arXiv subjects

Ariel Weiss

Publications and source records attributed to Ariel Weiss.

11 recordsLinked to original sources

arXiv2025

Distinguishing Siegel modular forms

Let $f$ and $f'$ be genus $2$ cuspidal Siegel paramodular newforms. We prove that if their Hecke eigenvalues $a_p$ and $a_p'$ satisfy a non-trivial polynomial relation $P(a_p, a_p') = 0$ for a set of primes $p$ of positive density, then $f$ is a scalar multiple of a quadratic twist of $f'$. This result extends the strong multiplicity one theorem, which handles the case $P(x,y) = x - y$, to arbitrary polynomial relations. Our proof analyses the image of the product Galois representation attached to the pair $(f, f')$: we show that this image is as large as possible, unless $f$ is a twist of $f'$. Our results also apply to elliptic modular forms. They therefore provide a unified method for distinguishing both elliptic and Siegel modular forms based on their Hecke data, including their Hecke eigenvalues, Satake parameters, Sato--Tate angles, and the coefficients of their $L$-functions. We apply our methods to recover and generalise a range of existing results and to prove new ones in both the elliptic and Siegel settings.

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arXiv2024

Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultants

Given a finite set $S$ of distinct primes, we propose a method to construct polylogarithmic motivic Chabauty-Kim functions for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$ using resultants. For a prime $p\not\in S$, the vanishing loci of the images of such functions under the $p$-adic period map contain the solutions of the $S$-unit equation. In the case $\vert S\vert=2$, we explicitly construct a non-trivial motivic Chabauty-Kim function in depth 6 of degree 18, and prove that there do not exist any other Chabauty-Kim functions with smaller depth and degree. The method, inspired by work of Dan-Cohen and the first author, enhances the geometric step algorithm developed by Corwin and Dan-Cohen, providing a more efficient approach.

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arXiv2024

Five-dimensional compatible systems and the Tate conjecture for elliptic surfaces

Let $(\rho_\lambda\colon G_{\mathbb Q}\to \operatorname{GL}_5(\overline{E}_\lambda))_\lambda$ be a strictly compatible system of Galois representations such that no Hodge--Tate weight has multiplicity $5$. Under mild assumptions, we show that if $\rho_{\lambda_0}$ is irreducible for some $\lambda_0$, then $\rho_\lambda$ is irreducible for all but finitely many priimes $\lambda$. More generally, if $(\rho_\lambda)_\lambda$ is essentially self-dual, we show that either $\rho_\lambda$ is irreducible for all but finitely many $\lambda$, or the compatible system $(\rho_\lambda)_\lambda$ decomposes as a direct sum of lower-dimensional compatible systems. We apply our results to study the Tate conjecture for elliptic surfaces. For example, if $X_0\colon y^2 + (t+3)xy + y= x^3$, we prove the codimension one $\ell$-adic Tate conjecture for all but finitely many $\ell$, for all but finitely many general, degree $3$, genus $2$ branched multiplicative covers of $X_0$. To prove this result, we classify the elliptic surfaces into four one-dimensional families and two isolated classes. For each of the four families, we prove, using perverse sheaf theory and a result of Cadoret--Tamagawa, that if the relevant $5$-dimensional Galois representation is irreducible for one surface in a family, then it is irreducible for all but finitely many surfaces in that family. We then verify this irreducibility for one representative of each family by making our irreducibility result explicit: for the compatible system arising from the transcendental part of $H^2_{\mathrm{et}}(X_{\overline{\mathbb Q}}, \mathbb{Q}_\ell(1))$ for a representative $X$, we formulate an algorithm that takes as input the characteristic polynomials of Frobenius, and terminates if and only if the compatible system is irreducible.

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arXiv2024

Maximal stable lattices in representations over discretely valued fields

Let $\rho\colon G\to \mathrm{GL}_n(K)$ be an continuous irreducible representation of a compact group over a complete discretely valued field $K$. Let $W_i,W_j$ be two irreducible subrepresentations of $\overline{\rho}^{ss}$, the semisimplification of the residual representation. We study the structure of the $G$-stable lattices $\Lambda\subseteq K^n$ with a view to understanding the question of when $\rho$ realises a non-split extension of $W_i$ by $W_j$. In particular, we introduce the notion of a maximal $G$-stable lattice and prove that any non-split extension of $W_i$ by $W_j$ that can be realised by $\rho$ can also be realised by a maximal lattice. As applications, we give a new proof and a strengthening of Bella\"iche's generalisation of Ribet's Lemma, which assures the abundancy of non-split extensions that can be realised by $\rho$. On the other hand, we also show that, if the representations $W_i, W_j$ occur with multiplicity one in $\overline{\rho}^{ss}$, then $\rho$ can realise at most one non-split extension of $W_i$ by $W_j$.

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arXiv2022

On the Lang--Trotter conjecture for Siegel modular forms

Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms. Using this result, we investigate the distribution of the Hecke eigenvalues $a_p$ of $f$, and obtain upper bounds for the sizes of the sets $\{p \le x : a_p = a\}$ for fixed $a\in\mathbf{C}$, in the spirit of the Lang--Trotter conjecture for elliptic curves.

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arXiv2021

Rank growth of elliptic curves over $N$-th root extensions

Fix an elliptic curve $E$ over a number field $F$ and an integer $n$ which is a power of $3$. We study the growth of the Mordell--Weil rank of $E$ after base change to the fields $K_d = F(\sqrt[2n]{d})$. If $E$ admits a $3$-isogeny, then we show that the average ``new rank'' of $E$ over $K_d$, appropriately defined, is bounded as the height of $d$ goes to infinity. When $n = 3$, we moreover show that for many elliptic curves $E/\mathbb{Q}$, there are no new points on $E$ over $\mathbb{Q}(\sqrt[6]d)$, for a positive proportion of integers $d$. This is a horizontal analogue of a well-known result of Cornut and Vatsal. As a corollary, we show that Hilbert's tenth problem has a negative solution over a positive proportion of pure sextic fields $\mathbb{Q}(\sqrt[6]{d})$. The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the ``correlation trick'', which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves.

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arXiv2021

On Ribet's Lemma for $\mathrm{GL}_2$ modulo prime powers

Let $\rho\colon G\to \mathrm{GL}_2(K)$ be a continuous representation of a compact group $G$ over a complete discretely valued field $K$, with ring of integers $\mathcal O$ and uniformiser $\pi$. We prove that $\operatorname{tr}\rho$ is reducible modulo $\pi^n$ if and only if $\rho$ is reducible modulo $\pi^n$. More precisely, there exist characters $\chi_1,\chi_2 \colon G\to(\mathcal O/\pi^n\mathcal O)^{\times}$ such that $\det(t - \rho(g))\equiv (t-\chi_1(g))(t-\chi_2(g))\pmod{\pi^n}$ for all $g\in G$, if and only if there exists a $G$-stable lattice $\Lambda\subset K^2$ such that $\Lambda/\pi^n\Lambda$ contains a $G$-invariant, free, rank one $\mathcal O/\pi^n\mathcal O$-submodule. Our result applies in the case that $\rho$ is not residually multiplicity free, in which case it answers a question of Bella\"iche--Chenevier. As an application, we prove an optimal version of Ribet's Lemma, which gives a condition for the existence of a $G$-stable lattice $\Lambda$ that realises a non-split extension of $\chi_2$ by $\chi_1$

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arXiv2021

Ranks of abelian varieties in cyclotomic twist families

Let $A$ be an abelian variety over a number field $F$, and suppose that $\mathbb Z[\zeta_n]$ embeds in $\mathrm{End}_{\bar F} A$, for some root of unity $\zeta_n$ of order $n = 3^m$. Assuming that the Galois action on the finite group $A[1-\zeta_n]$ is sufficiently reducible, we bound the average rank of the Mordell--Weil groups $A_d(F)$, as $A_d$ varies through the family of $\mu_{2n}$-twists of $A$. Combining this with the recently proved uniform Mordell--Lang conjecture, we prove near-uniform bounds for the number of rational points in twist families of bicyclic trigonal curves $y^3 = f(x^2)$, as well as in twist families of theta divisors of cyclic trigonal curves $y^3 = f(x)$. Our main technical result is the determination of the average size of a $3$-isogeny Selmer group in a family of $\mu_{2n}$-twists.

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arXiv2021

Elements of prime order in Tate-Shafarevich groups of abelian varieties over $\mathbb{Q}$

For each prime $p$, we show that there exist geometrically simple abelian varieties $A/\mathbb Q$ with non-trivial $p$-torsion in their Tate-Shafarevich groups. Specifically, for any prime $N\equiv 1 \pmod{p}$, let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point $P$ of order $p$, and let $B = A_f/\langle P \rangle$. Then the number of positive integers $d \leq X$, such that the Tate-Shafarevich group of $\hat B_d$ has non-trivial $p$-torsion, is $\gg X/\log X$, where $\hat B_d$ is the dual of the $d$-th quadratic twist of $B$. We prove this more generally for abelian varieties of $\mathrm{GL}_2$-type with a $p$-isogeny satisfying a mild technical condition. In the special case of elliptic curves, we give stronger results, including many examples where $\mathrm{Sha}(E_d)[p] \neq 0$ for an explicit positive proportion of integers $d$.

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arXiv2020

Lafforgue pseudocharacters and parities of limits of Galois representations

Let $F$ be a CM field with totally real subfield $F^+$ and let $\pi$ be a $C$-algebraic cuspidal automorphic automorphic representation of $\mathrm{U}(a,b)(\mathbf{A}_{F^+})$ whose archimedean components lie in the (non-degenerate limit of) discrete series. We attach to $\pi$ a Galois representation $R_\pi:\mathrm{Gal}(\overline F/ F^+)\to{}^C\mathrm{U}(a,b)(\overline{\mathbf Q}_\ell)$ such that, for any complex conjugation element $c$, $R_\pi(c)$ is as predicted by the Buzzard--Gee conjecture. As a corollary, we deduce that the Galois representations attached to certain irregular, $C$-algebraic (essentially) conjugate self-dual cuspidal automorphic representations of $\mathrm{GL}_n(\mathbf A_F)$ are odd in the sense of Bella\"iche--Chenevier.

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arXiv2018

On the images of Galois representations attached to low weight Siegel modular forms

Let $\pi$ be a cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbf{A_Q})$, whose archimedean component is a holomorphic discrete series or limit of discrete series representation. If $\pi$ is not CAP or endoscopic, then we show that its associated $\ell$-adic Galois representations are irreducible and crystalline for $100\%$ of primes $\ell$. If, moreover, $\pi$ is neither an automorphic induction nor a symmetric cube lift, then we show that, for $100\%$ of primes $\ell$, the image of its mod $\ell$ Galois representation contains $\mathrm{Sp}_4(\mathbf{F}_{\ell})$.

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