EXCEPTIONAL ZEROES OF P-ADIC L-FUNCTIONS OVER NON-ABELIAN FIELD EXTENSIONS

Suppose E is an elliptic curve over $\Bbb Q$, and p>3 is a split multiplicative prime for E. Let q ≠ p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur–Tate–Teitelbaum conjecture for E at the prime p, over number fields $K\subset \Bbb Q\big(\mu_{{q^{\infty}}},\;\!^{q^{\infty}\!\!\!\!}\sqrt{m}\big)$ such that p remains inert in $K\cap\Bbb Q(\mu_{{q^{\infty}}})^+$. The proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.

Congruences for convolutions of Hilbert modular forms

Let f be a primitive, cuspidal Hilbert modular form of parallel weight. We investigate the Rankin convolution L-values L(f,g,s), where g is a theta-lift modular form corresponding to a finite-order character. We prove weak forms of Kato's ‘false Tate curve’ congruences for these values, of the form predicted by conjectures in non-commmutative Iwasawa theory.

A NEW CONSTRUCTION OF p-ADIC RANKIN CONVOLUTIONS IN THE CASE OF POSITIVE SLOPE

Given two newforms f and g of respective weights k and l with l < k, Hida constructed a p-adic L-function interpolating the values of the Rankin convolution of f and g in the critical strip l ≤ s ≤ k. However, this construction works only if f is an ordinary form. Using a method developed by Panchishkin to construct p-adic L-function associated with modular forms, we generalize this construction to the case where the slope of f is small.