Search ResultsShowing 1-5 of 5
-
arXiv:2502.06078 (Published 2025-02-10)
Semi-Lie arithmetic fundamental lemma for the full spherical Hecke algebra
As an analog to the Jacquet-Rallis fundamental lemma that appears in the relative trace formula approach to the Gan-Gross-Prasad conjectures, the arithmetic fundamental lemma was proposed by Wei Zhang and used in an approach to the arithmetic Gan-Gross-Prasad conjectures. The Jacquet-Rallis fundamental lemma was recently generalized by Spencer Leslie to a statement holding for the full spherical Hecke algebra. In the same spirit, Li, Rapoport, and Zhang have recently formulated a conjectural generalization of the arithmetic fundamental lemma to the full spherical Hecke algebra. This paper formulates another analogous conjecture for the semi-Lie version of the arithmetic fundamental lemma proposed by Yifeng Liu. Then this paper produces explicit formulas for particular cases of the weighted orbital integrals in the two conjectures mentioned above, and proves the first non-trivial case of the conjecture.
-
arXiv:1608.04146 (Published 2016-08-14)
Avoiding algebraic integers of bounded house in orbits of rational functions over cyclotomic closures
Comments: 10 pagesCategories: math.NTLet $k$ be a number field with cyclotomic closure $k^{\mathrm{cyc}}$, and let $h \in k^{\mathrm{cyc}}(x)$. For $A \ge 1$ a real number, we show that \[ \{ \alpha \in k^{\mathrm{cyc}} : h(\alpha) \in \overline{\mathbb Z} \text{ has house at most } A \} \] is finite for many $h$. We also show that for many such $h$ the same result holds if $h(\alpha)$ is replaced by orbits $h(h(\cdots h(\alpha)))$. This generalizes a result proved by Ostafe that concerns avoiding roots of unity, which is the case $A=1$.
-
arXiv:1507.07122 (Published 2015-07-25)
Elliptic Curve Variants of the Least Quadratic Nonresidue Problem and Linnik's Theorem
Comments: 26 pagesCategories: math.NTLet $E_1$ 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 $L(s, \mathrm{Sym}^i E_1\otimes\mathrm{Sym}^j E_2)$ where $i,j\in\{0,1,2\}$, we prove an explicit result that can be stated succinctly as follows: there exists a prime $p\nmid N_{E_1}N_{E_2}$ 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 $I\subset[-1,1]$ 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 $p\nmid N_{E_1}$ such that $a_{E_1}(p)/(2\sqrt{p})\in I$ and \[ p < \left((21+o(1)) \cdot \mu^{-2}\log (N_{E_1}/\mu)\right)^2. \]
-
arXiv:1507.02629 (Published 2015-07-09)
Benford's Law for Traces of Frobenius on Newforms with Complex Multiplication
Comments: 6 pagesCategories: math.NTIn this paper, we present an amusing application of the equidistribution of Sato-Tate angles $\theta_p$ for newforms $f$ of positive, even weight with complex multiplication (CM) to studying Benford's Law-type behavior in the leading digits of the Frobenius traces of $f$. Specifically, if $a_f(p) = 2p^{\frac{k-1}{2}}\cos \theta_p$ denotes the trace of Frobenius of $f$ at a prime $p$ that splits in the CM field of $f$, we show that the sequence $\{a_f(p)\}$ does not obey Benford's Law with respect to arithmetic density in any base $b \ge 2$, but does satisfy Benford's Law with respect to logarithmic density. Our result builds on the work of Jameson, Thorner, and Ye, who proved an analogous result in the case when $f$ does not have complex multiplication.
-
arXiv:1506.09170 (Published 2015-06-30)
Linnik's Theorem for Sato-Tate Laws on Elliptic Curves with Complex Multiplication
Comments: 11 pagesCategories: math.NTLet $E/\mathbb{Q}$ be an elliptic curve with complex multiplication (CM), and for each prime $p$ of good reduction, let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius. By the Hasse bound, $a_E(p) = 2\sqrt{p} \cos \theta_p$ for a unique $\theta_p \in [0, \pi]$. In this paper, we prove that the least prime $p$ such that $\theta_p \in [\alpha, \beta] \subset [0, \pi]$ satisfies \[ p \ll \left(\frac{N_E}{\beta - \alpha}\right)^A, \] where $N_E$ is the conductor of $E$ and the implied constant and exponent $A > 2$ are absolute and effectively computable. Our result is an analogue for CM elliptic curves of Linnik's Theorem for arithmetic progressions, which states that the least prime $p \equiv a \pmod q$ for $(a,q)=1$ satisfies $p \ll q^L$ for an absolute constant $L > 0$.