We define modular equations describing the `-torsion subgroups of the Jacobian of a hyperelliptic curve. Over a finite base field, we prove factorization properties that extend the well-known results used in Atkin’s improvement of Schoof’s genus 1 point counting algorithm.

This paper is concerned with a compatible family of 4-dimensional -adic representations ρ of GQ := Gal(Q/Q) attached to the space of weight-3 cuspforms S3(Γ) on a noncongruence subgroup Γ ⊂ SL2(Z). For this representation we prove that: 1. It is automorphic: the L-function L(s, ρ∨ ) agrees with the L-function for an automorphic form for GL4(AQ), where ρ ∨ is the dual of ρ . 2. For each prime p ...

Research in room-temperature ionic liquids (RTILs) has exploded in the last two decades, which are marked by an exponential increase in the number of theoretical and experimental publications in this field. The typically good thermal stability, ionic conductivity, and solvability of RTILs make them superior electrolytes in many electrochemical applications, such as metal electrodeposition and e...

The Schoof-Elkies-Atkin algorithm counts the number of rational points on elliptic curves over finite fields. This paper presents a number of optimizations specific for the characteristic two case. We give a detailed description of the computation of modular polynomials over F2n and the search for an eigenvalue in the Elkies algorithm. With our implementation, we were able to count the number o...

Applications of Mobile Technology in business in the current environment are characterised by a critical reliance on a diversity of highly complex and often competing technology infrastructures and architectures. Classification models are often very descriptive and orient themselves more on the overt attributes than on the underlying qualities. Recent models, however, apply a wider set of conce...

Christofas Stergianos , Jason Atkin , Patrick Schittekat , Tomas Eric Nordlander , Chris Gerada 1 and Herve Morvan 1 1 Institute for Aerospace Technology, The University of Nottingham, Nottingham NG7 2RD United Kingdom {christofas.stergianos, jason.atkin, chris.gerada, herve.morvan}@nottingham.ac.uk 2 SINTEF ICT, Department of Applied Mathematics, Oslo, Norway {patrick.schittekat, tomas.nordlan...

Tabulating elliptic curves has been carried out since the earliest days of machine computation in number theory. After some historical remarks, we report on significant recent progress in enlarging the database of elliptic curves defined over Q to include all those of conductor N ≤ 130000. We also give various statistics, summarize the data, describe how it may be obtained and used, and mention...

Ramanujan’s famous partition congruences modulo powers of 5, 7, and 11 imply that certain sequences of partition generating functions tend `-adically to 0. Although these congruences have inspired research in many directions, little is known about the `-adic behavior of these sequences for primes ` ≥ 13. We show that these sequences are governed by “fractal” behavior. Modulo any power of a prim...