Mathematics

This paper presents algorithms for computing the length of a sum of squares and a Pythagoras element in a global field K of characteristic different from 2 . In the first part of the paper, we present algorithms for computing the length in a non-dyadic and dyadic (if K is a number field) completion of K . These two algorithms serve as subsidiary steps for computing lengths in global fields. In the second part of the paper we present a procedure for constructing an element whose length equals the Pythagoras number of a global field, termed a Pythagoras element.

It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for rank parity function is f(q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.

Let G be a connected reductive group over a totally real field F which is compact modulo center at archimedean places. We find congruences modulo an arbitrary power of p between the space of arbitrary automorphic forms on G( A F ) and that of automorphic forms with supercuspidal components at p, provided that p is larger than the Coxeter number of the absolute Weyl group of G . We illustrate how such congruences can be applied in the construction of Galois representations. Our proof is based on type theory for representations of p-adic groups, generalizing the prototypical case of GL(2) in [arXiv:1506.04022, Section 7] to general reductive groups. We exhibit a plethora of new supercuspidal types consisting of arbitrarily small compact open subgroups and characters thereof. We expect these results of independent interest to have further applications. For example, we extend the result by Emerton--Paškūnas on density of supercuspidal points from definite unitary groups to general G as above.

Based on an equation for the rank of an elliptic surface over Q which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank 0 when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain L -functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental evidence supporting it.

Let p be a prime number and K a finite extension of Q p . We state conjectures on the smooth representations of GL n (K) that occur in spaces of mod p automorphic forms (for compact unitary groups). In particular, when K is unramified, we conjecture that they are of finite length and predict their internal structure (extensions, form of subquotients) from the structure of a certain algebraic representation of GL n . When n=2 and K is unramified, we prove several cases of our conjectures, including new finite length results.

We establish a one-to-one correspondence between conjugacy classes of any Hecke group and irreducible systems of poles of rational period functions for automorphic integrals on the same group. We use this correspondence to construct irreducible systems of poles and to count poles. We characterize Hecke-conjugation and Hecke-symmetry for poles of rational period functions in terms of the transpose of matrices in conjugacy classes. We construct new rational period functions and families of rational period functions.

We show that for every positive integer k , there exist k consecutive primes having the property that if any digit of any one of the primes, including any of the infinitely many leading zero digits, is changed, then that prime becomes composite.

We improve on a construction of Mestre--Shioda to produce some families of curves X/Q of record rank relative to the genus g of X . Our first main result is that for any integer g�? with g??(mod3) , there exist infinitely many genus g hyperelliptic curves over Q with at least 8g+32 Q -points and Mordell--Weil rank �?g+15 over Q . Our second main theorem is that if g+1 is an odd prime and K contains the g+1 -th roots of unity, then there exist infinitely many genus g hyperelliptic curves over K with Mordell--Weil rank at least 6g over K .

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime numbers. They proved, in particular, an explicit upper bound for the lim inf of the height of elements in such fields. We generalize their result in an effective way to maximal Galois extensions of number fields with given local behaviour at finitely many places.

It is well-known for an elliptic curve with complex multiplication that the existence of a Q -rational model is equivalent to its field of moduli being equal to Q , or its endomorphism ring being the ring of integers of 9 possible fields ( ∗ ). Murabayashi and Umegaki proved analogous results for abelian surfaces. For three dimensional CM abelian varieties with rational fields of moduli, Chun narrowed down to a list of 37 possible CM fields. In this paper, we show that his list is exact. By constructing certain Hecke characters that satisfy a theorem of Shimura, we prove that precisely 28 isogeny classes of these abelian varieties have Q -models. Therefore the complete analogy to (∗) fails here.