arXiv subjects

Levent Alpöge

Publications and source records attributed to Levent Alpöge.

12 recordsLinked to original sources

arXiv2026

Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing

By adapting the techniques used in the IPM blowup result of Córdoba-Martínez-Zoroa with spatially smooth force, we prove finite-time blow-up for the IPM equation on $\mathbb T^2$ with a uniformly spacetime smooth force. In particular, we show there exist a smooth odd initial density, a smooth odd force $F\in C^\infty([0,1]\times\mathbb T^2)$, and a classical solution $ρ$ on $[0,1)$ whose density gradient and spatial velocity gradient diverge in $L^\infty$ as $t\uparrow 1$. Nevertheless, $ρ(t)$ converges in $C^η$ for every $0\leqη<1$.

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arXiv2026

More than two thirds of the zeta zeros are simple and on the critical line

We prove unconditionally that at least two thirds of the nontrivial zeros of the Riemann zeta function, counted with multiplicity, are simple and lie on the critical line, and that at least five sixths are distinct; the previous unconditional records are $\frac{5}{12}$ and $0.6603$. With the Montgomery--Taylor window the constants become $0.6725$ and $0.8362$. The argument makes Montgomery's 1973 deduction unconditional: the Riemann hypothesis, classically needed to read the zero side as a positive sum over real ordinates, is replaced by a rank-trace inequality applied to a finite compression of Weil's Hermitian form, with Sylvester's law of inertia handling off-line pairs. The analytic inputs are those of Aryan and of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh. The results extend to primitive Dirichlet $L$-functions and are formally verified in Lean 4.

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arXiv2025

Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field

We show that for any quadratic extension of number fields $K/F$, there exists an abelian variety $A/F$ of positive rank whose rank does not grow upon base change to $K$. This result implies that Hilbert's tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring $\mathcal{O}_K$ of integers of any number field $K$, there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over $\mathcal{O}_K$ has solutions in $\mathcal{O}_K$.

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arXiv2024

Integers expressible as the sum of two rational cubes

We prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible, thus proving a conjecture of Davenport. More generally, we prove that a positive proportion (in fact, at least one sixth) of elliptic curves in any cubic twist family have rank 0, and a positive proportion (in fact, at least one sixth) of elliptic curves with good reduction at 2 in any cubic twist family have rank 1. Our method involves proving that the average size of the 2-Selmer group of elliptic curves in any cubic twist family, having any given root number, is 3. We accomplish this by generalizing a parametrization, due to the second author and Ho, of elliptic curves with extra structure by pairs of binary cubic forms. We then use a novel combination of geometry-of-numbers methods and the circle method that builds on earlier work of Ruth and the first author. In particular, we make use of a new interpretation of the singular integral and series arising in the circle method in terms of real and $p$-adic Haar measures on the relevant group. We prove a uniformity estimate for integral points on the relevant quadric, which along with a sieve allows us to prove that the average size of the 2-Selmer group over the cubic twist family is 3. By suitably partitioning the subset of curves in the family with given root number, we effect a further sieve to show that the root number is equidistributed and that the same average, now taken over only those curves of given root number, is again 3. Finally, we apply the $p$-parity theorem of Dokchitser-Dokchitser and a $p$-converse theorem of Burungale-Skinner to conclude. We also prove the analogue of the above results for the sequence of square numbers: namely, we prove that a positive proportion of square integers are expressible as the sum of two rational cubes, and a positive proportion are not.

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arXiv2024

Quadrics in arithmetic statistics

We (re)introduce the circle method into arithmetic statistics. More specifically, we combine the circle method with Bhargava's counting technique in order to give a general method that allows one to treat arithmetic statistical problems in which one is trying to count orbits on a subvariety of affine space defined by the vanishing of a quadratic invariant. We explain this method by way of example by computing the average size of $2$-Selmer groups in the families $y^2 = x^3 + B$ and $y^2 = x^3 + B^2$. In the course of the argument we introduce a smoothed form of Bhargava's aforementioned method, as well as a trick with which we formally deduce that the above averages are $3$ from knowledge of the averages over "unconstrained" families.

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arXiv2024

Conditional algorithmic Mordell

We specify a Turing machine $T_{\text{Mordell}}$ with the following properties. 1. On input $(K,C/K)$, with $K/\mathbb{Q}$ a number field and $C/K$ a smooth projective hyperbolic curve, if $T_{\text{Mordell}}$ terminates, then it outputs $C(K)$. 2. The Hodge, Tate, and Fontaine-Mazur conjectures imply that $T_{\text{Mordell}}$ always terminates. Similarly we specify a Turing machine $T_{\text{Shafarevich}}$ with the following properties. 1. On input $(g, K,S, d)$, with $g, d \in \mathbb{Z}^+$, $K/\mathbb{Q}$ a number field, and $S$ a finite set of places of $K$, if $T_{\text{Shafarevich}}$ terminates, then it outputs the finitely many polarized $g$-dimensional abelian varieties $A/K$, with polarization of degree $d$, having good reduction outside $S$. 2. The Hodge, Tate, and Fontaine-Mazur conjectures imply that $T_{\text{Shafarevich}}$ always terminates.

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arXiv2022

The second moment of the number of integral points on elliptic curves is bounded

Let $K$ be a number field and $S$ a finite set of places of $K$ containing all archimedean places. In this paper, we show that the second moment of the number of $S$-integral points on elliptic curves over $K$ is bounded. In particular, we prove that, for any positive real number $r\leq \log_2 5 = 2.3219 \ldots$, the $r$-th moment of the number of $S$-integral points is bounded for the family of all integral short Weierstrass curves ordered by height, or for any positive density subfamily thereof. For certain other families of elliptic curves over $\mathbb{Q}$, such as those with one or two marked points, we prove that the average of the number of integral points is bounded; in fact, for the family with one marked point, the $r$-th moment is also bounded for all positive $r\leq \log_2 3$. The essential new ingredient in our proof is an upper bound on the number of $S$-integral points on an affine integral Weierstrass model $\mathcal{E}$ of an elliptic curve $E$ over $K$ depending on the rank of the curve, the class group and degree of $K$, and the number of primes of $K$ whose square divides the discriminant of $\mathcal{E}$. For example, over $\mathbb{Q}$, the bound for integral points on $\mathcal{E}$ is $2^{\mathrm{rank}{E(\mathbb{Q})}} O(1)^s$, where $s$ is the number of prime squares dividing the discriminant of $\mathcal{E}$. The theorems on moments then follow from averaging this new upper bound; crucially, we can bound the average of the $2^{\mathrm{rank}}$ term by using results on the average sizes of Selmer groups in the families. In order to prove the bounds for the $r$-th moment when $r= \log_2 5$ (and the analogous equality cases for the other families), we introduce a method to count orbits of coregular representations with possibly unbounded weights.

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arXiv2021

Modularity and effective Mordell I

We give an effective proof of Faltings' theorem for curves mapping to Hilbert modular stacks over odd-degree totally real fields. We do this by giving an effective proof of the Shafarevich conjecture for abelian varieties of $\mathrm{GL}_2$-type over an odd-degree totally real field. We deduce for example an effective height bound for $K$-points on the curves $C_a : x^6 + 4y^3 = a^2$ ($a\in K^\times$) when $K$ is odd-degree totally real. (Over $\overline{\mathbb{Q}}$ all hyperbolic hyperelliptic curves admit an étale cover dominating $C_1$.)

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arXiv2021

Un peu d'effectivité pour les variétés modulaires de Hilbert-Blumenthal

We prove a "height-free" effective isogeny estimate for abelian varieties of $\mathrm{GL}_2$-type. More precisely, let $g\in \mathbb{Z}^+$, $K$ a number field, $S$ a finite set of places of $K$, and $A,B/K$ $g$-dimensional abelian varieties with good reduction outside $S$ which are $K$-isogenous and of $\mathrm{GL}_2$-type over $\overline{\mathbb{Q}}$. We show that there is a $K$-isogeny $A\to B$ of degree effectively bounded in terms of $g$, $K$, and $S$ only. We deduce among other things an effective upper bound on the number of $S$-integral $K$-points on a Hilbert modular variety.

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arXiv2021

Note on a theorem of Professor X

Between his arrival in Frankfurt in $1922$ and and his proof of his famous finiteness theorem for integral points in $1929$, Siegel had no publications. He did, however, write a letter to Mordell in $1926$ in which he explained a proof of the finiteness of integral points on hyperelliptic curves. Recognizing the importance of this argument (and Siegel's views on publication), Mordell sent the relevant extract to be published under the pseudonym "X". The purpose of this note is to explain how to optimize Siegel's $1926$ technique to obtain the following bound. Let $K$ be a number field, $S$ a finite set of places of $K$, and $f\in \mathfrak{o}_{K,S}[t]$ monic of degree $d\geq 5$ with discriminant $Δ_f\in \mathfrak{o}_{K,S}^\times$. Then: $$\#|\{(x,y) : x,y\in \mathfrak{o}_{K,S}, y^2 = f(x)\}|\leq 2^{\mathrm{rank}\,\mathrm{Jac}(C_f)(K)}\cdot O(1)^{d^3\cdot ([K:\mathbb{Q}] + \#|S|)}.$$ This improves bounds of Evertse-Silverman and Bombieri-Gubler from $1986$ and $2006$, respectively. The main point underlying our improvement is that, informally speaking, we insist on "executing the descents in the presence of only one root (and not three) until the last possible moment".

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arXiv2021

A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cubic fields that we obtained in a previous work. Along the way, we also prove that a positive proportion of quartic rings of integers do not arise as the invariant order of an integral binary quartic form despite having no local obstruction.

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arXiv2021

A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so

We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of $2$-descent and $3$-descent in a certain family of Mordell curves $E_k \colon y^2 = x^3 + k$. As a by-product of our methods, we show that, for every $r \geq 0$, a positive proportion of curves $E_k$ have Tate--Shafarevich group with $3$-rank at least $r$.

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