arXiv subjects

Maria Rosaria Pati

Publications and source records attributed to Maria Rosaria Pati.

8 recordsLinked to original sources

arXiv2026

Anticyclotomic Iwasawa main conjectures for modular forms

Let $f$ be a newform of even weight at least $4$, level $N$ and trivial character. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and let $K$ be an imaginary quadratic field satisfying a generalized Heegner hypothesis relative to $N$. In this paper, we prove (under mild arithmetic assumptions) Iwasawa main conjectures for $f$ over the anticyclotomic $\mathbb Z_p$-extension of $K$ both in the definite setting and in the indefinite setting (in the second case, we prove a main conjecture à la Perrin-Riou for modular forms). Our strategy of proof follows the approach of Bertolini-Darmon via congruences combined with our previous results on an analogue for $f$ of Kolyvagin's conjecture on the non-triviality of his $p$-adic system of derived Heegner points on elliptic curves. As a second contribution, when $p$ splits in $K$ we prove an Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna and Brooks.

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arXiv2024

Kolyvagin's conjecture for modular forms

Our main result in this article is a proof (under mild technical assumptions) of an analogue for $p$-adic Galois representations attached to a newform $f$ of even weight $k\geq4$ of Kolyvagin's conjecture on the $p$-indivisibility of derived Heegner points on elliptic curves, where $p$ is a prime number that is ordinary for $f$. Our strategy, which is inspired by work of W. Zhang in weight $2$, is based on a variant for modular forms of the congruence method originally introduced by Bertolini-Darmon to prove one divisibility in the anticyclotomic Iwasawa main conjecture for rational elliptic curves. We adapt to higher (even) weight modular forms this approach via congruences, building crucially on results of Wang on the indivisibility of Heegner cycles over Shimura curves. Then we offer an application of our results on Kolyvagin's conjecture to the Tamagawa number conjecture for the motive of $f$ and describe other (standard) consequences on structure theorems for Bloch-Kato-Selmer groups, $p$-parity results and converse theorems for $f$. Since in the present paper we need $p>k+1$, our main theorem and its applications can be viewed as complementary to results obtained by the first and third authors in their article on the Tamagawa number conjecture for modular motives, where Kolyvagin's conjecture was proved (in a completely different way exploiting the arithmetic of Hida families) under the assumption that $k$ is congruent to $2$ modulo $2(p-1)$, which forces $p<k$. In forthcoming work, we will use results contained in this paper to prove (under analogous assumptions) the counterpart for an even weight newform $f$ of Perrin-Riou's Heegner point main conjecture for elliptic curves ("Heegner cycle main conjecture" for $f$).

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arXiv2024

On the Iwasawa Main Conjecture for generalized Heegner classes in a quaternionic setting

We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form $f$ and an imaginary quadratic field satisfying a "relaxed" Heegner hypothesis. Let $Λ$ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo--Vigni, we construct the $Λ$-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-function defined by Magrone.

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arXiv2023

On Shafarevich-Tate groups and analytic ranks in families of modular forms, II. Coleman families

This is the second article in a two-part project whose aim is to study algebraic and analytic ranks in $p$-adic families of modular forms. Let $f$ be a newform of weight $2$, square-free level $N$ and trivial character, let $A_f$ be the abelian variety attached to $f$, whose dimension will be denoted by $d_f$, and for every prime number $p\nmid N$ let $\boldsymbol f^{(p)}$ be a $p$-adic Coleman family through $f$ over a suitable open disc in the $p$-adic weight space. We prove that, for all but finitely many primes $p$ as above, if $A_f(\mathbb Q)$ has rank $r\in\{0,d_f\}$ and the $p$-primary part of the Shafarevich-Tate group of $A_f$ over $\mathbb Q$ is finite, then all classical specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have finite $p$-primary Shafarevich-Tate group and $r/d_f$-dimensional image of the relevant $p$-adic étale Abel-Jacobi map. As a second contribution, assuming the non-degeneracy of certain height pairings à la Gillet-Soulé between Heegner cycles, we show that, for all but finitely many $p$, if $f$ has analytic rank $r\in\{0,1\}$, then all classical specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have analytic rank $r$. This result provides some evidence for a conjecture of Greenberg on analytic ranks in families of modular forms.

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arXiv2021

L-invariants for cohomological representations of PGL(2) over arbitrary number fields

Let $π$ be a cuspidal, cohomological automorphic representation of an inner form $G$ of $\mathrm{PGL}_2$ over a number field $F$ of arbitrary signature. Further, let $\mathfrak{p}$ be a prime of $F$ such that $G$ is split at $\mathfrak{p}$ and the local component $π_\mathfrak{p}$ of $π$ at $\mathfrak{p}$ is the Steinberg representation. Assuming that the representation is non-critical at $\mathfrak{p}$ we construct automorphic $\mathcal{L}$-invariants for the representation $π$. If the number field $F$ is totally real, we show that these automorphic $\mathcal{L}$-invariants agree with the Fontaine-Mazur $\mathcal{L}$-invariant of the associated $p$-adic Galois representation. This generalizes a recent result of Spiess respectively Rosso and the first named author from the case of parallel weight $2$ to arbitrary cohomological weights.

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arXiv2019

Generalized Heegner cycles on Mumford curves

We study generalised Heegner cycles, originally introduced by Bertolini-Darmon-Prasanna for modular curves, in the context of Mumford curves. The main result of this paper relates generalized Heegner cycles with the two variable anticyclotomic $p$-adic $L$-function attached to a Coleman family $f_\infty$ and an imaginary quadratic field $K$. Our generalised Heegner cycles allow us to study the restriction of this function to non-central critical lines. The main result expresses the derivative along the weight variable of this anticyclotomic $p$-adic $L$-function restricted to non necessarily central critical lines as a combination of the image of generalized Heegner cycles under a $p$-adic Abel-Jacobi map. In studying generalised Heegner cycles in the context of Mumford curves, we also obtain an extension of a result of Masdeu for the (one variable) anticyclotomic $p$-adic $L$-function of a modular form $f$ and an imaginary quadratic field $K$ at non-central critical integers.

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arXiv2017

Exceptional zero formulae for anticyclotomic p-adic L-functions of elliptic curves in the ramified case

Iwasawa theory of modular forms over anticyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields has been studied by several authors, starting from the works of Bertolini-Darmon and Iovita-Spiess, under the crucial assumption that the prime $p$ is unramified in $K$. We start in this article the systematic study of anticyclotomic $p$-adic $L$-functions when $p$ is ramified in $K$. In particular, when $f$ is a weight $2$ modular form attached to an elliptic curve $E/\mathbb{Q}$ having multiplicative reduction at $p$, and $p$ is ramified in $K$, we show an analogue of the exceptional zeroes phenomenon investigated by Bertolini-Darmon in the setting when $p$ is inert in $K$. More precisely, we consider situations in which the $p$-adic $L$-function $L_p(E/K)$ of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ does not vanish identically but, by sign reasons, has a zero at certain characters $χ$ of the Hilbert class field of $K$. In this case we show that the value at $χ$ of the first derivative of $L_p(E/K)$ is equal to the formal group logarithm of the specialization at $p$ of a global point on the elliptic curve (actually, this global point is a twisted sum of Heegner points). This generalizes similar results of Bertolini-Darmon, available when $p$ is inert in $K$ and $χ$ is the trivial character.

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arXiv2015

Extensions of degree p^l of a p-adic field

Given a $p$-adic field $K$ and a prime number $\ell$, we count the total number of the isomorphism classes of $p^\ell$-extensions of $K$ having no intermediate fields. Moreover for each group that can appear as Galois group of the normal closure of such an extension, we count the number of isomorphism classes that contain extensions whose normal closure has Galois group isomorphic to the given group. Finally we determine the ramification groups and the discriminant of the composite of all $p^\ell$-extensions of K with no intermediate fields. The principal tool is a result, proved at the beginning of the paper, which states that there is a one-to-one correspondence between the isomorphism classes of extensions of degree $p^\ell$ of $K$ having no intermediate extensions and the irreducible $H$-submodules of dimension $\ell$ of $F^*/{F^*}^p$, where $F$ is the composite of certain fixed normal extensions of $K$ and $H$ is its Galois group over $K$.

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