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Elliptic Curve Variants of the Least Quadratic Nonresidue Problem and Linniks Theorem

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 Publication date 2015
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and research's language is English




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Let $E_1$ and $E_2$ be $overline{mathbb{Q}}$-nonisogenous, semistable elliptic curves over $mathbb{Q}$, having respective conductors $N_{E_1}$ and $N_{E_2}$ and both without complex multiplication. For each prime $p$, denote by $a_{E_i}(p) := p+1-#E_i(mathbb{F}_p)$ the trace of Frobenius. Under the assumption of the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power $L$-functions $L(s, mathrm{Sym}^i E_1otimesmathrm{Sym}^j E_2)$ where $i,jin{0,1,2}$, we prove an explicit result that can be stated succinctly as follows: there exists a prime $p mid N_{E_1}N_{E_2}$ such that $a_{E_1}(p)a_{E_2}(p)<0$ and [ p < big( (32+o(1))cdot log N_{E_1} N_{E_2}big)^2. ] This improves and makes explicit a result of Bucur and Kedlaya. Now, if $Isubset[-1,1]$ is a subinterval with Sato-Tate measure $mu$ and if the symmetric power $L$-functions $L(s, mathrm{Sym}^k E_1)$ are functorial and satisfy GRH for all $k le 8/mu$, we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime $p mid N_{E_1}$ such that $a_{E_1}(p)/(2sqrt{p})in I$ and [ p < left((21+o(1)) cdot mu^{-2}log (N_{E_1}/mu)right)^2. ]



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We present a short, self-contained, and purely combinatorial proof of Linniks theorem: for any $varepsilon > 0$ there exists a constant $C_varepsilon$ such that for any $N$, there are at most $C_varepsilon$ primes $p leqslant N$ such that the least positive quadratic non-residue modulo $p$ exceeds $N^varepsilon$.
We compute the statistics of $SL_{d}(mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of radially projected integral vectors from a large sphere into the unit sphere. Using the above result we generalize the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo $q$ and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set $-(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+x_{4}^{2}=N$ as $Ntoinfty$, where the normalized orthogonal lattices sit in a submanifold of the moduli space of rank-$3$ discrete subgroups of $mathbb{R}^{4}$.
74 - N. A. Carella 2020
Let $p$ be a large prime, and let $kll log p$. A new proof of the existence of any pattern of $k$ consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic nonresidues $n_p$ modulo $p$ shows that $n_pll (log p)(log log p)$.
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.
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