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Florian Sprung

Publications and source records attributed to Florian Sprung.

14 recordsLinked to original sources
arXiv2023

On the signed Selmer groups for motives at non-ordinary primes in $\mathbb{Z}_p^2$-extensions

Generalizing the work of Kobayashi and the second author for elliptic curves with supersingular reduction at the prime $p$, B\"uy\"ukboduk and Lei constructed multi-signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a number field $F$ for more general non-ordinary motives. In particular, their construction applies to abelian varieties over $F$ with good supersingular reduction at all the primes of $F$ above $p$. In this article, we scrutinize the case in which $F$ is imaginary quadratic, and prove a control theorem (that generalizes Kim's control theorem for elliptic curves) of multi-signed Selmer groups of non-ordinary motives over the maximal abelian pro-$p$ extension of $F$ that is unramified outside $p$, which is the $\mathbb{Z}_p^2$-extension of $F$. We apply it to derive a sufficient condition when these multi-signed Selmer groups are cotorsion over the corresponding two-variable Iwasawa algebra. Furthermore, we compare the Iwasawa $\mu$-invariants of multi-signed Selmer groups over the $\mathbb{Z}_p^2$-extension for two such representations which are congruent modulo $p$.

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arXiv2022

La matrice de logarithme en termes de chiffres $p$-adiques (The logarithm matrix in terms of $p$-adic digits)

Nous donnons une nouvelle description de la matrice de logarithme d'une forme modulaire en termes de distributions, généralisant le travail de Dion et Lei pour le cas $a_p=0$. Ce qui nous permet d'inclure le cas $a_p\ne 0$ est une nouvelle définition, celle d'une matrice de distributions, et la caractérisation de cette matrice par de chiffres $p$-adiques. On peut appliquer ces méthodes au cas correspondant d'une distribution à plusieurs variables. -- We give a new description of the logarithm matrix of a modular form in terms of distributions, generalizing the work of Dion and Lei for the case $a_p=0$. What allows us to include the case $a_p\ne 0$ is a new definition, that of a distribution matrix, and the characterization of this matrix by $p$-adic digits. One can apply these methods to the corresponding case of distributions in multiple variables.

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arXiv2022

Studying Hilbert's 10th problem via explicit elliptic curves

N.García-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. We improve their proportions and extend their results to the case of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{Dq})$, where $D$ belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.

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arXiv2022

On characteristic power series of dual signed Selmer groups

We relate the cardinality of the $p$-primary part of the Bloch-Kato Selmer group over $\mathbb{Q}$ attached to a modular form at a non-ordinary prime $p$ to the constant term of the characteristic power series of the signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. This generalizes a result of Vigni and Longo in the ordinary case. In the case of elliptic curves, such results follow from earlier works by Greenberg, Kim, the second author, and Ahmed-Lim, covering both the ordinary and most of the supersingular case.

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arXiv2018

Ranks of elliptic curves over Z_p^2-extensions

Let $E$ be an elliptic curve with good reduction at a fixed odd prime $p$ and $K$ an imaginary quadratic field where $p$ splits. We give a growth estimate for the Mordell-Weil rank of $E$ over finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$.

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arXiv2018

On the Iwasawa main conjectures for modular forms at non-ordinary primes

In this paper, we prove under mild hypotheses the Iwasawa main conjectures of Lei--Loeffler--Zerbes for modular forms of weight $2$ at non-ordinary primes. Our proof is based on the study of the two-variable analogues of these conjectures formulated by Büyükboduk--Lei for imaginary quadratic fields in which $p$ splits, and on anticyclotomic Iwasawa theory. As application of our results, we deduce the $p$-part of the Birch and Swinnerton-Dyer formula in analytic ranks $0$ or $1$ for abelian varieties over $\mathbb{Q}$ of ${\rm GL}_2$-type for non-ordinary primes $p>2$.

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arXiv2016

Zeta-polynomials for modular form periods

Answering problems of Manin, we use the critical $L$-values of even weight $k\geq 4$ newforms $f\in S_k(Γ_0(N))$ to define zeta-polynomials $Z_f(s)$ which satisfy the functional equation $Z_f(s)=\pm Z_f(1-s)$, and which obey the Riemann Hypothesis: if $Z_f(ρ)=0$, then $\operatorname{Re}(ρ)=1/2$. The zeros of the $Z_f(s)$ on the critical line in $t$-aspect are distributed in a manner which is somewhat analogous to those of classical zeta-functions. These polynomials are assembled using (signed) Stirling numbers and "weighted moments" of critical values $L$-values. In analogy with Ehrhart polynomials which keep track of integer points in polytopes, the $Z_f(s)$ keep track of arithmetic information. Assuming the Bloch--Kato Tamagawa Number Conjecture, they encode the arithmetic of a combinatorial arithmetic-geometric object which we call the "Bloch-Kato complex" for $f$. Loosely speaking, these are graded sums of weighted moments of orders of Šafarevič-Tate groups associated to the Tate twists of the modular motives.

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arXiv2015

On pairs of p-adic analogues of the conjectures of Birch and Swinnerton-Dyer

For a weight two modular form and a good prime $p$, we construct a vector of Iwasawa functions $(L_p^\sharp,L_p^\flat)$. In the elliptic curve case, we use this vector to put the $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer for ordinary [MTT] and supersingular [BPR] primes on one footing. Looking at $L_p^\sharp$ and $L_p^\flat$ individually leads to a stronger conjecture containing an extra zero phenomenon. We also give an explicit upper bound for the analytic rank in the cyclotomic direction and an asymptotic formula for the $p$-part of the analytic size of the Šafarevič-Tate group in terms of the Iwasawa invariants of $L_p^\sharp$ and $L_p^\flat$. A very puzzling phenomenon occurs in the corresponding formulas for modular forms. When $p$ is supersingular, we prove that the two classical $p$-adic $L$-functions ([AV75],[VI76]) have finitely many common zeros, as conjectured by Greenberg.

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arXiv2015

A formulation for p-adic versions of the Birch and Swinnerton-Dyer conjectures in the supersingular case

Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L^\sharp(E,T) and L^\flat(E,T). They are equivalent to the conjectures of Perrin-Riou and Bernardi. We also generalize work of Kurihara and Pollack to give a criterion for positive rank in terms of the value of the quotient between these functions, and derive a result towards a non-vanishing conjecture. We also generalize a conjecture of Kurihara and Pollack concerning the greatest common divisor of the two functions to the general supersingular case.

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arXiv2015

On pairs of p-adic L-functions for weight two modular forms

The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L_p^\sharp(f,T) and L_p^\flat(f,T) for a weight two modular form \sum a_n q^n and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed a_p=0. The Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: We bound the rank and estimate the growth of the Tate-Shafarevich group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.

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arXiv2011

The Shafarevich-Tate group in cyclotomic Z_p-extensions at supersingular primes

We study the asymptotic growth of the p-primary component of the Shafarevich-Tate group in the cyclotomic direction at any odd prime of good supersingular reduction, generalizing work of Kobayashi. This explains formulas obtained by Kurihara, Perrin-Riou, and Nasybullin in terms of Iwasawa invariants of modified Selmer groups.

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