Let $E$ be an elliptic curve without complex multiplication (CM) over a number field $K$ , and let $G_{E}(\ell )$ be the image of the Galois representation induced by the action of the absolute Galois group of  $K$ on the $\ell$ -torsion subgroup of  $E$ . We present two probabilistic algorithms to simultaneously determine $G_{E}(\ell )$ up to local conjugacy for all primes $\ell$ by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine $G_{E}(\ell )$ up to one of at most two isomorphic conjugacy classes of subgroups of $\mathbf{GL}_{2}(\mathbf{Z}/\ell \mathbf{Z})$ that have the same semisimplification, each of which occurs for an elliptic curve isogenous to $E$ . Under the GRH, their running times are polynomial in the bit-size $n$ of an integral Weierstrass equation for  $E$ , and for our Monte Carlo algorithm, quasilinear in $n$ . We have applied our algorithms to the non-CM elliptic curves in Cremona’s tables and the Stein–Watkins database, some 140 million curves of conductor up to  $10^{10}$ , thereby obtaining a conjecturally complete list of 63 exceptional Galois images $G_{E}(\ell )$ that arise for $E/\mathbf{Q}$ without CM. Under this conjecture, we determine a complete list of 160 exceptional Galois images $G_{E}(\ell )$ that arise for non-CM elliptic curves over quadratic fields with rational $j$ -invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the $j$ -invariant is irrational.

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

We provide a comprehensive study of the function $h=h(q)$ defined by

$$\begin{eqnarray}h=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{12j-1})(1-q^{12j-11})}{(1-q^{12j-5})(1-q^{12j-7})}\end{eqnarray}$$

$k=k(q)$

$$\begin{eqnarray}k=q\mathop{\prod }_{j=1}^{\infty }\frac{(1-q^{10j-1})(1-q^{10j-2})(1-q^{10j-8})(1-q^{10j-9})}{(1-q^{10j-3})(1-q^{10j-4})(1-q^{10j-6})(1-q^{10j-7})}.\end{eqnarray}$$

For a half-integral weight modular form $f=\sum _{n=1}^{\infty }a_{f}(n)n^{(k-1)/2}q^{n}$ of weight $k=\ell +1/2$ on $\unicode[STIX]{x1D6E4}_{0}(4)$ such that $a_{f}(n)$ ( $n\in \mathbb{N}$ ) are real, we prove for a fixed suitable natural number $r$ that $a_{f}(n)$ changes sign infinitely often as $n$ varies over numbers having at most $r$ prime factors, assuming the analog of the Ramanujan conjecture for Fourier coefficients of half-integral weight forms.

We study the meromorphic continuation and the spectral expansion of the opposite-sign Kloosterman sum zeta function

$$\begin{eqnarray}\displaystyle (2{\it\pi}\sqrt{mn})^{2s-1}\mathop{\sum }_{\ell =1}^{\infty }\frac{S(m,-n,\ell )}{\ell ^{2s}} & & \displaystyle \nonumber\end{eqnarray}$$

$m,n$

$s\in \mathbb{C}$

We consider the classical theta operator ${\it\theta}$ on modular forms modulo $p^{m}$ and level $N$ prime to $p$ , where $p$ is a prime greater than three. Our main result is that ${\it\theta}$ mod $p^{m}$ will map forms of weight $k$ to forms of weight $k+2+2p^{m-1}(p-1)$ and that this weight is optimal in certain cases when $m$ is at least two. Thus, the natural expectation that ${\it\theta}$ mod $p^{m}$ should map to weight $k+2+p^{m-1}(p-1)$ is shown to be false. The primary motivation for this study is that application of the ${\it\theta}$ operator on eigenforms mod $p^{m}$ corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the ${\it\theta}$ -operator mod $p^{m}$ gives an explicit weight bound on the twist of a modular mod $p^{m}$ Galois representation by the cyclotomic character.

We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

In terms of class field theory we give a necessary and sufficient condition for an integer to be representable by the quadratic form $x^{2}+xy+ny^{2}$ ($n\in \mathbb{N}$ arbitrary) under extra conditions $x\equiv 1\;\text{mod}\;m$, $y\equiv 0\;\text{mod}\;m$ on the variables. We also give some examples where their extended ring class numbers are less than or equal to $3$.

Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsom et al. for mock theta functions. Here we see that Mortenson's previous work on the dual nature of Appell–Lerch sums and partial theta functions and on constructing bilateral q-series with mixed mock modular behaviour is well suited for such radial limits. We present five more radial limit results, which follow from mixed mock modular bilateral q-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function of Gordon and McIntosh. The later bilateral series can be used to compute radial limits for many classical second-, sixth-, eighth- and tenth-order mock theta functions.

Following Jacquet, Lapid and Rogawski, we regularize trilinear periods. We use the regularized trilinear periods to compute Fourier–Jacobi periods of residues of Eisenstein series on metaplectic groups, which has an application to the Gan–Gross–Prasad conjecture.

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let ${\it\varrho}_{\flat }$ and ${\it\varrho}_{\sharp }$ be odd two-dimensional Artin representations for which ${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$ is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms $f$, $g$, and $h$ of respective weights two, one, and one, giving rise to $E$, ${\it\varrho}_{\flat }$, and ${\it\varrho}_{\sharp }$ via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain $p$-adic iterated integrals attached to the triple $(f,g,h)$, which are $p$-adic avatars of the leading term of the Hasse–Weil–Artin $L$-series $L(E,{\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp },s)$ when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on $E$—referred to as Starkpoints—which are defined over the number field cut out by ${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$. This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight-one forms. It is proved when $g$ and $h$ are binary theta series attached to a common imaginary quadratic field in which $p$ splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing $p$-adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintani-type cycles on ${\mathcal{H}}_{p}\times {\mathcal{H}}$), and extensions of $\mathbb{Q}$ with Galois group a central extension of the dihedral group $D_{2n}$ or of one of the exceptional subgroups $A_{4}$, $S_{4}$, and $A_{5}$ of $\mathbf{PGL}_{2}(\mathbb{C})$.

We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogs of Fourier–Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.

We prove certain depth bounds for Arthur’s endoscopic transfer of representations from classical groups to the corresponding general linear groups over local fields of characteristic 0, with some restrictions on the residue characteristic. We then use these results and the method of Deligne and Kazhdan of studying representation theory over close local fields to obtain, under some restrictions on the characteristic, the local Langlands correspondence for split classical groups over local function fields from the corresponding result of Arthur in characteristic 0.

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

The second author has recently introduced a new class of $L$-series in the arithmetic theory of function fields over finite fields. We show that the values at one of these $L$-series encode arithmetic information of a generalization of Drinfeld modules defined over Tate algebras that we introduce (the coefficients can be chosen in a Tate algebra). This enables us to generalize Anderson’s log-algebraicity theorem and an analogue of the Herbrand–Ribet theorem recently obtained by Taelman.

Let $\mathbf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let ${\rm\Gamma}=\mathbf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\geqslant 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to ${\rm\Gamma}$ from a Hecke cusp form in $S_{2k-8}(\text{SL}_{2}(\mathbb{Z}))$. We follow Ikeda’s idea of using Siegel’s Eisenstein series, their Fourier–Jacobi expansions, and the compatible family of Eisenstein series.

We propose to generalize the work of Régis Dupont for computing modular polynomials in dimension $2$ to new invariants. We describe an algorithm to compute modular polynomials for invariants derived from theta constants and prove heuristically that this algorithm is quasi-linear in its output size. Some properties of the modular polynomials defined from quotients of theta constants are analyzed. We report on experiments with our implementation.

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series: it is an automorphic representation, and it decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of $\mathit{SO}_{2n+1}$ of Bump, Friedberg, and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin–Selberg integrals. We describe one application, to a calculation of a co-period integral.

We prove that formal Fourier Jacobi expansions of degree two are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension two, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree two. Combining both results enables us to determine relations of special cycles in the second Chow group.