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Register a new userOn the $lambda$-invariant of Selmer groups arising from certain quadratic twists of Gross curves
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Jianing Li
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Let $q$ be a prime with $q equiv 7 mod 8$, and let $K=mathbb{Q}(sqrt{-q})$. Then $2$ splits in $K$, and we write $mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_infty$ be the unique $mathbb{Z}_2$-extension of $K$ unramified outside $mathfrak{p}$ with $n$-th layer $K_n$. For certain quadratic extensions $F/K$, we prove a simple exact formula for the $lambda$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $mathfrak{p}$ of the field $F_infty = FK_infty$. Equivalently, our result determines the exact $mathbb{Z}_2$-corank of certain Selmer groups over $F_infty$ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to $K$ of the Gross curve with complex multiplication defined over the Hilbert class field of $K$. We also discuss computations of the associated Selmer groups over $K_n$ in the case when the $lambda$-invariant is equal to $1$.
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The Mordell-Weil groups $E(mathbb{Q})$ of elliptic curves influence the structures of their quadratic twists $E_{-D}(mathbb{Q})$ and the ideal class groups $mathrm{CL}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) in mathbb{Z}^2$, we define a family of homomorphisms $Phi_{u,v}: E(mathbb{Q}) rightarrow mathrm{CL}(-D)$ for particular negative fundamental discriminants $-D:=-D_E(u,v)$, which we use to simultaneously address questions related to lower bounds for class numbers, the structures of class groups, and ranks of quadratic twists. Specifically, given an elliptic curve $E$ of rank $r$, let $Psi_E$ be the set of suitable fundamental discriminants $-D<0$ satisfying the following three conditions: the quadratic twist $E_{-D}$ has rank at least 1; $E_{text{tor}}(mathbb{Q})$ is a subgroup of $mathrm{CL}(-D)$; and $h(-D)$ satisfies an effective lower bound which grows asymptotically like $c(E) log (D)^{frac{r}{2}}$ as $D to infty$. Then for any $varepsilon > 0$, we show that as $X to infty$, we have $$#, left{-X < -D < 0: -D in Psi_E right } , gg_{varepsilon} X^{frac{1}{2}-varepsilon}.$$ In particular, if $ell in {3,5,7}$ and $ell mid |E_{mathrm{tor}}(mathbb{Q})|$, then the number of such discriminants $-D$ for which $ell mid h(-D)$ is $gg_{varepsilon} X^{frac{1}{2}-varepsilon}.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2.
This paper has been withdrawn Any real number $x$ in the unit interval can be expressed as a continued fraction $x=[n_1,...,n_{_N},...]$. Subsets of zero measure are obtained by imposing simple conditions on the $n_{_N}$. By imposing $n_{_N}le m forall Nin zN$, Jarnik defined the corresponding sets $E_m$ and gave a first estimate of $d_H(E_m)$, $d_H$ the Hausdorff dimension. Subsequent authors improved these estimates. In this paper we deal with $d_H(E_m)$ and $d_H(F_m)$, $F_m$ being the set of real numbers for which ${sum_{i=1}^N n_iover N}le m$.
We obtain the asymptotic formula with an error term $O(X^{frac{1}{2} + varepsilon})$ for the smoothed first moment of quadratic twists of modular $L$-functions. We also give a similar result for the smoothed first moment of the first derivative of quadratic twists of modular $L$-functions. The argument is largely based on Youngs recursive method [19,20].
Let $k$ be a field of characteristic $q$, $cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$ or $>2d+1$. Let $p e q$ be a prime number and $cactocac$ a finite geometrically textsc{Galois} and etale cover defined over $k$ with function field $K:=k(cac)$. Let $(tau,B)$ be the $K/k$-trace of $A/K$. We give an upper bound for the $bbz_p$-corank of the textsc{Selmer} group $text{Sel}_p(Atimes_KK)$, defined in terms of the $p$-descent map. As a consequence, we get an upper bound for the $bbz$-rank of the textsc{Lang-Neron} group $A(K)/tauB(k)$. In the case of a geometric tower of curves whose textsc{Galois} group is isomorphic to $bbz_p$, we give sufficient conditions for the textsc{Lang-Neron} group of $A$ to be uniformly bounded along the tower.
In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $Gamma_0(N)^+$, where $N>1$ is a square-free integer. After we prove that $Gamma_0(N)^+$ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an average Weyls law for the distribution of eigenvalues of Maass forms, from which we prove the classical Weyls law as a special case. The groups corresponding to $N=5$ and $N=6$ have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $Gamma_0(5)^+$ than for $Gamma_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyls laws. In addition, we employ Hejhals algorithm, together with recently developed refinements from [31], and numerically determine the first $3557$ of $Gamma_0(5)^+$ and the first $12474$ eigenvalues of $Gamma_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.
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