In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{{\mathbb{Q}}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in {\mathbb{Z}}^+$, we wish to determine the least degree of a point on the modular curve $X_1(N)$ associated to any $E \in \mathcal{E}$. We consider the cases where $\mathcal{E}$ is rational, i.e., contains an elliptic curve with rational j-invariant, or where $\mathcal{E}$ consists of elliptic curves with complex multiplication (CM). If $N=\ell^k$ is a power of a single prime, we give a complete characterisation upon restricting to points of odd degree, and also in the case where $\mathcal{E}$ is CM. We include various partial results in the more general setting.

We study the boundedness of the Mordell–Weil rank and the growth of the v-primary part of the Tate–Shafarevich group of p-supersingular abelian varieties of $\mathrm {GL}_2$-type with real multiplication over $\mathbb Z_p$-extensions of number fields, where v is a prime lying above p. Building on the work of Iovita and Pollack in the case of elliptic curves, under precise ramification and splitting conditions on p, we construct explicit systems of local points using the theory of Lubin–Tate formal groups. We then define signed Coleman maps, which in turn allow us to formulate and analyse signed Selmer groups. Assuming these Selmer groups are cotorsion, we prove that the Mordell–Weil groups are bounded over any subextensions of the ${\mathbb Z}_p$-extension and provide an asymptotic formula for the growth of the v-primary part of the Tate–Shafarevich groups. Our results extend those of Kobayashi, Pollack, and Sprung on p-supersingular elliptic curves.

This paper addresses the prediction of positive rank for elliptic curves without the need to find a point of infinite order or compute L-functions. While the most common method relies on parity conjectures, a recent technique introduced by Dokchitser, Wiersema and Evans predicts positive rank based on the value of a certain product of Tamagawa numbers, raising questions about its relationship to parity. We show that their method is a subset of the parity conjectures approach: whenever their method predicts positive rank, so does the use of parity conjectures. To establish this, we extend previous work on Brauer relations and regulator constants to a broader setting involving combinations of permutation modules known as K-relations. A central ingredient in our argument is demonstrating a compatibility between Tamagawa numbers and local root numbers.

Let E be an elliptic curve over the finite field $\mathbb {F}_p$, and $P \in E(\mathbb {F}_p)$ be an ${\mathbb F}_p$-rational point. We obtain nontrivial estimates for multiplicative character sums associated with the division polynomials $\psi _n(P)$ twisted by several multiplicative functions, where $\psi _n(P)$ denotes the nth division polynomial evaluated at P.

Let $(F_r M)_{r\in {\mathbb Z}}$ be a finite filtration on an object M of a neutral Tannakian category ${\mathbf {T}}$ in characteristic zero. Let ${\mathfrak {u}}(M)={\mathfrak {u}}^F(M)$ be the Lie algebra of the subgroup of the Tannakian fundamental group of M that acts trivially on the associated graded $\mathrm {Gr}^FM$. The filtration $F_\bullet M$ induces a filtration on the internal Hom ${\underline {\mathrm{Hom}}}(M,M)$, which in turn induces a filtration $F_\bullet {\mathfrak {u}}(M)$ on ${\mathfrak {u}}(M)$. This filtration on ${\mathfrak {u}}(M)$ is concentrated in negative degrees. In this article, we give a description of the graded piece $\mathrm {Gr}^F_{-1}{\mathfrak {u}}( M)$ in terms of the extensions $F_{r+1}M/F_{r-1}M\in \mathrm {Ext}^1(\mathrm {Gr}^F_{r+1}M, \mathrm {Gr}^F_{r}M)$. In particular, these extensions determine $\mathrm {Gr}^F_{-1}{\mathfrak {u}}(M)$. Note that here we neither assume the filtration is functorial, nor we assume that $\mathrm {Gr}^FM$ is semisimple. The problem of studying ${\mathfrak {u}}(M)$ in this generality is motivated by the desire to understand Tannakian groups associated with a mixed motive and its realizations, including realizations for which semisimplicity of realizations of pure motives is not known and realizations that lack an interesting functorial weight filtration. We also give two related applications. The first is an equivalent condition in the generality described above for when ${\mathfrak {u}}(M)$ coincides with its trivial upper bound $F_{-1}{\underline {\mathrm{Hom}}}(M,M)$. This result generalizes earlier criteria for maximality of ${\mathfrak {u}}(M)$ obtained by various authors in special contexts or under limiting conditions. In the second application, we apply the results to the setting of a neutral Tannakian category with a functorial weight filtration $W_\bullet $. Combining the constructions of [7] with our general maximality criterion, we prove a result about the structure of the set of isomorphism classes of objects M for which $\mathrm {Gr}^WM$ is isomorphic to a given graded object A and ${\mathfrak {u}}^W(M)=W_{-1}{\underline {\mathrm{Hom}}}(M,M)$.

We show that the moduli spaces of Scholze’s p-adic shtukas with framing satisfy a p-adic rigid analytic version of Borel’s extension theorem. In particular, this holds for local Shimura varieties, for all local Shimura data $(G, [b],\{\mu \})$, even for exceptional groups G, and extends work of Oswal-Shankar-Zhu-Patel who proved a p-adic Borel extension property for Rapoport-Zink spaces. As a corollary, we deduce that all these spaces satisfy a p-adic rigid analytic version of Brody hyperbolicity.

In the 17th century, a letter from Fermat to Mersenne considered the problem of finding the Pythagorean triples in which the hypotenuse and the sum of the other two sides are squares. We call this question the Fermat–Mersenne problem. Number-theoretic solutions of the problem are known, but they are complicated. We solve the Fermat–Mersenne problem by using a geometric approach that provides new insight. The argument involves stereographically projecting points onto an elliptic curve and applying complex multiplication on that curve.

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemerédi–Trotter-type theorem over finite fields, with an application to polynomial expansion.

Let p and q be two primes with $(p,q)\equiv (1,5)$ or $(7,3) \pmod 8$. Lagrange [‘Nombres congruents et courbes elliptiques’, Séminaire Delange-Pisot-Poitou. Théorie des Nombres 16(1) (1974–1975), Article no. 16] and Qin [‘Congruent numbers, quadratic forms and $K_2$’, Math. Ann. 383(3–4) (2022), 1647–1686] showed that if $(\frac {q}{p})=-1$, then $2pq$ is not a congruent number. By using Qin’s method, we prove that if $(p,q)\equiv (1,5) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-pq)\not \equiv p-1 \pmod {16}$, then $2pq$ is not a congruent number; if $(p,q)\equiv (7,3) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-2pq)\not \equiv p+1 \pmod {16}$, then $2pq$ is not a congruent number. Here, $h(-d)$ denotes the class number of the imaginary quadratic field $\mathbb {Q}(\sqrt {-d})$.

The notion of $\theta $-congruent numbers generalises the classical congruent number problem. A positive integer n is $\theta $-congruent if it is the area of a rational triangle with an angle $\theta $ whose cosine is rational. Das and Saikia [‘On $\theta $-congruent numbers over real number fields’, Bull. Aust. Math. Soc. 103(2) (2021), 218–229] established criteria for numbers to be $\theta $-congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $\theta $-congruent and properly $\theta $-congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to six, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree six and examine the exceptional cases $n=1, 2, 3$ and $6$.

We extend the construction of the p-adic L-function interpolating unitary Friedberg–Jacquet periods in previous work of the author to include the p-adic variation of Maass–Shimura differential operators. In particular, we develop a theory of nearly overconvergent automorphic forms in higher degrees of coherent cohomology for unitary Shimura varieties generalising previous work for modular curves. The construction of this p-adic L-function can be viewed as a higher-dimensional generalisation of the work of Bertolini–Darmon–Prasanna and Castella–Hsieh, and the inclusion of this extra variable arising from the p-adic iteration of differential operators will play a key role in relating values of this p-adic L-function to p-adic regulators of special cycles on unitary Shimura varieties.

Let $f \in \mathbb{Q}[x]$ be a square-free polynomial of degree at least $3$, $m_i$, $i=1,2,3$, odd positive integers, and $a_i$, $i=1,2,3$, non-zero rational numbers. We show the existence of a rational function $D\in \mathbb{Q}(v_1,v_2,v_3,v_4)$ such that the Jacobian of the quadratic twist of $y^2=f(x)$ and the Jacobian of the $m_i$-twist, respectively, $2m_i$-twist, of $y^2=x^{m_i}+a_i^2$, $i=1,2,3$, by $D$ are all of positive Mordell–Weil ranks. As an application, we present families of hyperelliptic curves with large Mordell–Weil rank.

Under the Generalised Riemann Hypothesis (GRH), any element in the multiplicative group of a number field K that is globally primitive (i.e., not a perfect power in $K^*$) is a primitive root modulo a set of primes of K of positive density.

For elliptic curves $E/K$ that are known to have infinitely many primes ${\mathfrak{p}}$ of cyclic reduction, possibly under GRH, a globally primitive point $P\in E(K)$ may fail to generate any of the point groups $E(k_{\mathfrak{p}})$. We describe this phenomenon in terms of an associated Galois representation $\rho_{E/K, P}\,:\,G_K\to\mathrm{GL}_3({\widehat {{\mathbf{Z}}}})$, and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive.

We prove an André–Oort-type result for a family of hypersurfaces in ${\mathbb{C}}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel–Tatuzawa lower bound for the class number. We prove that, for $m, n \in {\mathbb{Z}}_{\gt0}$, there exists an effective constant $c(m, n)\gt0$ with the following property: if pairwise distinct singular moduli $x_1, \ldots, x_n$ with respective discriminants $\Delta_1, \ldots, \Delta_n$ are such that $a_1 x_1^m + \cdots + a_n x_n^m \in {\mathbb{Q}}$ for some $a_1, \ldots, a_n \in {\mathbb{Q}} \setminus \{0\}$ and $\# \{ \Delta_i \;:\; {\mathbb{Q}}(\sqrt{\Delta_i}) = K_*\} \leq 1$, then $\max_i \lvert \Delta_i \rvert \leq c(m, n)$. In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples $(x_1, x_2, x_3)$ of singular moduli such that $a_1 x_1 + a_2 x_2 + a_3 x_3 \in {\mathbb{Q}}$ for some $a_1, a_2, a_3 \in {\mathbb{Q}} \setminus \{0\}$.

In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at hyperspecial level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in an earlier work. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and the non-admissibility of the cohomology in general. Via p-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the supersingular locus of a Shimura variety with no level at p. In the case $n=3$ or $4$, we give a complete description of the cohomology of the supersingular locus in terms of automorphic representations.

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers.

For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let S be a Shimura variety. Let $\pi :D \to \Gamma \backslash D = S$ realize S as a quotient of D, a homogeneous space for the action of a real algebraic group G, by the action of $\Gamma < G$, an arithmetic subgroup. Let $S' \subseteq S$ be a special subvariety of S realized as $\pi (D')$ for $D' \subseteq D$ a homogeneous space for an algebraic subgroup of G. Let $X \subseteq S$ be an irreducible subvariety of S not contained in any proper weakly special subvariety of S. Assume that the intersection of X with $\pi (gD')$ is persistently likely as g ranges through G with $\pi (gD')$ a special subvariety of S, meaning that whenever $\zeta :S_1 \to S$ and $\xi :S_1 \to S_2$ are maps of Shimura varieties (regular maps of varieties induced by maps of the corresponding Shimura data) with $\zeta $ finite, $\dim \xi \zeta ^{-1} X + \dim \xi \zeta ^{-1} \pi (gD') \geq \dim \xi S_1$. Then $X \cap \bigcup _{g \in G, \pi (g D') \text { is special }} \pi (g D')$ is dense in X for the Euclidean topology.

Let F be a totally real field. Let $\mathsf {A}$ be a simple modular self-dual abelian variety defined over F. We study the growth of the corank of Selmer groups of $\mathsf {A}$ over $\mathbb {Z}_p$-extensions of a complex multiplication (CM) extension of F. We propose an extension of Mazur’s growth number conjecture for elliptic curves to this new setting. We provide evidence supporting an affirmative answer by studying special cases of this problem, generalising previous results on elliptic curves and imaginary quadratic fields.

Let K be an imaginary quadratic field, and let $E/\mathbb {Q}$ be an elliptic curve with complex multiplication by $\mathcal {O}_K$. Let $K_\infty /K$ be the anticyclotomic $\mathbb {Z}_p$-extension of K and $K_n$ be the intermediate layers. Under additional assumptions on Kobayashi’s signed Selmer groups, we prove an asymptotic formula for .

We show the Harris–Viehmann conjecture under some Hodge–Newton reducibility condition for a generalisation of the diamond of a non-basic Rapoport–Zink space at infinite level, which appears as a cover of the non-semi-stable locus in the Hecke stack. We show also that the cohomology of the non-semi-stable locus with coefficients coming from a cuspidal Langlands parameter vanishes. As an application, we show the Hecke eigensheaf property in Fargues’ conjecture for cuspidal Langlands parameters in the $ {\mathrm {GL}}_2$-case.

A 2009 article of Allcock and Vaaler explored the $\mathbb {Q}$-vector space $\mathcal {G} := \overline {\mathbb {Q}}^\times /{\overline {\mathbb {Q}}^\times _{\mathrm {tors}}}$, showing how to represent it as part of a function space on the places of $\overline {\mathbb {Q}}$. We establish a representation theorem for the $\mathbb {R}$-vector space of $\mathbb {Q}$-linear maps from $\mathcal {G}$ to $\mathbb {R}$, enabling us to classify extensions to $\mathcal {G}$ of completely additive arithmetic functions. We further outline a strategy to construct $\mathbb {Q}$-linear maps from $\mathcal {G}$ to $\mathbb {Q}$, i.e., elements of the algebraic dual of $\mathcal {G}$. Our results make heavy use of Dirichlet’s S-unit Theorem as well as a measure-like object called a consistent map, first introduced by the author in previous work.