t) whose L-functions vanish at the central point s=1/2. This is in contrast with the situation over the rational numbers. Moreover, using a counting result of Poonen on the number of squarefree values of squarefree polynomials over the function field, I give a lower bound on the number of such characters which grows to infinity when the conductor is allowed to be arbitrarily large. Although the analogous statement of Chowla's conjecture does not hold over the function field, it is still believed that 100% of the quadratic characters satisfy the condition that their L-functions do not vanish at the central point s=1/2. So in order to approach this conjecture, joint with J. Ellenberg and M. Shusterman, we use the idea of reduction to give an upper bound on the number of quadratic characters whose L-functions vanish at a given point of the critical line. This upper bound gets better when the size of the constant field is large and the density of such characters goes to 0 when the size of the constant field grows to infinity. Geometrically, we realize the number of hyperelliptic curves whose Jacobians admit some fixed number as an eigenvalue of the Frobenius action on its l-torsion subgroup can be counted by the number of rational points of a twisted Hurwitz scheme over finite fields. Using an earlier result of Ellenberg--Venkatesh--Westerland on the homological stability for Hurwitz spaces, we give an upper bound on the number of rational points of the twisted Hurwitz scheme to get the result. The previous work are all related to studying Weil integers realized as Frobenius eigenvalues for curves over finite fields. From Honda-Tate theory, it is known that every Weil integer appears as a Frobenius eigenvalue for some abelian variety over finite fields. To show the same holds for Jacobian varieties, it suffices to show that every abelian variety over the finite field is covered by a Jacobian variety. This result can be deduced from Poonen's work on the Bertini theorem over finite fields. But there was not an effective bound on the dimension of the Jacobian variety with respect to the degree and dimension of the abelian variety and this is the topic of the last part of my thesis. Given an abelian variety in a projective space over a finite field, joint with J. Bruce, we show the existence of a smooth curve whose Jacobian admits a dominant map to the given abelian variety with an explicit upper bound on its genus. Applying this to simple abelian varieties combined with the theory of Honda-Tate, one can deduce the existence of smooth curves whose Jacobians admit some fixed Weil integer as an eigenvalue with an upper bound on its genus.

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