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Balakrishnan, J.S.; Elkies, N.; Hassett, B.; Poonen, B.; Sutherland, A.V.; Voight, J. (Ed.)In this article, we discuss whether a single congruent number t can have two (or more) distinct corresponding triangles with the same hypotenuse. We describe and carry out computational experimentation providing evidence that this does not occur.more » « less
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Abstract We study two polynomial counting questions in arithmetic statistics via a combination of Fourier analytic and arithmetic methods. First, we obtain new quantitative forms of Hilbert’s Irreducibility Theorem for degree $$n$$ polynomials $$f$$ with $$\textrm {Gal}(f) \subseteq A_n$$. We study this both for monic polynomials and non-monic polynomials. Second, we study lower bounds on the number of degree $$n$$ monic polynomials with almost prime discriminants, as well as the closely related problem of lower bounds on the number of degree $$n$$ number fields with almost prime discriminants.more » « less
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Abstract The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ C . We develop and present a new technique to compute the expected value of tr(Θ C n ) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [ Rud10 ] and Chinis [ Chi16 ]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [ BDF + 16 ] and [ Zha ]. We extend [ BDF + 16 ] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L -functions L (1/2 + it , χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers.more » « less
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