The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeros of families of automorphic L-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups U (N ). This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeros near the central point. Iwaniec, Luo, and Sarnak studied the one-level densities of cuspidal newforms of weight k and level N . They showed in the limit as kN → ∞ that these families have one-level densities agreeing with orthogonal type for test functions with Fourier transform supported in (−2, 2). Exceeding (−1, 1) is important as the three orthogonal groups are indistinguishable for support up to (−1, 1) but are distinguishable for any larger support. We study the other family of GL 2 automorphic forms over Q: Maass forms. To facilitate the analysis, we use smooth weight functions in the Kuznetsov formula which, among other restrictions, vanish to order 2M at the origin. For test functions with Fourier transform supported inside −2 + 2 2M +1 , 2 − 2 2M +1 , we unconditionally prove the one-level density of the low-lying zeros of level 1 Maass forms, as the eigenvalues tend to infinity, agrees only with that of the scaling limit of orthogonal matrices.2010 Mathematics Subject Classification. 11M26 (primary), 11M41, 15A52 (secondary).
We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of 2-descent and 3-descent in a certain family of Mordell curves E k : y 2 = x 3 + k. As a byproduct of our methods, we show that, for every r ≥ 0, a positive proportion of curves E k have Tate-Shafarevich group with 3-rank at least r.
We give an effective proof of Faltings' theorem for curves mapping to Hilbert modular stacks over odd-degree totally real fields.We do this by giving an effective proof of the Shafarevich conjecture for abelian varieties of GL2-type over an odd-degree totally real field.We deduce for example an effective height bound for K-points on the curves Ca : x 6 + 4y 3 = a 2 (a ∈ K × ) when K is odd-degree totally real.(Over Q all hyperbolic hyperelliptic curves admit an étale cover dominating C1.) Introduction.Faltings' theorem is one of the classic ineffective results in mathematics. His (first) method of proof is roughly to realize, following Parshin, a given curve inside a moduli space, and then to prove a finiteness result for integral points on that moduli space (compactness provides integrality). He is forced to work with the entire moduli space of principally polarized abelian varieties because of a construction of Kodaira that he invokes.This forces him to consider quite general Galois representations. We will deal only with Galois representations valued in GL 2 by working only with curves mapping to Hilbert modular varieties 1 (or, slightly more generally, curves over which there is a non-isotrivial family of GL 2 -type abelian varieties). We will further put ourselves in a situation where it is known that there are motives attached to the conjecturally corresponding automorphic forms by working only over odd-degree totally real fields.In that situation we will prove that all relevant abelian varieties become modular over a computable 2 finite list of odd-degree totally real extensions, and then deduce a height bound on all such abelian varieties using the usual construction 1 We will use this terminology for the fine moduli spaces, which are, without level structures imposed, not varieties but stacks.2 We remind the reader that to say that a quantity is computable, or, equivalently, effectively computable, is to say that there is a Turing machine that terminates on all inputs and which, on input the relevant parameters -in this case the base number field and bounds on the dimensions and conductors of the relevant abelian varieties -outputs said quantity.
The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of L-functions near the central point (as the conductors tend to zero) agrees with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo and Sarnak proved that the behavior of zeros near the central point of holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions φ . We prove similar results for families of cuspidal Maass forms, the other natural family of GL 2 /Q L-functions. For suitable weight functions on the space of Maass forms, the limiting behavior agrees with the expected orthogonal group. We prove this for supp( φ ) ⊆ (−3/2, 3/2) when the level N tends to infinity through the square-free numbers; if the level is fixed the support decreases to being contained in (−1, 1), though we still uniquely specify the symmetry type by computing the 2-level density.
In this paper, we show that the second moment of the number of integral points on elliptic curves over Q is bounded. In particular, we prove that, for any 0 < s < log 2 5 = 2.3219 . . ., the s-th moment of the number of integral points is bounded for many families of elliptic curvese.g., for the family of all integral short Weierstrass curves ordered by naive height, for the family of only minimal such Weierstrass curves, for the family of semistable curves, or for subfamilies thereof defined by finitely many congruence conditions. For certain other families of elliptic curves, such as those with a marked point or a marked 2-torsion point, the same methods show that for 0 < s < log 2 3 = 1.5850 . . ., the s-th moment of the number of integral points is bounded.The main new ingredient in our proof is an upper bound on the number of integral points on an affine integral Weierstrass model of an elliptic curve depending only on the rank of the curve and the number of square divisors of the discriminant. We obtain the bound by studying a bijection first observed by Mordell between integral points on these curves and certain types of binary quartic forms. The theorems on moments then follow from Hölder's inequality, analytic techniques, and results on bounds on the average sizes of Selmer groups in the families.Remark 1.2. If v p (∆ A,B ) = 2 or 3, the factor in Theorem 1.1 for p may be improved to 4 (rather than 5). This results from a slightly more careful analysis of the p-adic argument at the end of Bombieri-Schmidt [BS87]; see Remark 3.3.
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