PhD defence
Central Values of L-Functions of Twisted Modular Forms of Composite Level
- C.K.L. Dombrowsky
- Date
- Wednesday 18 March 2026
- Time
- Location
-
Academy Building
Rapenburg 73
2311 GJ Leiden
Supervisor(s)
Summary
How many integers (x,y) satisfy the equation: y^2+xy=x^3+x^2+2x+1 is equal to 0? It turns out that only such pair exists: x is 2 and y is -1.
We now ask this questions over bigger sets of numbers: We fix an integer D (e.g. 5) and assume that the numbers x an y are of the form a+b sqrt(D) where a and b are rationals (e.g. x and y can be of the form 1/3+7 sqrt(5)). A famous result predicts that either there exists less than 16 pairs satisfying this equation or infinitely many – but even finding out if we have a finite or an infinite number of such pairs is a hard question to answer.
It turns out that for every value of D, we can construct a (so called L-)function which when evaluated at 1 is zero if and only if there are infinitely many pairs (x,y) of the form a+b\sqrt(D) satisfying the above equation. In this thesis, we describe an easy to apply criterion which predicts when L evaluated at 1 vanishes. We also study generalizations of this phenomenon and provide a criterion when the product of two such L-functions is zero.
PhD dissertations
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