This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where n+1 is a prime p or a power of a prime p^k, and show that these systems can have a focus order n^2-n; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order n^2-1 for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too.

Let E be a quadratic algebra over a number field F. Let E(g, s) be an Eisenstein series on GL2(E), and let F be a cuspidal automorphic form on GL2(F). We will consider in this paper the following automorphic integral: ∫ZAGL2(F)∖GL2(AF)F(g)E(g,s)dg. This is in some sense the complementary case to the well-known Rankin–Selberg integral and the triple product formula. We will approach this integral by Waldspurger’s formula, giving a criterion about when the integral is automatically zero, and otherwise the L-functions it represents. We will also calculate the local integrals at some ramified places, where the level of the ramification can be arbitrarily large.

In this short paper we give the sub-local upper bound for the sup norm of an automorphic form on PGL_n, whose associated automorphic representation has finite conductor C(π)=p^c with c→∞, and its local component at the place of ramification is a minimal vector belonging to an irreducible representation with generic induction datum.

In this paper we prove the equidistribution of the restriction of the mass of automorphic newforms to a nonsplit torus in the depth aspect. This result is better than the current known results on the similar problem in the eigenvalue aspect. The method is relatively elementary and makes use of the known effective QUE result in the depth aspect.

For primes q≡7 mod 16, the present manuscript shows that elementary methods enable one to prove surprisingly strong results about the Iwasawa theory of the Gross family of elliptic curves with complex multiplication by the ring of integers of the field K=Q(√(-q)), which are in perfect accord with the predictions of the conjecture of Birch and Swinnerton-Dyer. We also prove some interesting phenomena related to a classical conjecture of Greenberg, and give a new proof of an old theorem of Hasse.

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We study elliptic curves of the form x^3+y^3=2^p and x^3+y^3=2p^2 where p is any odd prime satisfying p ≡ 2 mod 9 or p ≡ 5 mod 9. We first show that the 3-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their 2-Selmer group to the 2-rank of the ideal class group of Q(\sqrt[3]{p}) to obtain some examples of elliptic curves with rank one and non-trivial 2-part of the Tate-Shafarevich group.

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