arXiv subjects

Antonio Lei

Publications and source records attributed to Antonio Lei.

At least 19 recordsLinked to original sources

arXiv2026

Iwasawa theory of CM elliptic curves at potentially supersingular primes

We develop a plus and minus Iwasawa theory for CM elliptic curves with potentially supersingular reduction at a prime $p\ge5$. We construct local points that satisfy suitable "jumping trace" relations using the Honda--Demchenko theory applied to height-two formal groups over local fields with small ramification degree. These local points allow us to study plus and minus Selmer groups of CM elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension, to construct the corresponding plus and minus $p$-adic $L$-functions, and to prove an Iwasawa main conjecture relating these objects. As an application, we obtain asymptotic growth formulae for the $p$-primary part of the Tate--Shafarevich groups.

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arXiv2026

Kolyvagin's conjecture at non-ordinary primes

Let $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner hypothesis. In the case where $p$ is inert in $K$, we further assume that $\mathcal{A}_f$ is an elliptic curve. We develop an Euler-characteristic formula for signed Selmer groups over anticyclotomic $\mathbb{Z}_p$-extensions that applies when the corresponding Selmer modules have arbitrary $Λ$-rank. Assuming one inclusion in the signed Iwasawa main conjecture, we apply this formula to prove Kolyvagin's conjecture on the non-vanishing of the Kolyvagin system attached to Heegner points. Our results extend to the non-ordinary setting the results of Wei Zhang, Burungale--Castella--Grossi--Skinner, Castella--Sano and Kim in the ordinary case, and complement the works of Sweeting and Kim in the non-ordinary case under different hypotheses. We also study the effect of the exceptional zero phenomenon on the Iwasawa main conjecture in the inert case.

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arXiv2026

Recovering Laplacian Lattices from $L$-Functions of Graphs

We introduce $L$-functions associated with characters of the Jacobian of a finite graph, as a graph-theoretic analogue of the $L$-functions arising from unramified coverings of algebraic curves. These $L$-functions are defined using the Riemann--Roch structure on the graph and extend Lorenzini's two-variable zeta function. We show that if two graphs without bridges have isomorphic Jacobians and their $L$-functions agree under the induced correspondence of characters, then their Laplacian lattices coincide. We also show that Lorenzini's zeta function is invariant under contraction of bridges, explaining the necessity of the bridge-free hypothesis in the main theorem. Finally, we give examples showing that neither the Jacobian nor the Lorenzini zeta function alone determine the Laplacian lattice.

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arXiv2026

Congruences of $p$-adic $L$-functions of modular forms at non-ordinary primes

We present an analogue of Greenberg-Vatsal's and Emerton-Pollack-Weston's results on congruences of $p$-adic $L$-functions for $p$-non-ordinary cuspidal eigenforms $f$ and $g$ of equal weight that are $p$-congruent. In particular, we prove that the Iwasawa invariants of the analytic and algebraic signed $p$-adic $L$-functions of $f$ and $g$ are related by explicit formulae under appropriate hypotheses. We also show under the same assumptions that provided the algebraic and analytic $μ$-invariants vanish, the signed Iwasawa main conjecture is true for $f$ if and only if it is true for $g$.

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arXiv2026

A canonical construction of signed $p$-adic $L$-functions for non-ordinary modular forms of weight $\leq p+1$

Fix an odd prime $p$ and let $f$ be a $p$-non-ordinary cuspidal eigen-newform of weight $2\leq k\leq p+1$. We construct a pair of bounded $p$-adic $L$-functions associated to $f$ by decomposing the unbounded $p$-adic $L$-functions in terms of an explicit logarithm-type matrix whose definition does not require $p$-adic Hodge theory. Using this decomposition, we compute asymptotic formulas for the Iwasawa invariants of Mazur--Tate elements attached to non-ordinary forms of weight $\leq p+1$. As a corollary, we obtain a relation between the signed Iwasawa invariants of $p$-non-ordinary and $p$-congruent cuspforms of weights 2 and $p+1$, generalizing previous results in the $a_p=0$ case.

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arXiv2026

On the Iwasawa Invariants of Mazur--Tate elements of elliptic curves at additive primes

We investigate the $λ$-invariants of Mazur--Tate elements of elliptic curves defined over the field of rational numbers at primes of additive reduction. We explain their growth and how these invariants relate to other better understood invariants depending on the potential reduction type. We give examples and a conjecture for the additive potentially supersingular case, supported by computational data from Sage in this setting. Further, we extend our results to $λ$-invariants of Mazur--Tate elements of cuspidal Hecke eigenforms associated with potentially ordinary $p$-adic Galois representations.

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arXiv2026

Mazur-Tate elements of non-ordinary modular forms with Serre weight larger than two

Fix an odd prime $p$ and let $f$ be a non-ordinary eigen-cuspform of weight $k$ and level coprime to $p$. Assuming $p>k-1$, we compute asymptotic formulas for the Iwasawa invariants of the Mazur-Tate elements attached to $f$ in terms of the corresponding invariants of the signed $p$-adic $L$-functions. By combining this with a version of mod $p$ multiplicity one, we also obtain descriptions of the $λ$-invariants of Mazur-Tate elements attached to certain higher weight modular forms with Serre weight $<p+1$, generalizing results of Pollack and Weston in the Serre weight 2 case.

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arXiv2026

Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes II

Let $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at a prime $p\geq 5$, and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ equals $-1$. When $p$ splits in $K$, Castella and Wan formulated the plus/minus Heegner point main conjectures for $E$ along the anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved them for semistable curves. We generalize their results to two settings: 1. For $p$ split in $K$, we formulate Sprung-type main conjectures for $\mathrm{GL}_2$-type abelian varieties at non-ordinary primes and prove them under some conditions. 2. For $p$ inert in $K$, we formulate, relying on the work of the first-named author with Kobayashi and Ota, plus/minus Heegner point main conjectures for elliptic curves, and prove the minus main conjecture for semistable curves. The latter yields a $p$-converse to the Gross--Zagier and Kolyvagin theorem for semistable elliptic curves $E$ at supersingular primes $p\geq 5$, complementing the pioneering $p$-converse theorems of Skinner and Zhang. Our method relies on Howard's framework of bipartite Euler systems, Zhang's resolution of Kolyvagin's conjecture and the recent proof of cyclotomic main conjecture at non-ordinary primes.

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arXiv2026

Bowen--Franks groups and minus class groups of cyclotomic number fields with prime conductor

Let $p$ be an odd rational prime and consider the cyclotomic number field $K = \mathbb{Q}(ζ_{p})$ of conductor $p$. We construct a directed graph $Y$ on $p-1$ vertices for which the torsion part of the corresponding Bowen--Franks group is closely related to the minus part of the class group of $K$. In particular, both groups have the same cardinality up to an explicit power of $p$. Furthermore, they are both $\mathrm{Gal}(K/\mathbb{Q})$-modules, and we prove the equality of the cardinalities of their isotypic components after tensoring them with the valuation ring of an appropriate $\ell$-adic field for $\ell \nmid p-1$.

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arXiv2026

Iwasawa theory for abelian towers of digraphs

Let $p$ and $\ell$ be prime numbers, and $d\ge1$ an integer. We formulate and prove Iwasawa main conjectures of the Picard groups and Bowen--Franks groups in $\mathbb{Z}_p^d$-towers of digraphs. In particular, we relate the $\ell$ parts of these groups to certain $p$-adic $L$-functions defined using a voltage assignment. In the case where $\ell$ is not equal to $p$, we make use of the recent work of Bandini--Longhi to define the appropriate characteristic ideals. We also prove the growth of the $\ell$-part of these groups, generalizing classical results of Sinnott and Washington on ideal class groups of number fields. Finally, we introduce the concept of defect, which compare certain algebraic and analytic ranks related to Bowen--Franks groups and study their asymptotic behaviour in a $\mathbb{Z}_p^d$-tower.

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arXiv2025

On the maximality of the $λ$-invariants of Mazur--Tate elements

Let $E$ be an elliptic curve with good ordinary reduction at an odd prime $p$. Assuming that Greenberg's $μ=0$ conjecture holds, we show that the $λ$-invariants of the Mazur--Tate elements attached to $E$ either stabilise to the $λ$-invariant of the $p$-adic $L$-function or they attain the largest possible value at all finite levels. We characterise the latter phenomenon:\ it occurs if and only if $\ord_p\left(\frac{L(E',1)}{Ω_{E'}}\right)$ is negative for some $E'$ that is isogenous to $E$. Furthermore, we relate this condition to congruences with boundary symbols coming from Eisenstein series. We also study the extension of these results to Hecke eigenforms of weight two.

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arXiv2025

On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $\mathbf{Z}_p$-towers

Let $p$ be an odd prime number and let $K$ be an imaginary quadratic field in which $p$ is split. Let $f$ be a modular form with good reduction at $p$. We study the variation of the Bloch--Kato Selmer groups and the Bloch--Kato--Shafarevich--Tate groups of $f$ over the anticyclotomic $\mathbf{Z}_p$-extension $K_\infty$ of $K$. In particular, we show that under the generalized Heegner hypothesis, if the $p$-localization of the generalized Heegner cycle attached to $f$ is primitive and certain local conditions hold, then the Pontryagin dual of the Selmer group of $f$ over $K_\infty$ is free over the Iwasawa algebra. Consequently, the Bloch--Kato--Shafarevich--Tate groups of $f$ vanish. This generalizes earlier works of Matar and Matar--Nekovář on elliptic curves. Furthermore, our proof applies uniformly to the ordinary and non-ordinary settings.

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arXiv2025

Trito-non-ordinary Iwasawa theory of diagonal cycles

Our goal in this paper is to introduce and study the Euler system of signed diagonal cycles associated with a trito-non-ordinary triple product of the form $f^B \times g^B \times \mathbf{h}^B$, where $f^B$ (resp. $g^B$) is a $p$-ordinary (resp. non-ordinary) eigenform on an indefinite quaternion algebra $B_{/\mathbb{Q}}$ of weight $2$, and $\mathbf{h}^B$ is a primitive Hida ($p$-ordinary) family. When $B=\mathrm{M}_2(\mathbb{Q})$ is split and $\mathbf{h}=\mathbf{h}^B$ has CM by an imaginary quadratic field, this allows us to develop the signed anticyclotomic Iwasawa theory for the base change $\mathrm{BC}_{K/\mathbb{Q}}(π_f)\times \mathrm{BC}_{K/\mathbb{Q}}(π_g)\times ψ$, where $ψ$ is a Hecke character of $K$. We formulate a signed Perrin-Riou-style Iwasawa main conjecture in this setting, and obtain a result on one inclusion in this conjecture. Our methods also allow us to extend Hsieh's construction of the balanced triple-product $p$-adic $L$-function to the trito-non-ordinary scenario, and to define its signed counterparts.

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arXiv2025

On the growth of Tate-Shafarevich groups of $p$-supersingular abelian varieties of ${\rm GL}_2$-type over $\mathbb{Z}_p$-extensions of number fields

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 ${\rm GL}_2$-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-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.

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arXiv2025

The growth of Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic $\mathbb{Z}_p$-extensions at inert primes

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be an imaginary quadratic field. Consider an odd prime $p$ at which $E$ has good supersingular reduction with $a_p(E)=0$ and which is inert in $K$. Under the assumption that the signed Selmer groups are cotorsion modules over the corresponding Iwasawa algebra, we prove that the Mordell-Weil ranks of $E$ are bounded over any subextensions of the anticyclotomic $\mathbb{Z}_p$-extension of $K$. Additionally, we provide an asymptotic formula for the growth of the $p$-parts of the Tate-Shafarevich groups of $E$ over these extensions.

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arXiv2025

On $\mathbb{Z}_p$-towers of graph coverings arising from a constant voltage assignment

We investigate properties of $\mathbb{Z}_p$-towers of graph coverings that arise from a constant voltage assignment. We prove the existence and uniqueness (up to isomorphisms) of such towers. Furthermore, we study the Iwasawa invariants of these towers, and apply our results to towers of isogeny graphs enhanced with level structures, as well as towers arising from volcano graphs.

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arXiv2025

Mazur's Growth Number Conjecture in the Rank One Case

Let $p\geq 5$ be a prime number. Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. Let $K$ be an imaginary quadratic field where $p$ splits, and such that the generalized Heegner hypothesis holds. Under mild hypotheses, we show that if the $p$-adic height of the Heegner point of $\mathsf{E}$ over $K$ is non-zero, then Mazur's conjecture on the growth of Selmer coranks in the $\mathbb{Z}_p^2$-extension of $K$ holds.

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arXiv2025

Isogeny graphs with level structures arrising from the Verschiebung map

We enhance an isogeny graph of elliptic curves by incorporating level structures defined by bases of the kernels of iterates of the Verschiebung map. We extend several previous results on isogeny graphs with level structures defined by geometric points to these graphs. Firstly, we prove that these graphs form $\mathbb{Z}_p$-towers of graph coverings as the power of the Verschiebung map varies. Secondly, we prove that the connected components of these graphs display a volcanic structure.

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