The aim of this paper is to find a formula for the double Laplace transform of the truncated variation of a Brownian motion with drift. In order to find the double Laplace transform, we also prove some identities for the Brownian motion with drift, which may be of independent interest.

In this paper, the semilocal convergence for ameliorated super-Halley methods in Banach spaces is considered. Different from the results in [J. M. Gutiérrez and M. A. Hernández, Comput. Math. Appl. 36 (1998) 1–8], these ameliorated methods do not need to compute a second derivative, the computation for inversion is reduced and the $R$ -order is also heightened. Under a weaker condition, an existence–uniqueness theorem for the solution is proved.

The Li coefficients $\unicode[STIX]{x1D706}_{F}(n)$ of a zeta or $L$ -function $F$ provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the $\unicode[STIX]{x1D70F}$ -Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport–Heilbronn zeta function. The behavior of the $\unicode[STIX]{x1D70F}$ -Li coefficients varies depending on whether the function in question has any zeros in the half-plane $\text{Re}(z)>\unicode[STIX]{x1D70F}/2.$ We investigate analytically and numerically the behavior of these coefficients for such functions in both the $n$ and $\unicode[STIX]{x1D70F}$ aspects.

Let $UY_{n}(q)$ be a Sylow $p$ -subgroup of an untwisted Chevalley group $Y_{n}(q)$ of rank $n$ defined over  $\mathbb{F}_{q}$ where $q$ is a power of a prime $p$ . We partition the set $\text{Irr}(UY_{n}(q))$ of irreducible characters of $UY_{n}(q)$ into families indexed by antichains of positive roots of the root system of type $Y_{n}$ . We focus our attention on the families of characters of $UY_{n}(q)$ which are indexed by antichains of length $1$ . Then for each positive root $\unicode[STIX]{x1D6FC}$ we establish a one-to-one correspondence between the minimal degree members of the family indexed by $\unicode[STIX]{x1D6FC}$ and the linear characters of a certain subquotient $\overline{T}_{\unicode[STIX]{x1D6FC}}$ of $UY_{n}(q)$ . For $Y_{n}=A_{n}$ our single root character construction recovers, among other things, the elementary supercharacters of these groups. Most importantly, though, this paper lays the groundwork for our classification of the elements of $\text{Irr}(UE_{i}(q))$ , $6\leqslant i\leqslant 8$ , and $\text{Irr}(UF_{4}(q))$ .

We consider a smooth system of two homogeneous quadratic equations over $\mathbb{Q}$ in $n\geqslant 13$ variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over $\mathbb{R}$ . In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists and, if so, compute one.

Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava and Shankar studying the average sizes of $n$ -Selmer groups, have given new upper bounds on the average algebraic rank in families of elliptic curves over $\mathbb{Q}$ , ordered by height. We describe databases of elliptic curves over $\mathbb{Q}$ , ordered by height, in which we compute ranks and $2$ -Selmer group sizes, the distributions of which may also be compared to these theoretical results. A striking new phenomenon that we observe in our database is that the average rank eventually decreases as height increases.

Consider two ordinary elliptic curves $E,E^{\prime }$ defined over a finite field $\mathbb{F}_{q}$ , and suppose that there exists an isogeny $\unicode[STIX]{x1D713}$ between $E$ and $E^{\prime }$ . We propose an algorithm that determines $\unicode[STIX]{x1D713}$ from the knowledge of $E$ , $E^{\prime }$ and of its degree $r$ , by using the structure of the $\ell$ -torsion of the curves (where  $\ell$  is a prime different from the characteristic  $p$ of the base field). Our approach is inspired by a previous algorithm due to Couveignes, which involved computations using the $p$ -torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of  $\tilde{O} (r^{2})p^{O(1)}$ base field operations. On the other hand, the cost of our algorithm is $\tilde{O} (r^{2})\log (q)^{O(1)}$ , for a large class of inputs; this makes it an interesting alternative for the medium- and large-characteristic cases.

We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order $7$ in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.

Since its introduction in 2010 by Lyubashevsky, Peikert and Regev, the ring learning with errors problem (ring-LWE) has become a popular building block for cryptographic primitives, due to its great versatility and its hardness proof consisting of a (quantum) reduction from ideal lattice problems. But, for a given modulus $q$ and degree $n$ number field $K$ , generating ring-LWE samples can be perceived as cumbersome, because the secret keys have to be taken from the reduction mod $q$ of a certain fractional ideal ${\mathcal{O}}_{K}^{\vee }\subset K$ called the codifferent or ‘dual’, rather than from the ring of integers ${\mathcal{O}}_{K}$ itself. This has led to various non-dual variants of ring-LWE, in which one compensates for the non-duality by scaling up the errors. We give a comparison of these versions, and revisit some unfortunate choices that have been made in the recent literature, one of which is scaling up by ${|\unicode[STIX]{x1D6E5}_{K}|}^{1/2n}$ with $\unicode[STIX]{x1D6E5}_{K}$ the discriminant of $K$ . As a main result, we provide, for any $\unicode[STIX]{x1D700}>0$ , a family of number fields $K$ for which this variant of ring-LWE can be broken easily as soon as the errors are scaled up by ${|\unicode[STIX]{x1D6E5}_{K}|}^{(1-\unicode[STIX]{x1D700})/n}$ .

Lattice sieving is asymptotically the fastest approach for solving the shortest vector problem (SVP) on Euclidean lattices. All known sieving algorithms for solving the SVP require space which (heuristically) grows as $2^{0.2075n+o(n)}$ , where $n$ is the lattice dimension. In high dimensions, the memory requirement becomes a limiting factor for running these algorithms, making them uncompetitive with enumeration algorithms, despite their superior asymptotic time complexity.

We generalize sieving algorithms to solve SVP with less memory. We consider reductions of tuples of vectors rather than pairs of vectors as existing sieve algorithms do. For triples, we estimate that the space requirement scales as $2^{0.1887n+o(n)}$ . The naive algorithm for this triple sieve runs in time $2^{0.5661n+o(n)}$ . With appropriate filtering of pairs, we reduce the time complexity to $2^{0.4812n+o(n)}$ while keeping the same space complexity. We further analyze the effects of using larger tuples for reduction, and conjecture how this provides a continuous trade-off between the memory-intensive sieving and the asymptotically slower enumeration.

We report on our project to find explicit examples of K3 surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for point counting on surfaces defined over finite fields. For this, we describe algorithms that are $p$ -adic in nature.

We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows a Jacobi eigenform to be generated directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms, it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.

We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the naive height into an archimedean and a non-archimedean term. Our main contribution is an algorithm for the computation of the non-archimedean term that requires no integer factorization and runs in quasi-linear time.

In this paper we describe how to compute smallest monic polynomials that define a given number field  $\mathbb{K}$ . We make use of the one-to-one correspondence between monic defining polynomials of  $\mathbb{K}$ and algebraic integers that generate  $\mathbb{K}$ . Thus, a smallest polynomial corresponds to a vector in the lattice of integers of  $\mathbb{K}$ and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of  $\mathbb{K}$ . This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.

In this paper, we present novel algorithms for finding small relations and ideal factorizations in the ideal class group of an order in an imaginary quadratic field, where both the norms of the prime ideals and the size of the coefficients involved are bounded. We show how our methods can be used to improve the computation of large-degree isogenies and endomorphism rings of elliptic curves defined over finite fields. For these problems, we obtain improved heuristic complexity results in almost all cases and significantly improved performance in practice. The speed-up is especially high in situations where the ideal class group can be computed in advance.

In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in $\mathbb{F}_{p^{6}}$ with the number field sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special- $\mathfrak{q}$ strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various polynomial selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special- $\mathfrak{q}$ and various enumeration strategies.

Brauer classes of a global field can be represented by cyclic algebras. Effective constructions of such algebras and a maximal order therein are given for $\mathbb{F}_{q}(t)$ , excluding cases of wild ramification. As part of the construction, we also obtain a new description of subfields of cyclotomic function fields.

We describe the construction of a database of genus- $2$ curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated $L$ -function. This data has been incorporated into the $L$ -Functions and Modular Forms Database (LMFDB).