More Like this

1. Counting imaginary quadratic fields with an ideal class group of 5-rank at least 2

   https://doi.org/10.1007/s11139-025-01184-6  

   Bartz, Kollin; Levin, Aaron; Thamminana, Aman_Dhruva (August 2025, The Ramanujan Journal)

   Abstract We prove that there are$$\gg \frac{X^{\frac{1}{3}}}{(\log X)^2}$$ $\gg \frac{X^{\frac{1}{3}}}{{(logX)}^2}$imaginary quadratic fieldskwith discriminant$$|d_k|\le X$$ $|d_k|\le X$and an ideal class group of 5-rank at least 2. This improves a result of Byeon, who proved the lower bound$$\gg X^{\frac{1}{4}}$$ $\gg X^{\frac{1}{4}}$in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus 2 curveCover$$\mathbb {Q}$$ $Q$such thatChas a rational Weierstrass point and the Jacobian ofChas a rational torsion subgroup of 5-rank 2. We deduce the main result from the existence of the curveCand a quantitative result of Kulkarni and the second author. 

   more » « less
2. Abelian varieties of prescribed order over finite fields

   https://doi.org/10.1007/s00208-024-03084-4  

   van_Bommel, Raymond; Costa, Edgar; Li, Wanlin; Poonen, Bjorn; Smith, Alexander (March 2025, Mathematische Annalen)

   Abstract Given a prime powerqand$$n \gg 1$$ $n\gg 1$, we prove that every integer in a large subinterval of the Hasse–Weil interval$$[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$$ $[{(\sqrt{q}-1)}^{2n},{(\sqrt{q}+1)}^{2n}]$is$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some ordinary geometrically simple principally polarized abelian varietyAof dimensionnover$${\mathbb {F}}_q$$ $F_q$. As a consequence, we generalize a result of Howe and Kedlaya for$${\mathbb {F}}_2$$ $F_2$to show that for each prime powerq, every sufficiently large positive integer is realizable, i.e.,$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some abelian varietyAover$${\mathbb {F}}_q$$ $F_q$. Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse–Weil interval. A separate argument determines, for fixedn, the largest subinterval of the Hasse–Weil interval consisting of realizable integers, asymptotically as$$q \rightarrow \infty $$ $q\to \infty$; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if$$q \le 5$$ $q\le 5$, then every positive integer is realizable, and for arbitraryq, every positive integer$$\ge q^{3 \sqrt{q} \log q}$$ $\ge q^{3\sqrt{q}logq}$is realizable. 

   more » « less
3. Refined Chabauty–Kim computations for the thrice-punctured line over $$\mathbb {Z}[1/6]$$

   https://doi.org/10.1007/s40993-024-00597-4  

   Lüdtke, Martin (March 2025, Research in Number Theory)

   Abstract The Chabauty–Kim method and its refined variant by Betts and Dogra aim to cut out theS-integral points$$X(\mathbb {Z}_S)$$ $X\left(Z_S\right)$on a curve inside thep-adic points$$X(\mathbb {Z}_p)$$ $X\left(Z_p\right)$by producing enough Coleman functions vanishing on them. We derive new functions in the case of the thrice-punctured line whenScontains two primes. We describe an algorithm for computing refined Chabauty–Kim loci and verify Kim’s Conjecture over$$\mathbb {Z}[1/6]$$ $Z[1/6]$for all choices of auxiliary prime $$p < 10{,}000$$ $p<10,000$. 

   more » « less
4. Universality and phase transitions in low moments of secular coefficients of critical holomorphic multiplicative chaos

   https://doi.org/10.1007/s00440-025-01443-z  

   Gu, Haotian; Zhang, Zhenyuan (November 2025, Probability Theory and Related Fields)

   Abstract We investigate the low moments$$\mathbb {E}[|A_N|^{2q}],\, 0 $E\left[\right|A_N{|}^{2q}],0<q⩽1$of secular coefficients$$A_N$$ $A_N$of the critical non-Gaussian holomorphic multiplicative chaos, i.e. coefficients of$$z^N$$ $z^N$in the power series expansion of$$\exp (\sum _{k=1}^\infty X_kz^k/\sqrt{k})$$ $exp({\sum }_{k=1}^{\infty }{X}_k{z}^k/\sqrt{k})$, where$$\{X_k\}_{k\geqslant 1}$$ ${\left\{X_k\right\}}_{k⩾1}$are i.i.d. rotationally invariant unit variance complex random variables. Inspired by Harper’s remarkable result on random multiplicative functions, Soundararajan and Zaman recently showed that if each$$X_k$$ $X_k$is standard complex Gaussian,$$A_N$$ $A_N$features better-than-square-root cancellation:$$\mathbb {E}[|A_N|^2]=1$$ $E\left[\right|A_N{|}^2]=1$and$$\mathbb {E}[|A_N|^{2q}]\asymp (\log N)^{-q/2}$$ $E\left[\right|A_N{|}^{2q}]\asymp {(logN)}^{-q/2}$for fixed$$q\in (0,1)$$ $q\in (0,1)$as$$N\rightarrow \infty $$ $N\to \infty$. We show that this asymptotics holds universally if$$\mathbb {E}[e^{\gamma |X_k|}]<\infty $$ $E\left[e^{\gamma |X_k|}\right]<\infty$for some$$\gamma >2q$$ $\gamma >2q$. As a consequence, we establish the universality for the tightness of the normalized secular coefficients$$A_N(\log (1+N))^{1/4}$$ $A_N{(log(1+N))}^{1/4}$, generalizing a result of Najnudel, Paquette, and Simm. Another corollary is the almost sure regularity of some critical non-Gaussian holomorphic chaos in appropriate Sobolev spaces. Moreover, we characterize the asymptotics of$$\mathbb {E}[|A_N|^{2q}]$$ $E\left[\right|A_N{|}^{2q}]$for$$|X_k|$$ $|X_k|$following a stretched exponential distribution with an arbitrary scale parameter, which exhibits a completely different behavior and underlying mechanism from the Gaussian universality regime. As a result, we unveil a double-layer phase transition around the critical case of exponential tails. Our proofs combine Harper’s robust approach with a careful analysis of the (possibly random) leading terms in the monomial decomposition of$$A_N$$ $A_N$. 

   more » « less
5. Towards a Schinzel–Wójcik theorem for number fields

   https://doi.org/10.1007/s40879-025-00819-8  

   Pollack, Paul (April 2025, European Journal of Mathematics)

   Abstract Schinzel and Wójcik have shown that for every$$\alpha ,\beta \in \mathbb {Q}^{\times }\hspace{0.55542pt}{\setminus }\hspace{1.111pt}\{\pm 1\}$$ $\alpha ,\beta \in Q^{\times }\backslash \{\pm 1\}$, there are infinitely many primespwhere$$v_p(\alpha )=v_p(\beta )=0$$ $v_p\left(\alpha \right)=v_p\left(\beta \right)=0$and where$$\alpha $$ $\alpha$and$$\beta $$ $\beta$generate the same multiplicative group modp. We prove a weaker result in the same direction for algebraic numbers$$\alpha , \beta $$ $\alpha ,\beta$. Let$$\alpha , \beta \in \overline{\mathbb {Q}} ^{\times }$$ $\alpha ,\beta \in {\overline{Q}}^{\times }$, and suppose$$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\alpha )|\ne 1$$ $|N_{Q(\alpha ,\beta )/Q}\left(\alpha \right)|\ne 1$and$$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\beta )|\ne 1$$ $|N_{Q(\alpha ,\beta )/Q}\left(\beta \right)|\ne 1$. Then for some positive integer$$C = C(\alpha ,\beta )$$ $C=C(\alpha ,\beta )$, there are infinitely many prime idealsPof Equation missing<#comment/>where$$v_P(\alpha )=v_P(\beta )=0$$ $v_P\left(\alpha \right)=v_P\left(\beta \right)=0$and where the group$$\langle \beta \bmod {P}\rangle $$ $⟨\beta modP⟩$is a subgroup of$$\langle \alpha \bmod {P}\rangle $$ $⟨\alpha modP⟩$with$$[\langle \alpha \bmod {P}\rangle \,{:}\, \langle \beta \bmod {P}\rangle ]$$ $[⟨\alpha modP⟩:⟨\beta modP⟩]$dividingC. A key component of the proof is a theorem of Corvaja and Zannier bounding the greatest common divisor of shiftedS-units. 

   more » « less