相关论文: Secondary terms in the first moment of $|{\rm Sel}…

We prove the existence of secondary terms of order X^{5/6} in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic fields this confirms a conjecture of Datskovsky-Wright and Roberts.…

We prove that the smoothed counting function of the set of quartic fields, satisfying any finite set of local conditions, can be written as a linear combination of $X,X^{5/6}\log X,X^{5/6}$, upto an error term of $O(X^{13/16+o(1)})$. For…

We prove the existence of secondary terms of order $X^{5/6}$ in the asymptotic formulas for the average size of the genus number of cubic fields and for the number of cubic fields with a given genus number, establishing improved error…

The leading order term for the average, over quadratic discriminants satisfying the so-called Heegner condition, of the Neron-Tate height of Heegner points on a rational elliptic curve E has been determined in [12]. In addition, the second…

We combine a sieve method together with good uniformity estimates to prove a secondary term for the asymptotic estimate of $S_3\times A$ extensions over $\mathbb{Q}$ when $A$ is an odd abelian group with minimal prime divisor greater than…

In this note, we prove a power-saving remainder term for the function counting $S_3$-sextic number fields. We also give a prediction on the second main term. We also present numerical data on counting functions for $S_3$-sextic number…

Michel proved that for a one-parameter family of elliptic curves over Q(T) with non-constant j(T) that the second moment of the number of solutions modulo p is p^2 + O(p^{3/2}). We show this bound is sharp by studying y^2 = x^3 + Tx^2 + 1.…

We prove significant power savings for the error term when counting abelian extensions of number fields (as well as the twisted version of these results for nontrivial Galois modules). In some cases over $\mathbb{Q}$, these results reveal…

For a fixed elliptic curve $E$ without complex multiplication, $a_p := p+1 - \#E(\mathbb{F}_p)$ is $O(\sqrt{p})$ and $a_p/2\sqrt{p}$ converges to a semicircular distribution. Michel proved that for a one-parameter family of elliptic curves…

We show how the Selberg $\Lambda^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, we give such an error term for the counting function of…

In this paper, we prove that when elliptic curves over $\mathbb{Q}$ are ordered by height, the second moment of the size of the $2$-Selmer group is at most $15$. This confirms a conjecture of Poonen and Rains.

We prove an asymptotic formula with a power-saving error term for a specific weighted second moment of $\mathrm{GL}(2)\times \mathrm{GL}(2)$ Rankin-Selberg $L$-function, $L(1/2,\pi\otimes \pi_0)$ over any number field $F$ where $\pi$ runs…

We study the counting function of cubic function fields. Specifically, we derive an asymptotic formula for this counting function including a secondary term and an error term of order $\mathcal{O}\big(X^{2/3+\epsilon}\big)$, which matches…

In this thesis, we prove the existence of a secondary term for the count of cubic extensions of the function field $\mathbb{F}_q(t)$ of fixed absolute norm of discriminant. We show that the number of cubic extensions with absolute norm of…

Let $\mathcal E: y^2 = x^3 + A(T)x + B(T)$ be a nontrivial one-parameter family of elliptic curves over $\mathbb{Q}(T)$, with $A(T), B(T) \in \mathbb Z(T)$, and consider the $k$\textsuperscript{th} moments $A_{k,\mathcal{E}}(p) := \sum_{t…

It is proved that, if $k\ge2$ is a fixed integer and $1 \ll H \le X/2$, then $$ \int_{X-H}^{X+H}\Delta^4_k(x)\d x \ll_\epsilon X^\epsilon\Bigl(HX^{(2k-2)/k} + H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}\Bigr), $$ where $\Delta_k(x)$ is the error…

We determine the smoothed counts of $S_4$-quartic fields with bounded discriminant, satisfying any finite specified set of local conditions, as the sum of two main terms with a power saving error term. We also prove an analogous result for…

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is \textit{nice} at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal…

The moments of the coefficients of elliptic curve L-functions are related to numerous arithmetic problems. Rosen and Silverman proved a conjecture of Nagao relating the first moment of one-parameter families satisfying Tate's conjecture to…

We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet $L$-functions, $L(1/2,\chi_d)$, and also of the $L$-functions associated to quadratic twists of an…