Recent zbMATH articles in MSC 11Ghttps://zbmath.org/atom/cc/11G2024-03-13T18:33:02.981707ZUnknown authorWerkzeugAlgebraic number theory. Abstracts from the workshop held June 25--30, 2023https://zbmath.org/1528.110022024-03-13T18:33:02.981707ZSummary: Algebraic Number Theory is an area of Mathematics that has a legendary history and lies at the interface of Algebra and Number Theory. The last four decades of the last century witnessed rapid developments that led to connections with other areas such as Algebraic Geometry, Representation Theory, Harmonic Analysis, Iwasawa theory, to mention a few. In the last two decades, emergent areas such as \(p\)-adic Analysis, \(p\)-adic Geometry (\(p\) is a prime number) led to additional new facets. More recent developments in Arithmetic Geometry via Perfectoid Spaces and other emerging areas have added newer facets. The lectures in this workshop present current developments in these diverse areas.Formulas for moments of class numbers in arithmetic progressionshttps://zbmath.org/1528.110222024-03-13T18:33:02.981707Z"Bringmann, Kathrin"https://zbmath.org/authors/?q=ai:bringmann.kathrin"Kane, Ben"https://zbmath.org/authors/?q=ai:kane.ben"Pujahari, Sudhir"https://zbmath.org/authors/?q=ai:pujahari.sudhirThe well-known identity \(\sum_{t\in\mathbb{Z}} H(4_p-t^2)= 2p\) where \(p\) is a prime and \(H\) denotes the Hurwitz class number is generalized to
\[
H_{2,m,3}(n) := \sum_{\substack{t\in\mathbb{Z}\\ t\equiv m\bmod 3}} t^2 H(4n-t^2)\quad\text{for }m\in\{0,1,2\}.
\]
A special result for \(p\equiv2\bmod 3\) is \(H_{2,1,3}(p)=\frac{p(p+1)}{2}\). The proof uses non-holomorphic modular forms and their growth towards the cusps and evaluations of generalized quadratic Gauss sums.
Reviewer: Meinhard Peters (Münster)The Gross-Zagier-Zhang formula over function fieldshttps://zbmath.org/1528.110322024-03-13T18:33:02.981707Z"Qiu, Congling"https://zbmath.org/authors/?q=ai:qiu.conglingThe conjecture of Birch and Swinnerton-Dyer establishes that
\[
\mathrm{rank}_{\mathbb Z} A(F)=\mathrm{ord}_{s=1}L(s,A),
\]
where \(A\) is an elliptic curve defined over a global field \(F\). When \(F={\mathbb Q}\), \textit{B. H. Gross} and \textit{D. B. Zagier} [Invent. Math. 84, 225--320 (1986; Zbl 0608.14019)] established a formula relating the Néron-Tate height of a Heegner point on \(A\) associated to an imaginary quadratic field \(E\), and the derivative \(L'(1, A_E)\). The analog for global function fields of odd characteristic was established by \textit{U. Tipp} and \textit{H.-G. Rück} [Doc. Math. 5, 365--444 (2000; Zbl 1012.11039)]. The work of Gross and Zagier was generalized by \textit{S. Zhang} to Shimura curves over totally real number fields [Ann. Math. (2) 153, No. 1, 27--147 (2001; Zbl 1036.11029); Asian J. Math. 5, No. 2, 183--290 (2001; Zbl 1111.11030)].
In the paper under review, the author generalizes the analog of the Gross-Zagier-Zhang formula to global function fields of arbitrary characteristic (Theorem 1.2.1). As a consequence, it is proved the Waldspurger formula [\textit{J. L. Waldspurger}, Compos. Math. 54, 173--242 (1985; Zbl 0567.10021)] (Theorem 1.2.3).
As an application of the Gross-Zagier-Zhang formula, the Birch and Swinnerton-Dyer conjecture is proved for elliptic curves of analytic rank 1 in arbitrary positive characteristic (Theorem 1.2.2).
One of the main tools used to prove the Gross-Zagier-Zhang formula and the Waldspurger formula, is the relative trace formulas of \textit{H. Jacquet} [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 2, 185--229 (1986; Zbl 0605.10015); Compos. Math. 63, 315--389 (1987; Zbl 0633.10029)] and \textit{H. Jacquet} and \textit{C. Nan} [Bull. Soc. Math. Fr. 129, No. 1, 33--90 (2001; Zbl 1069.11017)] and \textit{W. Zhang}'s arithmetic analogs [Invent. Math. 188, No. 1, 197--252 (2012; Zbl 1247.14031)].
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)\(p\)-adic Eichler-Shimura maps for the modular curvehttps://zbmath.org/1528.110392024-03-13T18:33:02.981707Z"Rodríguez Camargo, Juan Esteban"https://zbmath.org/authors/?q=ai:rodriguez-camargo.juan-estebanIn his paper [Math. Ann. 278, 133--149 (1987; Zbl 0646.14026)], \textit{G. Faltings} described the Hodge-Tate decomposition of the étale cohomology of the modular curve \(Y_{K_p}\), where \(K_p \subset \mathrm{GL}_2(\mathbb{Q}_p)\) is an open compact subgroup and \(Y_{K_p}\) is the rigid analytic varieties attached to the modular curves. More precisely, let \(E\) be the universal elliptic curve over \(Y_{K_p}\). This admits an extension to a semi-abelian adic space \(E^{sm}\) over \(X_{K_p}\). Let \(e: X_{K_p} \rightarrow E^{sm}\) be the unit section, \(\omega_E=e^{*}\Omega_{E^{sm}/X}^1\) the modular sheaf and \(T_p E=\varprojlim_{n} E[p^n]\) the Tate module over \(Y_{K_p}\). Let \(k \geq 0\) be an integer. There exists a Galois and Hecke equivariant isomorphism
\[
H_{\text{ét}}^1(Y_{K_p, \mathbb{C}_p}, \mathrm{Sym}^k T_p(E))\otimes_{\mathbb{Q}_p} \mathbb{C}_p(1)=H_{\text{an}}^0(X_{K_p, \mathbb{C}_p}, \omega_E^{k+2}) \oplus H_{\text{an}}^1(X_{K_p, \mathbb{C}_p}, \omega_E^{-k})(k+1)
\]
called the Eichler-Shimura decomposition.
The first result of the present paper is a new proof of Faltings's Eichler-Shimura decomposition using Bernstein-Gelfand-Gelfand (BGG) methods and the Hodge-Tate period map. The author's proof is the proétale analogue of the BGG decomposition for the de Rham cohomology of \textit{G. Faltings} and \textit{C.-L. Chai} [Degeneration of abelian varieties. Berlin etc.: Springer-Verlag (1990; Zbl 0744.14031)]. The second goal of this paper is the interpolation of the above Eichler-Shimura decomposition. He constructs over-convergent Eichler-Shimura maps for the modular curves providing ``the second half'' of the over-convergent Eichler-Shimura map of Andreatta, Iovita and Stevens [\textit{F. Andreatta} et al., J. Inst. Math. Jussieu 14, No. 2, 221--274 (2015; Zbl 1379.11062)]. He gets over-convergent modular sheaves whose cohomology is the object of study in higher Coleman theory and shows that the small-slope part of the Eichler-Shimura maps interpolates the classical \(p\)-adic Eichler-Shimura decompositions. He also proves that over-convergent Eichler-Shimura maps are compatible with Poincaré and Serre pairings.
Reviewer: Lei Yang (Beijing)Cube sum problem for integers having exactly two distinct prime factorshttps://zbmath.org/1528.110422024-03-13T18:33:02.981707Z"Majumdar, Dipramit"https://zbmath.org/authors/?q=ai:majumdar.dipramit"Shingavekar, Pratiksha"https://zbmath.org/authors/?q=ai:shingavekar.pratikshaSummary: Given an integer \(n>1\), it is a classical Diophantine problem that whether \(n\) can be written as a sum of two rational cubes. The study of this problem, considering several special cases of \(n\), has a copious history that can be traced back to the works of
\textit{J. J. Sylvester} [Am. J. Math. 2, 280--285 (1879; JFM 11.0141.01)],
\textit{P. Satgé} [Invent. Math. 87, 425--439 (1987; Zbl 0616.14023)],
\textit{E. S. Selmer} [Acta Math. 85, 203--362 (1951; Zbl 0042.26905)] etc., and up to the recent works of \textit{L. Alpöge} et al. [``Integers expressible as the sum of two rational cubes'', Preprint, arXiv:2210.10730]. In this article, we consider the cube sum problem for cube-free integers \(n\) which are coprime to 3 and have exactly two distinct prime factors.On the surjectivity of \(\bmod\, \ell\) representations associated to elliptic curveshttps://zbmath.org/1528.110432024-03-13T18:33:02.981707Z"Zywina, David"https://zbmath.org/authors/?q=ai:zywina.davidSummary: Let \(E\) be an elliptic curve over the rationals that does not have complex multiplication. For each prime \(\ell \), the action of the absolute Galois group on the \(\ell \)-torsion points of \(E\) can be given in terms of a Galois representation \(\rho_{E,\ell }\colon \operatorname{Gal}({\overline{\mathbb{Q}}}/\mathbb{Q}) \rightarrow \operatorname{GL}_2(\mathbb{F}_\ell )\). An important theorem of Serre says that \(\rho_{E,\ell }\) is surjective for all sufficiently large \(\ell \). In this paper, we describe a simple algorithm based on Serre's proof that can quickly determine the finite set of primes \(\ell >13\) for which \(\rho_{E,\ell }\) is not surjective. We will also give some improved bounds for Serre's theorem.
{{\copyright} 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}Study of elliptic curve over a finite ring \(\mathbb{F}_{3^d}[\varepsilon]\), \(\varepsilon^4=\varepsilon^3\)https://zbmath.org/1528.110442024-03-13T18:33:02.981707Z"Selikh, Bilel"https://zbmath.org/authors/?q=ai:selikh.bilelSummary: Let \(\mathbb{F}_{3^d}\) be a finite field of order \(3^d\) with \(d\in\mathbb{N}^*\). In this paper, we study the elliptic curve over the finite ring \(\mathbb{F}_{3^d}[\varepsilon]:=\mathbb{F}_{3^d}[X]/ (X^4-X^3)\), where \(\varepsilon^4=\varepsilon^3\) of characteristic 3 given by the homogeneous Weierstrass equation of the form \[Y^2Z=X^3 +aX^2Z+bZ^3,\text{ where }a,b\in\mathbb{F}_{3d}[\varepsilon].\] Such that we study the arithmetic operations of this ring and define the elliptic curve over it. Next, we show that \(E_{\Pi_0(a),\Pi_0(b)} (\mathbb{F}_{3^d})\) and \(E_{\Pi_1(a),\Pi_1(b)}(\mathbb{F}_{3^d})\) are two elliptic curves over the finite field \(\mathbb{F}_{3^d}\), such that \(\Pi_0\) is a canonical projection and \(\Pi_1\) is a sum projection of coordinate of element in \(\mathbb{F}_{3^d}[\varepsilon]\) and we conclude by given a classification of elements in elliptic curve over the finite ring \(\mathbb{F}_{3^d}[\varepsilon]\).Laguerre type polynomials for rational function fields and applicationshttps://zbmath.org/1528.110452024-03-13T18:33:02.981707Z"Oukhaba, Hassan"https://zbmath.org/authors/?q=ai:oukhaba.hassan"El Kati, Mohamed"https://zbmath.org/authors/?q=ai:el-kati.mohamedIn this intriguing paper currently under review, the authors introduce a sequence of \({\mathbb F}_{q}\)-linear polynomials \({\mathcal L}_{n, \phi}\), where \(n\in {\mathbb N}\), that exhibit properties similar to those satisfied by the classical Laguerre polynomials. The construction of these polynomials, denoted as \({\mathcal L}_{n, \phi}\), relies on the theory of Carlitz modules.
To present their results, consider a fixed prime number \(p\) and let \(q\) be a power of \(p\). Let \(k = {\mathbb F}_{q}(T)\) be the rational function field over the finite field \({\mathbb F}_{q}\) with \(q\) elements. Additionally, define \(\tau= (x\mapsto x^{q})\) as the \(q\)-th power Frobenius map. The Carlitz module can be described as the homomorphism of \({\mathbb F}_{q}\)-algebras \(\phi : {\mathbb F}_{q}[T] \to k\{\tau\}\), which is determined by the image of \(T\), denoted by \(\phi_{T}\), satisfying \(\phi_{T} = \tau + T \tau^{0}\).
As it is customary, we denote \(R\{\!\{\tau\}\!\}\) (respectively, \(R\{\tau\}\)) as the \(R\)-algebra of (twisted) infinite power series (respectively, twisted polynomials) generated by \(\tau\) over \(R\) subject to the relation \(\tau r = r^{q} \tau, \; \text{ for all } \; r\in R.\)
There are two roles played by the polynomials \({\mathcal L}_{n, \phi}\): (i) as a polynomial in \(k \{\tau\}\), which is the ring of \({\mathbb F}_{q}\)-linear endomorphisms of the additive group \({\mathbf G}_{a}\) over \(k\), and (ii) as a polynomial in \(k[X]\), with \(X\) an indeterminate over \(k.\)
Let us start by (i) defining \({\mathcal L}_{n, \phi}\) as a polynomial in \(k \{\tau\}\). Let \(\Psi\) be the \(\bar{k}\)-linear operators defined on \(\bar{k}\{\!\{\tau\}\!\}\) given by \(\Psi\left(\sum_{i\ge 0} c_{i} \tau^{i}\right) = \sum_{i\ge 1} [i] \tau^{i-1}\) where \([i] = T^{q^{i}} - T\) for non-negative integers \(i\). Now, for any non-negative integer \(n\), we define the power series \({\mathcal L}_{n, \phi} = e_{\phi}(\tau) \Psi^{n}\left(\log_{\phi}(\tau) \cdot \tau^{n}\right)\), where \(\Psi^{n}\) denotes the \(n\)-th iterates of \(\Psi\), and \(e_{\phi}(\tau) (\mathrm{resp.} \log_{\phi}(\tau) )\) is an element in \(k\{\!\{\tau\}\!\}\) representing the exponential function (resp. logarithm function) of the Carlitz module \(\phi.\) It is shown that \({\mathcal L}_{n, \phi}\) satisfies the recursive relation \({\mathcal L}_{n+1, \phi} = {\mathcal L}_{n, \phi}[n+1] - \tau {\mathcal L}_{n, \phi} + \Delta \left({\mathcal L}_{n, \phi}\right)\) for any non-negative integer \(n\), where \(\Delta\) is the \(\bar{k}\)-derivation of \(\bar{k}\{\!\{\tau\}\!\}\) given by \(\Delta(f) = \Psi(f) \cdot \tau\) for \(f\in \bar{k}\{\!\{\tau\}\!\}.\)
From this recursive relation, it's not hard to deduce inductively that \({\mathcal L}_{n, \phi}\) is a polynomial of degree \(n\) in \(\tau\) with coefficients in \({\mathbb F}_{q}[T]\).
Analogous to the case of classical Laguerre polynomials, the sequence of polynomials \({\mathcal L}_{n, \phi}\) are solutions to a ``second order equations'' of the \(\bar{k}\)-derivation \(\Delta\). Specifically, for any non-negative integer \(n\) we have
\[
\Delta^{2}\left({\mathcal L}_{n, \phi}\right) - \tau \Delta\left({\mathcal L}_{n, \phi}\right) + \left({\mathcal L}_{n, \phi}\right)\left([n+1]-[1]\right) = \Lambda\left(\tau{\mathcal L}_{n, \phi} - {\mathcal L}_{n, \phi}\tau\right),
\]
where \(\Lambda(f) = \Delta(f) - \tau f.\)
For (ii), let's consider the polynomials \({\mathcal L}_{n, \phi}(X)\) in which \(\tau^{i}\) is replaced by \(X^{q^{i}}.\) Thus, \({\mathcal L}_{n, \phi}(X)\) are \({\mathbb F}_{q}\)-linear polynomials with coefficients in \({\mathbb F}_{q}[T].\) The authors study the Galois group over \(k\) of \({\mathcal L}_{n, \phi}(X).\) They show (Theorem~2) that for any positive integer \(n\), the polynomial \({\mathcal L}_{n, \phi}(X)/X\) is separable and irreducible in \({\mathbb F}_{q}[T][X]\). Moreover, the Galois group over \(k\) of \({\mathcal L}_{n, \phi}(X)\) is isomorphic to the general linear group \(\mathrm{GL}_{n}({\mathbb F}_{q})\). Hence, \({\mathcal L}_{n, \phi}(X)\) gives a direct example of polynomials with Galois group \(\mathrm{GL}_{n}({\mathbb F}_{q})\). Additionally, a consequence of this result is that \(\mathrm{GL}_{n}({\mathbb F}_{q})\) is realizable as a Galois group over \({\mathbb F}_{q^{e}}(T)\), for any positive integer \(e\).
The main ingredients of the proof of Theorem~2 in this paper are to study the decomposition subgroups at irreducible polynomials of \({\mathbb F}_{q}[T]\) of degree, respectively, equal to 1, \(n-1\) or \(n.\) For the polynomials of degree 1, tools from non-Archimedean analysis are used. In particular, the Newton polygons of the polynomials \({\mathcal L}_{n, \phi}(X)/X\) are computed and methods generally used in the theory of Lubin-Tate formal groups are applied to determine the decomposition groups at the degree 1 polynomials in question.
Reviewer: Liang-Chung Hsia (Taipei)A geometric approach to some systems of exponential equationshttps://zbmath.org/1528.110462024-03-13T18:33:02.981707Z"Aslanyan, Vahagn"https://zbmath.org/authors/?q=ai:aslanyan.vahagn-a"Kirby, Jonathan"https://zbmath.org/authors/?q=ai:kirby.jonathan"Mantova, Vincenzo"https://zbmath.org/authors/?q=ai:mantova.vincenzoThis paper gravitates around Zilber's model theoretic study of the complex exponential function, more precisely around which systems of equations involving polynomials and exponentials have solutions in \(\mathbb C\).
From the Introduction : ``Zilber formulated a precise conjecture that captures the idea that every system of equations should have a solution unless that would contradict Schanuel's conjecture, which we call his \textit{Exponential Algebraic Closedness} conjecture or \textit{EAC} conjecture (...) The EAC conjecture is expressed in geometric terms. Let \(\mathbb G^n_\mathrm{m}\) be the algebraic torus of dimension \(n.\) Since we are exclusively working over \(\mathbb C\), we shall identify \(\mathbb G_\mathrm{m}\) with its complex points, so \(\mathbb G_\mathrm{m}=\mathbb G_\mathrm{m}(\mathbb C)=\mathbb C^\times.\)
Conjecture 1.1 (EAC [\textit{B. Zilber}, Ann. Pure Appl. Logic 132, No. 1, 67--95 (2005; Zbl 1076.03024)]). Let \(V\subseteq \mathbb C^n\times \mathbb G^n_\mathrm{m}\) be a free and rotund variety. Then there is a point \(z\in\mathbb C^n\) such that \((z,\mathrm{exp}(z))\in V. \)
(\dots) we now know that EAC directly implies the quasiminimality of \((\mathbb C, +, \times, \mathrm {exp})\) [\textit{M. Bays} and \textit{J. Kirby}, Algebra Number Theory 12, No. 3, 493--549 (2018; Zbl 1522.03144)], Theorem 1.5]).
Apart from the classical exponential function, one can consider other periodic functions such as the Weierstrass \(\wp \)-functions and their derivatives. (...) In this generality, the EAC conjecture becomes the following.
Conjecture 1.3 (EAC for semiabelian varieties). Let \(S\) be a complex semiabelian variety of dimension \(n\), and write exp\(_S:\mathbb C^n\to S\) for its exponential map. Let \(V\subseteq \mathbb C^n\times S\) be a free and rotund subvariety. Then there is \(z\in\mathbb C^n\) such that \((z, \mathrm{exp} _S(z))\in V. \)
(\dots) At least when \(S\) is simple, EAC for \(S\) also implies that the structure \((\mathbb C, +, \times, \mathrm{exp} _S)\) is quasi-minimal [\textit{M. Bays} and \textit{J. Kirby}, Algebra Number Theory 12, No. 3, 493--549 (2018; Zbl 1522.03144)], Theorem 1.9]. (\dots) In this paper, we establish the following family of instances of Conjecture 1.3 in the case of abelian varieties.
Theorem 1.4 Let \(A\) be a complex abelian variety of dimension \(n\). Let \(V\subseteq \mathbb C^n\times A\) be an algebraic subvariety with dominant projection to \(\mathbb C^n\), that is, its projection to \(\mathbb C^n\) has dimension \(n.\) Then there is \(z\in\mathbb C^n\) such that \((z,\mathrm{exp} _A(z))\in V. \)
(...) The analogous theorem for algebraic tori was proven by Brownawell and Masser.''
The authors use similar but more geometric methods than Brownawell and Masser. They first illustrate their methods in section 2 of the paper, with the following equation, and hands-on calculations : \(\wp'(\wp(z)^2)=z\), where \(\wp\) is any Weierstrass \(\wp\)-function.
Reviewer: Luc Bélair (Montréal)An introduction to abelian varietieshttps://zbmath.org/1528.110472024-03-13T18:33:02.981707Z"Hindry, Marc"https://zbmath.org/authors/?q=ai:hindry.marc"Rebolledo, Marusia"https://zbmath.org/authors/?q=ai:rebolledo.marusia"Roberts, David"https://zbmath.org/authors/?q=ai:roberts.david-p|roberts.david-lindsay|roberts.david-j|roberts.david-l|roberts.david-m|roberts.david-b|roberts.david-w|roberts.david-michael|roberts.david-c|roberts.david-w.1(no abstract)A \(p\)-adic approach to singular moduli on Shimura curveshttps://zbmath.org/1528.110482024-03-13T18:33:02.981707Z"Giampietro, Sofia"https://zbmath.org/authors/?q=ai:giampietro.sofia"Darmon, Henri"https://zbmath.org/authors/?q=ai:darmon.henri\textit{B. H. Gross} and \textit{D. B. Zagier} [J. Reine Angew. Math. 355, 191--220 (1985; Zbl 0545.10015)] gave an explicit formula for the factorization of the norm of differences of singular moduli. In the present paper under review the authors define a rational invariant \(\mathcal J_N(D_1,D_2)\) associated to singular moduli of discriminants \(D_1\) and \(D_2\) on the genus-zero Shimura curves \(X_N\) of discriminant \(N=6, 10\) or \(22\) as the norm of a \(p\)-adic invariant \(J^{(p)}_N(\tau_1,\tau_2)\) and explore the similar factorization property of \(\mathcal J_N(D_1,D_2)\).
More precisely, let \(p\) be a prime dividing \(N\) and write \(N=pq\). Let \(B\) be the definite quaternion \(\mathbb Q\)-algebra of discriminant \(q\), and fix an embedding \(B\to M_2(\mathbb Q_p)\). Let \(R\) be a maximal \(\mathbb Z[1/p]\)-order, and \(\Gamma_N^{(p)}\subset \mathrm{SL}_2(\mathbb Q_p)\) be the image of norm-\(1\) elements of \(R^\times\). By Cerednik-Drinfeld's theorem, \(X_N(\mathbb C_p)\) admits a \(p\)-adic uniformization by the Drinfeld upper half plane \(\mathcal H_p\) with group \(\Gamma_N^{(p)}\).
Let \(D_1,D_2<0\) be two coprime fundamental discriminants with \((D_i/q)=-1\), and fix an embedding \(O_{D_i} \to R\) of the imaginary quadratic order \(O_{D_i}\) of discriminant \(D_i\). This gives rise to two \(p\)-adic conjugate complex multiplication points \(\{\tau_i,\tau_i'\}\) in \(\mathcal H_p\). Let
\[
[a,b,c,d]:=\frac{c-a}{c-b} \cdot \frac{d-b}{d-a}
\]
be the cross ratio of four elements, and define
\[
J^{(p)}_N(\tau_1,\tau_2):=\prod_{\gamma\in \Gamma_N^{(p)}} [\gamma \tau_1,\gamma \tau_1', \tau_2, \tau_2']\in \mathbb C_p.
\]
One can show that \(J^{(p)}_N(\tau_1,\tau_2)\) is algebraic over \(\mathbb Q\) and is contained in the compositum \(H_1H_2\) of the ring class fields \(H_i\) of \(O_{D_i}\). The authors define
\[
\mathcal J_N(D_1,D_2):=N_{H_1H_2/\mathbb Q} (J^{(p)}_N(\tau_1,\tau_2)).
\]
The authors give a simple recursive algorithm to compute \(J^{(p)}_N\), following the approach described in the thesis of \textit{I. Negrini} [On the computation of \(p\)-adic theta functions arising from the Hurwitz quaternions. Milan: Università degli Studi di Milano. (Master Thesis) (2017)]. Based on the computational results, the authors give a few results, observations, conjectures and remarks. The last section gives a comparison to \textit{E. Errthum}'s work [Can. J. Math. 63, No. 4, 826--861 (2011; Zbl 1233.11067)].
Reviewer: Chia-Fu Yu (Taipei)The Coleman-Oort conjecture: reduction to three key caseshttps://zbmath.org/1528.110492024-03-13T18:33:02.981707Z"Moonen, Ben"https://zbmath.org/authors/?q=ai:moonen.benSummary: We show that the Coleman-Oort conjecture can be reduced to three particular cases. As an application, we extend a result of \textit{X. Lu} and \textit{K. Zuo} [J. Math. Pures Appl. (9) 108, No. 4, 532--552 (2017; Zbl 1429.14016)], to the effect that for \(g \geqslant 8\) the Coleman-Oort conjecture is true on the hyperelliptic locus.
{{\copyright} 2022 The Authors. \textit{Bulletin of the London Mathematical Society} is copyright {\copyright} London Mathematical Society.}\(L\)-functions of noncommutative torihttps://zbmath.org/1528.110502024-03-13T18:33:02.981707Z"Nikolaev, Igor V."https://zbmath.org/authors/?q=ai:nikolaev.igor-vladimirovich|nikolaev.igor-vasilievichSummary: We introduce an analog of the \(L\)-function for noncommutative tori. It is proved that such a function coincides with the Hasse-Weil \(L\)-function of an elliptic curve with complex multiplication. As a corollary, one gets a localization formula for the noncommutative tori with real multiplication.On smooth plane models for modular curves of Shimura typehttps://zbmath.org/1528.110512024-03-13T18:33:02.981707Z"Anni, Samuele"https://zbmath.org/authors/?q=ai:anni.samuele"Assaf, Eran"https://zbmath.org/authors/?q=ai:assaf.eran"Lorenzo García, Elisa"https://zbmath.org/authors/?q=ai:lorenzo-garcia.elisaA modular curve \(X_{\Gamma}\) is the compactification, by normalisation, of the quotient space of the complex upper half plane by the action of a subgroup \(\Gamma\) of \(\mathrm{SL}_2(\mathbb{Z})\), and it admits the structure of a compact Riemann surface. Furthermore, \(X_{\Gamma}(\mathbb{C})\) is a projective complex algebraic curve. In this paper, the problem of computing equations for such curves and their projective embeddings is investigated. It is proved that there are finitely many modular curves which admit a smooth plane model over the rationals. There is no modular curve which admits a smooth plane model of degree greater or equal to 19. Moreover, there is no modular curve of Shimura type which admits a smooth plane model of degree 5, 6 or 7. A modular curve of Shimura type which admits a smooth plane model of degree 8 must be a twist of one of four curves.
Reviewer: Dimitros Poulakis (Thessaloniki)Torsion properties of modified diagonal classes on triple products of modular curveshttps://zbmath.org/1528.110522024-03-13T18:33:02.981707Z"Lilienfeldt, David T.-B. G."https://zbmath.org/authors/?q=ai:lilienfeldt.david-t-b-gIn this interesting paper the author studies the modified diagonal cycle on the triple product of modular curves. The modified diagonal cycle is a canonical nullhomologous cycle one can define on the $n$-fold product of curves. This was first introduced by \textit{B. H. Gross} and \textit{C. Schoen} [Ann. Inst. Fourier 45, No. 3, 649--679 (1995; Zbl 0822.14015)] and further studied by \textit{B. H. Gross} and \textit{S. S. Kudla} [Compos. Math. 81, No. 2, 143--209 (1992; Zbl 0807.11027)]. The most interesting case is when $n=3$. According to the conjectures of Bloch and Beilison the height of this cycle is expected to be related to the $L$-value of the 3rd cohomology group of the triple product of the curve. Precisely, the value of the derivative at $s=2$ is a non-zero multiple of the height of the modified diagonal cycle. This is now a theorem, though it is still unpublished, of \textit{X. Yuan} et al. [``Triple product \(L\)-series and Gross-Kudla-Schoen cycles'', Preprint, \url{http://math.mit.edu/ wz2113/math/online/triple.pdf}], when the curve is a Shimura curve or a modular curve. In that case, since the motive of the curve decomposes in to motives of modular forms, the $L$-function in question is related to the Rankin triple product $L$-function of cusp forms of weight 2 for the Shimura curve.
In this paper the author studies the triple product of the modular curve $X_0(p)$. Let $f_1$, $f_2$ and $f_3$ be three cusp forms of weight 2. Let $F=f_1\otimes f_2\otimes f_3$ be the associated cusp form of weight (2, 2, 2) for $\Gamma_0(p)^3$. He considers the case when the triple product $L$-function has global root number $W(F)=+1$. This implies that $L$-function vanishes to even multiplicity at the point $s=2$. Assuming the conjectures, this would imply that the height of the $F$-component of the modified diagonal cycle is torsion. Using the Atkin-Lerner involution as a correspondence on $X_0(p)$ and it action on the modified diagonal cycle he is able to show that indeed this is the case. This is very satisfactory as it shows that the Atkin-Lerner involution -- which is a object coming from the theory of modular forms -- implies something about the geometric side of the conjecture very directly. He studies two subcases -- the case when the modular curve is an elliptic curve and the general case. In the case when it is an elliptic curve, there is only one form $f$ and one knows that $L(F,s)=0$. One can also show independently that the modified diagonal cycle is torsion. So this is consistent with the Bloch-Beilinson conjectures. The other case he studies is $X_0(37)$, which is a hyperelliptic curve. In this case, from the hyperelliptic involution one can conclude that the modified diagonal cycle is torsion. However, he provides a different argument as the Atkin-Lerner involution is not the same as the hyperelliptic involution. Finally, he applies this to torsion of Chow-Heegner points, which are points on elliptic curves determined by these exceptional cycles.
Reviewer: Ramesh Sreekantan (Bangalore)On the supersingular locus of the Shimura variety for \(\mathrm{GU}(2,2)\) over a ramified primehttps://zbmath.org/1528.110532024-03-13T18:33:02.981707Z"Oki, Yasuhiro"https://zbmath.org/authors/?q=ai:oki.yasuhiroSummary: We study the structure of the supersingular locus of the Rapoport-Zink integral model of the Shimura variety for \(\mathrm{GU}(2,2)\) over a ramified odd prime with the special maximal parahoric level. We prove that the supersingular locus equals the disjoint union of two basic loci, one of which is contained in the flat locus, and the other is not. We also describe explicitly the structure of the basic loci. More precisely, the former one is purely \(2\)-dimensional, and each irreducible component is birational to the Fermat surface. On the other hand, the latter one is purely \(1\)-dimensional, and each irreducible component is birational to the projective line.Curves over finite fieldshttps://zbmath.org/1528.110542024-03-13T18:33:02.981707Z"Abdón, Miriam"https://zbmath.org/authors/?q=ai:abdon.miriam"Carvalho, Cícero"https://zbmath.org/authors/?q=ai:carvalho.cicero-f"Panario, Daniel"https://zbmath.org/authors/?q=ai:panario.daniel(no abstract)Genus two curves with full \(\sqrt{3}\)-level structure and Tate-Shafarevich groupshttps://zbmath.org/1528.110552024-03-13T18:33:02.981707Z"Bruin, Nils"https://zbmath.org/authors/?q=ai:bruin.nils"Flynn, E. Victor"https://zbmath.org/authors/?q=ai:flynn.eugene-victor"Shnidman, Ari"https://zbmath.org/authors/?q=ai:shnidman.ariBruin, Flynn and Shnidman study here the moduli space \(H_3\) of principally polarized abelian surfaces with real multiplication by \(\sqrt{3}\) and full \(\sqrt{3}\)-level structure. This \(H_3\) turns out to be a rational surface. More precisely, the present authors realize the universal abelian surface over an open subset of \(H_3\) as the (relative) Jacobian of an explicit family \(C\rightarrow\mathbb{P}^2\smallsetminus\Delta\) of genus two curves.
As a first application, they show the following result. For \(x\in(\mathbb{P}^2\smallsetminus\Delta)(\mathbb{Q})\), put \(A_x=\mathrm{Jac}(C_x)/\bigl<P_x\bigr>\) where \(P_x\) is the marked point of order \(3\). Fix \(r\geqslant1\) and order points in \(\mathbb{P}^2(\mathbb{Q})\) by height. Then for \(100\%\) of points \(x\in\mathbb{P}^2(\mathbb{Q})\), a positive proportion of the quadratic twists \(A_{x,d}\) of \(A_x\) satisfy \(\#\Sha(A_{x,d})[3]\geqslant3^r\). Here \(d\) varies through squarefree integers and \(\Sha(A)[3]\) denotes the \(3\)-torsion subgroup of the Tate-Shafarevich group of \(A\).
As a second application, they consider \(y=(1:2:-1)\) and give an explicit bound on the average Mordell-Weil rank of the twists \(\mathrm{Jac}(C_y)_d\), as well as statistical results on the number of \(\mathbb{Q}\)-rational points of \(C_{y,d}\).
Reviewer: Pascal Autissier (Bordeaux)Moments of Kloosterman sums, supercharacters, and elliptic curveshttps://zbmath.org/1528.110712024-03-13T18:33:02.981707Z"Sayed, Fahim"https://zbmath.org/authors/?q=ai:sayed.fahim"Kalita, Gautam"https://zbmath.org/authors/?q=ai:kalita.gautamAuthors' abstract: In this paper, we use a supercharacter theory for \(\mathbb{F}_p^2\) induced by the action of a subgroup of \(\mathrm{GL}_2\left(\mathbb{F}_p\right)\) to express the fifth and seventh power moments of Kloosterman sums in terms of traces of Frobenius endomorphism for certain families of elliptic curves.
Reviewer: Alexey Ustinov (Khabarovsk)Sums of multiplicative coefficients twisted by Frobenius traceshttps://zbmath.org/1528.110732024-03-13T18:33:02.981707Z"Zhou, Yuxuan"https://zbmath.org/authors/?q=ai:zhou.yuxuanLet \(\mathbb{E}\) be a smooth projective curve over a finite field \(\mathbb{F}_q\) of \(q\) elements. The Frobenius traces \(A_{\mathbb{E}}(n)\in\mathbb{Z}\) associated with \(q\) satisfy the Weil type bound \(|A_{\mathbb{E}}(n)|\leq 2gq^{n/2}\), where \(g\geq 1\) is the genus of \(\mathbb{E}\), and the normalized Frobenius traces \(a_{\mathbb{E}}(n)=A_{\mathbb{E}}(n)/(2gq^{n/2})\in[-1,1]\). In this paper, the author obtains upper bound of order \(N/\log N\) for the weighted sums \(\sum_{n\leq N}f(n)a_{\mathbb{E}}(n)\) and \(\sum_{n\leq N}\lambda_\pi(n)a_{\mathbb{E}}(n)\), where \(f\) belongs to a family of multiplicative functions satisfying some certain conditions, and \(\pi\) is an automorphic irreducible cuspidal representation of \(\mathrm{GL}_m\) over \(\mathbb{Q}\) with unitary central character and admitting an \(L\)-function with coefficients \(\lambda_\pi(n)\).
Reviewer: Mehdi Hassani (Zanjan)Residual supersingular Iwasawa theory over quadratic imaginary fieldshttps://zbmath.org/1528.111142024-03-13T18:33:02.981707Z"Hamidi, Parham"https://zbmath.org/authors/?q=ai:hamidi.parhamSummary: Let \(p\) be an odd prime. Let \(E\) be an elliptic curve defined over a quadratic imaginary field, where \(p\) splits completely. Suppose \(E\) has supersingular reduction at primes above \(p\). Under appropriate hypotheses, we extend the results of [\textit{F. A. E. Nuccio Mortarino Majno di Capriglio} and \textit{S. Ramdorai}, Rend. Semin. Mat. Univ. Padova 149, 83--129 (2023; Zbl 1517.11139)] to \(\mathbb{Z}_p^2\)-extensions. We define and study the fine double-signed residual Selmer groups in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed \(\mu\)-invariants of one elliptic curve implies the vanishing of the signed \(\mu\)-invariants of the other. Finally, we show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.Explicit uniform bounds for Brauer groups of singular \(K3\) surfaceshttps://zbmath.org/1528.140252024-03-13T18:33:02.981707Z"Balestrieri, Francesca"https://zbmath.org/authors/?q=ai:balestrieri.francesca"Johnson, Alexis"https://zbmath.org/authors/?q=ai:johnson.alexis"Newton, Rachel"https://zbmath.org/authors/?q=ai:newton.rachel\(K3\) surfaces provide a remarkable example of algebraic varieties for which the Hasse principle does not hold. Although a complete explanation for the failure of the Hasse principle for \(K3\) surfaces is still missing, we know one reason that might occur: the \textit{Brauer-Manin obstruction}. This obstruction to the Hasse principle can be explained in terms of the Brauer group.
More precisely, let \(k\) be a number field and \(X\) a \(K3\) surface over \(k\). The \textit{Brauer group} of \(X\), denoted by \(\mathrm{Br}\,X\) is defined as \(H^2(X,\mathbf{G}_m)\). Let us fix an algebraic closure \(\overline{k}\) of \(k\); let \(X(\mathbf{A}_k)\) denote the set of adelic points of \(X\). Manin showed that there is a pairing on \(\mathrm{Br}\,X \times X(\mathbf{A}_k)\) whose left kernel, denoted by \(X(\mathbf{A}_k)^{\mathrm{Br}}\), fits in the following chain of inclusions:
\[
X(k)\subseteq X(\mathbf{A}_k)^{\mathrm{Br}} \subseteq X(\mathbf{A}_k) \subseteq X( \overline{k} )\; .
\]
If \(X(\mathbf{A}_k)\neq \emptyset\) and \(X(\mathbf{A}_k)^{\mathrm{Br}}=\emptyset\) then we say that \(X\) has the Brauer-Manin obstruction to the Hasse principle. This approach to the study of rational points of a \(K3\) surface led to an increasing interest in their Brauer group. The Brauer group of a \(K3\) surface \(X\) over \(k\) fits in the following sequence, where \(\overline{X}=X\times_k \overline{k}\).
\[
\begin{tikzcd} \mathrm{Br}\, k \arrow[r]{}{\alpha} & \mathrm{Br}\,X \arrow[r]{}{\beta} & \mathrm{Br}\,\overline{X} \end{tikzcd}
\]
The above sequence gives a filtration of the Brauer group: \(\mathrm{Br}_1\, X\) and \(\mathrm{Br}_0\, X\) denote \(\ker \beta\) and \(\mathrm{im} \alpha\), respectively. Clearly \(\mathrm{Br}_0\, X \subseteq \mathrm{Br}_1\, X \subseteq \mathrm{Br}\, X\) and hence we can consider the following quotients: \(\mathrm{Br}_1\, X/ \mathrm{Br}_0\, X\) and \(\mathrm{Br}\, X/ \mathrm{Br}_1\, X\), called the algebraic and transcendental parts of \(\mathrm{Br}\, X\), respectively.
Let \(X\) be a \(K3\) surface defined over a number field \(k\) and assume that \begin{center} the Néron-Severi lattice of \(X\) is isometric to the Néron-Severi lattice of the Kummer surface associated to a product of isogenous (not necessarily full) CM elliptic curves over~\(\overline{k}\). \end{center} Then, in the paper under review, the authors give an explicit upper bound for the size of the transcendental part of the Brauer group of \(X\). The bound is explicit in terms of the discriminant of the Néron-Severi lattice and of \([k:\mathbb{Q}]\). The bound is not expected to be sharp and it can be greatly improved if one allows for some extra assumptions, like:
\begin{itemize}
\item that \(X\) is isomorphic to the Kummer surface associated to a product of isogenous (not necessarily full) CM elliptic curves, or
\item the Generalised Riemann Hypothesis.
\end{itemize}
Under the GRH, the authors obtain analogous results for
\begin{itemize}
\item singular \(K3\) surfaces whose Néron-Severi lattice is isometric to the Néron-Severi lattice of the Kummer surface associated to the product of two elliptic curves over \(\overline{k}\), and
\item for Kummer surfaces associated to a product of \textit{non}-isogenous (not necessarily full) CM elliptic curves over \(\overline{k}\).
\end{itemize}
The proof of the main result is based on the study of the transcendental part of the Brauer group of abelian surfaces obtained as product of elliptic curves.
Historically, the transcendental part of the Brauer group is the hardest part to compute, making the above result particularly relevant. The explicit nature of the result implies that if \(X\) is a singular \(K3\) surface over \(k\) that is also geometrically Kummer, then the set \(X(\mathbf{A}_k)^{\mathrm{Br}}\) is effectively computable.
Reviewer: Dino Festi (Milano)Equidistribution, potential theory and arithmetic applicationshttps://zbmath.org/1528.300072024-03-13T18:33:02.981707Z"Burgos, José Ignacio"https://zbmath.org/authors/?q=ai:burgos.jose-ignacio"Menares, Ricardo"https://zbmath.org/authors/?q=ai:menares.ricardo(no abstract)An efficient key recovery attack on SIDHhttps://zbmath.org/1528.940382024-03-13T18:33:02.981707Z"Castryck, Wouter"https://zbmath.org/authors/?q=ai:castryck.wouter"Decru, Thomas"https://zbmath.org/authors/?q=ai:decru.thomasSummary: We present an efficient key recovery attack on the Supersingular Isogeny Diffie-Hellman protocol (SIDH). The attack is based on Kani's ``reducibility criterion'' for isogenies from products of elliptic curves and strongly relies on the torsion point images that Alice and Bob exchange during the protocol. If we assume knowledge of the endomorphism ring of the starting curve then the classical running time is polynomial in the input size (heuristically), apart from the factorization of a small number of integers that only depend on the system parameters. The attack is particularly fast and easy to implement if one of the parties uses 2-isogenies and the starting curve comes equipped with a non-scalar endomorphism of very small degree; this is the case for SIKE, the instantiation of SIDH that recently advanced to the fourth round of NIST's standardization effort for post-quantum cryptography. Our Magma implementation breaks \texttt{SIKEp434}, which aims at security level 1, in about ten minutes on a single core.
For the entire collection see [Zbl 1525.94005].A direct key recovery attack on SIDHhttps://zbmath.org/1528.940702024-03-13T18:33:02.981707Z"Maino, Luciano"https://zbmath.org/authors/?q=ai:maino.luciano"Martindale, Chloe"https://zbmath.org/authors/?q=ai:martindale.chloe"Panny, Lorenz"https://zbmath.org/authors/?q=ai:panny.lorenz"Pope, Giacomo"https://zbmath.org/authors/?q=ai:pope.giacomo"Wesolowski, Benjamin"https://zbmath.org/authors/?q=ai:wesolowski.benjaminSummary: We present an attack on SIDH utilising isogenies between polarized products of two supersingular elliptic curves. In the case of arbitrary starting curve, our attack (discovered independently from [\textit{W. Castryck} and \textit{T. Decru}, Lect. Notes Comput. Sci. 14008, 423--447 (2023; Zbl 1528.94038)]) has subexponential complexity, thus significantly reducing the security of SIDH and SIKE. When the endomorphism ring of the starting curve is known, our attack (here derived from [loc. cit.]) has polynomial-time complexity assuming the generalised Riemann hypothesis. Our attack applies to any isogeny-based cryptosystem that publishes the images of points under the secret isogeny, for example, Séta and B-SIDH. It does not apply to CSIDH, CSI-FiSh, or SQISign.
For the entire collection see [Zbl 1525.94005].Breaking SIDH in polynomial timehttps://zbmath.org/1528.940752024-03-13T18:33:02.981707Z"Robert, Damien"https://zbmath.org/authors/?q=ai:robert.damienSummary: We show that we can break SIDH in (classical) polynomial time, even with a random starting curve \(E_0\).
For the entire collection see [Zbl 1525.94005].New algorithms for the Deuring correspondence. Towards practical and secure SQISign signatureshttps://zbmath.org/1528.940952024-03-13T18:33:02.981707Z"De Feo, Luca"https://zbmath.org/authors/?q=ai:de-feo.luca"Leroux, Antonin"https://zbmath.org/authors/?q=ai:leroux.antonin"Longa, Patrick"https://zbmath.org/authors/?q=ai:longa.patrick"Wesolowski, Benjamin"https://zbmath.org/authors/?q=ai:wesolowski.benjaminSummary: The Deuring correspondence defines a bijection between isogenies of supersingular elliptic curves and ideals of maximal orders in a quaternion algebra. We present a new algorithm to translate ideals of prime-power norm to their corresponding isogenies -- a central task of the effective Deuring correspondence. The new method improves upon the algorithm introduced in [\textit{L. De Feo} et al., Lect. Notes Comput. Sci. 12491, 64--93 (2020; Zbl 1511.94176)] as a building-block of the SQISign signature scheme. SQISign is the most compact post-quantum signature scheme currently known, but is several orders of magnitude slower than competitors, the main bottleneck of the computation being the ideal-to-isogeny translation. We implement the new algorithm and apply it to SQISign, achieving a more than two-fold speedup in key generation and signing with a new choice of parameter. Moreover, after adapting the state-of-the-art \(\mathbb{F}_{p^2}\) multiplication algorithms by Longa to implement SQISign's underlying extension field arithmetic and adding various improvements, we push the total speedups to over three times for signing and four times for verification.
In a second part of the article, we advance cryptanalysis by showing a very simple distinguisher against one of the assumptions used in SQISign. We present a way to impede the distinguisher through a few changes to the generic KLPT algorithm. We formulate a new assumption capturing these changes, and provide an analysis together with experimental evidence for its validity.
For the entire collection see [Zbl 1525.94005].